Corrections to: Exact Bias Correction for Linear Adjustment of Randomized Controlled Trials

Five-unit example: exact expectations of the components of the bias of $\widehat{\mathrm{ATE}}_I$ .
Quantity	Third component as defined	Third component as printed
First component	$3/40$	$3/40$
Second component	$11/80$	$11/80$
Third component	$-1/80$	$1/80$
Sum of the three components	$1/5$	$9/40$
Bias of $\widehat{\mathrm{ATE}}_I$ , by enumeration	$1/5$	$1/5$

Exact variance of $\widehat{\operatorname{Cov}}_t(a,b)$ and the two bounds in three examples, rounded to four decimals; the exact values are in the text. A valid upper bound is at least as large as the variance.
Example	Exact variance	Bound as printed	Centered bound
$n=6$ , $n_t=4$ , arbitrary $a$ and $b$	3.4598	2.6590	6.5234
$n=14$ , $n_t=7$ , $a_ib_i=1$ for all $i$	2.9458	2.6620	6.0435
$n=16$ , $n_T=8$ , $a=h$ and $b=y(1)$	0.2032	0.1899	0.4200

Lemma A.1, the variance bound, as printed for arbitrary $n$ -vectors $a$ , $b$ . The witness is the validator's and is not in the paper

p. 1511 (PDF p. 10), Appendix A.2; the vectors $a$ , $b$ and the covariance estimators are introduced on p. 1510 (PDF p. 9)

Appendix A.2, Lemma A.1, pp. 1510–1511 (PDF pp. 9–10). The six-unit counterexample has all the stated means, coefficients, moments, expectation, variance, bound and positive gap. Covariance and its variance are unchanged by constant shifts; the printed bound changes under centering, and its centered value bounds the variance.

Historical component credits

Counterexample proved AronowAudit.P005.covariance_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This row is a witness refuting the variance inequality, not a positive proof of that inequality; ancillary moment diagnostics are not required for that credited component.
Proved AronowAudit.P005.univ_six Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.design_six_four Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.expectation_six_four Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:07cd11b18d2e901d8689802e3c6c37169e97b4a892e9bfc2bae1df9f9e502988

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1510 (PDF p. 9), Appendix A.2] “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$. The following lemma characterizes the mean and variance of these covariance estimators.” [Printed, p. 1511 (PDF p. 10)] “LEMMA A.1: Let $t \in \{T, C\}$ and $n \ge 4$. Then, $$\mathbf{E}\big[\widehat{\mathrm{Cov}}_t(a,b)\big] = \frac{(n_t-1)n}{n_t(n-1)}(\overline{ab} - \bar a\bar b),$$ $$\mathrm{Var}\big(\widehat{\mathrm{Cov}}_t(a,b)\big) \le \frac{2(n-n_t)}{n_t(n-1)}\frac1n\sum_{i=1}^n(a_ib_i-\overline{ab})^2 + \frac{2(n-n_t)n}{n_t^3(n-1)}\frac1n\sum_{i=1}^n a_i^2 \times \frac1n\sum_{i=1}^n b_i^2.$$” [Counterexample to the variance bound] Take $n = 6$, $t = T$, $n_T = 4$ (so $n_C = 2$), $a = (0, -3, -1, 2, -4, -3)$ and $b = (2, 1, 3, -3, 1, 1)$. Every hypothesis of the lemma holds: $a$ and $b$ are fixed vectors in $\mathbb{R}^6$, $t \in \{T, C\}$, $n = 6 \ge 4$, and the treated group is a uniformly chosen subset of size 4 (15 equally likely subsets). Then: (i) $\bar a = -\frac32$, $\bar b = \frac56$, $\overline{ab} = -\frac{19}{6}$, $\overline{ab} - \bar a\bar b = -\frac{23}{12}$, and $\mathbf{E}[\widehat{\mathrm{Cov}}_T(a,b)] = -\frac{69}{40} = \frac{(n_T-1)n}{n_T(n-1)}(\overline{ab} - \bar a\bar b)$: the expectation identity of the lemma holds. (ii) $\mathrm{Var}(\widehat{\mathrm{Cov}}_T(a,b)) = \frac{16607}{4800}$. (iii) The printed right-hand side equals $\frac15\cdot\frac{113}{36} + \frac{3}{40}\cdot\frac{13}{2}\cdot\frac{25}{6} = \frac{113}{180} + \frac{65}{32} = \frac{3829}{1440}$, where $\frac{2(n-n_T)}{n_T(n-1)} = \frac15$, $\frac{2(n-n_T)n}{n_T^3(n-1)} = \frac{3}{40}$, $\frac1n\sum_i(a_ib_i - \overline{ab})^2 = \frac{113}{36}$, $\frac1n\sum_ia_i^2 = \frac{13}{2}$ and $\frac1n\sum_ib_i^2 = \frac{25}{6}$. (iv) $\frac{16607}{4800} - \frac{3829}{1440} = \frac{11531}{14400} > 0$, so $\mathrm{Var}(\widehat{\mathrm{Cov}}_T(a,b))$ exceeds the printed bound: the variance inequality of Lemma A.1 is false for this instance. (v) [Why it fails] The left-hand side is unchanged when constants are added to $a$ and to $b$; the printed right-hand side is not. With $a^c = a - \bar a$ and $b^c = b - \bar b$, $\widehat{\mathrm{Cov}}_T(a^c, b^c) = \widehat{\mathrm{Cov}}_T(a, b)$ for each of the 15 subsets, and the printed right-hand side evaluated at $(a^c, b^c)$ is $\frac{835}{128} \ge \frac{16607}{4800}$.

: SAP.P005a.T1WholeStatement

Theorem: SAP.P005a.T1

Statement hash: b350726246439bb0472eae147ecc7154387bd6083af07f9ca818b383a28ec8df

Statement definition: SAP.P005a.T1WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:15:03Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.design_six_four: observed used

AronowAudit.P005.expectation_six_four: observed used

AronowAudit.P005.univ_six: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T2

Lemma A.1, as repaired: the variance bound with centred vectors (Sol's smallest repair and the validator's derivation)

p. 1511 (PDF p. 10), Appendix A.2

Appendix A.2, Lemma A.1, p. 1511 (PDF p. 10). For either nonempty arm, the printed expectation and the variance bound with both vectors centered hold. Covariance is location invariant, the centered sample-product mean has the simple-random-sampling variance, the product of centered sums has the stated exact expectation and variance and their bound, and the generic variance-difference inequality gives the repaired bound. The printed bound is recovered when both population means are zero.

Historical component credits

Proved, conditional on a published theorem AronowAudit.P005.Wave1.corrected_covariance_lemma_conditional Assumes as a hypothesis: L. Lei and P. Ding, "Regression adjustment in completely randomized experiments with a diverging number of covariates", Biometrika 108 (2021), Supplementary Material, Lemma A.5 Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Its literature premise is not a canonical hypothesis; the unconditional centered bound and expectation are instead supplied by existing supporting declarations.
Proved AronowAudit.P005.covariance_location_invariant Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This harmless generalization includes exactly the centering operation needed by the correction.
Proved AronowAudit.P005.variance_difference_bound Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Empty-design totalization adds cases but does not replace the nonempty sampling case the target needs.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:5c6443506628f488b4ba342d66180dd1a8ca7057aedd752d50358c6aebfb1ef7

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] Lemma A.1 and the definition of the covariance estimators, as transcribed in P005-T1. [Printed, proof of Lemma A.1, p. 1511] “To upper-bound the variance, we first note that the covariance estimator is location invariant and we can assume, without loss of generality, $\bar a = 0$ and $\bar b = 0$.$^{9}$ We then have $$\mathrm{Var}(\overline{ab}_T - \bar a_T\bar b_T) \le 2\,\mathrm{Var}(\overline{ab}_T) + 2\,\mathrm{Var}(\bar a_T\bar b_T) = 2\,\mathrm{Var}(\overline{ab}_T) + \frac{2}{n_T^4}\mathrm{Var}\Big(\sum_{i,j\in[T]}a_ib_j\Big)$$ $$\le \frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_{i=1}^n(a_ib_i-\overline{ab})^2 + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_{i=1}^n a_i^2\,\frac1n\sum_{i=1}^n b_i^2,$$ where the last inequality follows from the variance calculation of simple random samplings (first term) and Lemma A.5 in Lei and Ding (2021) (second term).” Footnote 9: “This is required by the conditions of Lemma A.5 in Lei and Ding (2021).” [Sol's corrected text (validated.json, P005-E002, smallest_repair), verbatim, first part] “For arbitrary fixed $a,b$, define $a_i^c=a_i-\bar a$ and $b_i^c=b_i-\bar b$. Keep the printed expectation identity. For $m=n_t$ and $n\ge4$, the variance bound is \[\operatorname{Var}(\widehat{\operatorname{Cov}}_t(a,b))\le\frac{2(n-m)}{m(n-1)}\frac1n\sum_i(a_i^c b_i^c-\overline{a^c b^c})^2+\frac{2(n-m)n}{m^3(n-1)}\left(\frac1n\sum_i(a_i^c)^2\right)\left(\frac1n\sum_i(b_i^c)^2\right).\]” [Corrected statement, with every hypothesis] Let $n \ge 4$, $1 \le n_T \le n-1$ and $n_C = n - n_T$; let $[T]$ be uniformly distributed over the subsets of $\{1, \ldots, n\}$ of size $n_T$ and $[C]$ its complement; let $t \in \{T, C\}$; let $a, b \in \mathbb{R}^n$ be arbitrary fixed vectors and put $a_i^c = a_i - \bar a$, $b_i^c = b_i - \bar b$. Then (i) $\mathbf{E}\big[\widehat{\mathrm{Cov}}_t(a,b)\big] = \frac{(n_t-1)n}{n_t(n-1)}(\overline{ab} - \bar a\bar b)$ (as printed; it needs no centring); (ii) $\mathrm{Var}\big(\widehat{\mathrm{Cov}}_t(a,b)\big) \le \frac{2(n-n_t)}{n_t(n-1)}\frac1n\sum_{i=1}^n\big(a_i^cb_i^c - \overline{a^cb^c}\big)^2 + \frac{2(n-n_t)n}{n_t^3(n-1)}\Big(\frac1n\sum_{i=1}^n(a_i^c)^2\Big)\Big(\frac1n\sum_{i=1}^n(b_i^c)^2\Big)$. Equivalently: the printed bound holds for all $a$, $b$ with $\bar a = 0$ and $\bar b = 0$, and for general $a$, $b$ after both vectors are centred on its right-hand side. [The steps of the validator's proof; each is an exact statement for fixed $n \ge 4$ and $m = n_t$, with $S$ a uniform subset of size $m$, $u = a^c$, $v = b^c$] (a) Location invariance: $\overline{ab}_S - \bar a_S\bar b_S = \overline{uv}_S - \bar u_S\bar v_S$ for every subset $S$. (b) $\mathrm{Var}(\overline{uv}_S) = \frac{n-m}{m(n-1)}\frac1n\sum_i(u_iv_i - \overline{uv})^2$ (the variance of a sample mean under simple random sampling). (c) Let $Z = (\sum_{i\in S}u_i)(\sum_{i\in S}v_i)$, $A = \sum_iu_i^2$, $B = \sum_iv_i^2$, $G = \sum_iu_iv_i$, $H = \sum_iu_i^2v_i^2$, $\alpha = \frac{m(n-m)}{n(n-1)}$ and $\ell = \frac{m(m-1)(n-m)(n-m-1)}{n(n-1)(n-2)(n-3)}$. Because $\sum_iu_i = \sum_iv_i = 0$: $\mathbf{E}Z = \alpha G$ and $\mathrm{Var}\,Z = (\alpha - 6\ell)H + \ell AB + (2\ell - \alpha^2)G^2 \le \alpha AB$. Hence $\mathrm{Var}(\bar u_S\bar v_S) = m^{-4}\,\mathrm{Var}\,Z \le \frac{(n-m)n}{m^3(n-1)}\big(\frac1n\sum_iu_i^2\big)\big(\frac1n\sum_iv_i^2\big)$. (This is the step the paper takes from Lemma A.5 of Lei and Ding (2021); it requires centred vectors, as footnote 9 says.) (d) $\mathrm{Var}(U - V) \le 2\,\mathrm{Var}(U) + 2\,\mathrm{Var}(V)$, applied with $U = \overline{uv}_S$ and $V = \bar u_S\bar v_S$; with (a)–(c) this gives (ii).

: SAP.P005a.T2WholeStatement

Theorem: SAP.P005a.T2

Statement hash: f428426168d8739e2465c27f0d73938996a8e0b19176728b77d6f935632cb73d

Statement definition: SAP.P005a.T2WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:15:03Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.Wave1.corrected_covariance_lemma_conditional: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T3

Proof of Theorem 3.2 (Appendix B.2), item 2: the chain “First notice, by Lemma A.1 and Assumption 1, …”, as printed. The witness is the validator's and is not in the paper

p. 1517 (PDF p. 16), Appendix B.2, item 2

Appendix B.2, item 2, p. 1517 (PDF p. 16). The standardized regular sixteen-unit counterexample satisfies all stated moment and allocation conditions. Its leverage vector, mean and constant leverage-product, three-category probabilities and covariance values, exact variance, both failed bounds and positive gaps, and the valid third-expression and centered-bound comparisons have the stated numerical values.

Historical component credits

Proved AronowAudit.P005.Wave1.leverage_variance_counterexample_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The credited component is the finite failure of the first two bounds. To satisfy sequence language, insert this n=16 population into the standardized scalar-grid sequence of R037-15; K=1 and fixed allocation bounds remain valid, and all-assignment regularity holds on the relevant tail.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:8d0005d2108e1ecdd2266b7ee73d1ae0a1ea3a54c00c449dc48a37758167bf2f

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1517 (PDF p. 16), proof of Theorem 3.2, item 2] “2. Since the first-order term in (11) is canceled, we characterize the variance of the form $\frac1n(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T)$. First notice, by Lemma A.1 and Assumption 1, $$\mathrm{Var}\big[(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T)\big] \le \frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_{i=1}^n\big(h_iy_i(1)-\overline{hy(1)}\big)^2 + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_{i=1}^n y_i^2(1)\frac1n\sum_{i=1}^n h_i^2$$ $$\le \frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_{i=1}^n h_i^2y_i^2(1) + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_{i=1}^n y_i^2(1)\frac1n\sum_{i=1}^n h_i^2$$ $$\le \frac{2(n-n_T)}{n_T(n-1)}n^2\kappa^2\frac1n\sum_{i=1}^n y_i^2(1) + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_{i=1}^n y_i^2(1)\frac1n\sum_{i=1}^n h_i\times n\kappa = O\Big(\kappa^2n+\frac{\kappa d}{n}\Big).$$ Thus, $\mathrm{Var}\big(\frac1n(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T)\big) = O\big(\frac{\kappa^2}{n}+\frac{\kappa d}{n^3}\big)$.” [Counterexample to the first inequality of the chain, with an admissible leverage vector] Take $n = 16$, $n_T = n_C = 8$, $d = 1$, $c = \sqrt{8/71}$, $x = (8c, -8c, c, -c, c, -c, \ldots, c, -c)$ (two units with $|x_i| = 8c$ followed by seven pairs $(c, -c)$), $y(1) = (\frac{1}{64}, \frac{1}{64}, 1, \ldots, 1)$ and $y(0) = (0, \ldots, 0)$. Then: (i) [The paper's conditions hold at this $n$] $\bar x = 0$ and $\overline{x^2} = \frac{2\cdot64 + 14}{16}\cdot\frac{8}{71} = 1$ (Assumption 2); $\frac1n\sum_iy_i(1)^2 = \frac{28673}{32768} \le 1$, $\frac1n\sum_iy_i(0)^2 = 0$ and $\frac1n\sum_ix_i^2 = 1$ (the bounds of Assumption 1 with $K = 1$); $p_T = \frac12$ (the bound of Assumption 3 with, for example, $p_{\min} = \frac14$, $p_{\max} = \frac34$); no value of $x$ occurs more than 7 times, so every arm of 8 units has positive within-arm variance of $x$ and $\widehat L$, $\widehat L_T$, $\widehat L_C$ are invertible under every assignment (footnote 4). (ii) $h = x^2 = (\frac{512}{71}, \frac{512}{71}, \frac{8}{71}, \ldots, \frac{8}{71})$, $\bar h = 1 = d$, $\kappa = \frac{32}{71}$, and $h_iy_i(1) = \frac{8}{71}$ for every $i$, so that $\frac1n\sum_i(h_iy_i(1) - \overline{hy(1)})^2 = 0$. (iii) Over the $\binom{16}{8} = 12870$ equally likely treated sets, $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T$ depends on the set only through the number $k \in \{0, 1, 2\}$ of the two high-leverage units that it contains: $\Pr(k = 0) = \frac{7}{30}$, $\Pr(k = 1) = \frac{8}{15}$, $\Pr(k = 2) = \frac{7}{30}$, with values $0$, $-\frac{27783}{36352}$ and $-\frac{11907}{9088}$. Hence $\mathrm{Var}\big[\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T\big] = \frac{1678565511}{8259174400}$. (iv) The first printed bound, $\frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_i(h_iy_i(1) - \overline{hy(1)})^2 + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_iy_i^2(1)\frac1n\sum_ih_i^2$, equals $\frac{117645319}{619438080}$, and the variance exceeds it by $\frac{329883773}{24777523200} > 0$. (v) The second printed expression, $\frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_ih_i^2y_i^2(1) + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_iy_i^2(1)\frac1n\sum_ih_i^2$, equals $\frac{334349}{1744896}$, and the variance exceeds it by $\frac{287940733}{24777523200} > 0$. (vi) [What does not fail] The third printed expression equals $\frac{60758087}{9678720}$ and exceeds the variance; and the bound of Lemma A.1 evaluated at the centred vectors $h - d$ and $y(1) - \overline{y(1)}$ is $\frac{5550293259}{13214679040} \ge \frac{1678565511}{8259174400}$. So the first “$\le$” of the printed chain is false for covariates and outcomes that satisfy the paper's conditions, and the first two right-hand sides are not upper bounds on the variance; the chain as printed does not establish its conclusion, which is restored by P005-T4.

: SAP.P005a.T3WholeStatement

Theorem: SAP.P005a.T3

Statement hash: 69f5ec612f56b22d57d0b54f1f29bbaa21c97dfdce9dad518fb11c36029f38b9

Statement definition: SAP.P005a.T3WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:15:03Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T4

Proof of Theorem 3.2 (Appendix B.2), item 2, as repaired: the variance chain with $h$ and $y(1)$ centred (Sol's smallest repair and the validator's derivation)

p. 1517 (PDF p. 16), Appendix B.2, item 2

Appendix B.2, item 2, p. 1517 (PDF p. 16). For either arm the centered leverage and outcome moments give every inequality in the repaired finite variance chain, including the intermediate raw leverage second moment. The two allocation coefficients obey the stated bounds. Under Assumptions 1–3 the numerical right-hand side and the actual variance are O(nκ²+κd/n), and the variance after division by n is O(κ²/n+κd/n³). Dimension zero is allowed, including along a sequence.

Historical component credits

Proved AronowAudit.P005.Wave1.corrected_leverage_variance_chain_proof Limited support; baseline fidelity: narrower; manifest: used; used in Solution; {'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/Gap/GP005B4c.lean:500, the conclusion includes mean(u²) = mean(h²) − d² and mean(u²) ≤ nκd, but omits mean(u²) ≤ mean(h²) and mean(h²) ≤ nκd. P005-T4(b) explicitly asserts both inequalities, and this row claims the whole fixed-n component T4(a)–(e). In particular, the raw-second-moment bound is a separate, stronger conclusion than the exported centered bound; its derivability from the hypotheses does not supply the missing conclusion. The centering, finite-sample assumptions and quantifier over both arms otherwise match the corrected claim.', 'suggested_fix': 'Extend the conclusion with popMean (fun i => u i^2) ≤ popMean (fun i => leverage P i^2) and popMean (fun i => leverage P i^2) ≤ (n : ℝ) * kappa P * d, retaining the existing identity and variance chain.', 'second_review': 'P005/Wave1/Gap/GP005B4c.lean:500 exports the centered second-moment identity and terminal bound but omits the two intermediate inequalities expressly required by P005-T4(b), mean(u²) ≤ mean(h²) ≤ nκd. The omission needs a statement repair, but it is minor: the already exported mean(h) = d and 0 ≤ h_i ≤ nκ give mean(h²) ≤ nκ mean(h) = nκd by averaging h_i² ≤ nκh_i, while the exported identity and d² ≥ 0 give the other inequality. The fixed-n assumptions, both-arm quantifier, and centered variance chain otherwise match the credited T4(a)–(e) component.', 'severity': 'minor'}
Proved AronowAudit.P005.Wave1.corrected_leverage_variance_order_proof Limited support; baseline fidelity: narrower; manifest: not used; not used in Solution; {'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/Gap/GP005B5.lean:350, the conclusion gives Big-O bounds only for the actual unscaled and scaled covariance variances. P005-T4(f) additionally asserts that the explicit right-hand side of T4(e) is O(nκ² + κd/n), and states two coefficient comparisons using the allocation lower bound; none of these conclusions is present. Calling these assertions proof details in the earlier record does not remove them from the supplied target. The restriction to positive covariate dimension excludes a trivial zero-leverage case, and finite initial sequence indices do not affect the asymptotic assertions.', 'suggested_fix': 'For a fixed uniform outcome-moment bound K, add IsBigO atTop U_t (fun n => (n : ℝ) * kappa (P n)^2 + kappa (P n) * d n / n), where U_t is the final numerical right-hand side of T4(e) with arm size s_n = armSize n (m n) t. Also export the comparisons 2(n−s)/(s(n−1)) ≤ 2/(a(n−1)) and 2(n−s)n/(s³(n−1)) ≤ 2/(a³n(n−1)) when 0 < a < s/n and n ≥ 4; take a = p_min for treatment and a = 1−p_max for control. Retain the existing variance conclusions.', 'second_review': 'P005/Wave1/Gap/GP005B5.lean:350 exports the actual covariance-variance orders, whereas P005-T4(f) also expressly requires the order of the numerical right-hand side U of (e) and the two allocation coefficient comparisons. Those conclusions are absent, but their repair is elementary: take a = lo for treatment or a = 1−hi for control from P005/P005_Formalization.lean:332; s ≥ an and n−s ≤ n give the stated coefficient bounds, and for n ≥ 4 they yield 0 ≤ U ≤ 4K(a⁻¹+a⁻³)(nκ²+κd/n). Thus the missing order is immediate deterministic arithmetic under the existing assumptions, warranting minor severity; the additional positive-dimension restriction also excludes only the trivial zero-leverage case.', 'severity': 'minor'}
Proved AronowAudit.P005.mean_add Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_sub Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_mul Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_const Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_nonneg Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_mono Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_square_centering Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.leverage_nonneg Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.leverage_le_maximum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.leverage_population_mean Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.centered_leverage_moment_bounds Limited support; baseline fidelity: narrower; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:1d5458b1dd90f172654a9da46d25c2b4a6b28e44cd42c978e532c46966e8b71b

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] The chain of item 2 on p. 1517, as transcribed in P005-T3. [Sol's corrected text (validated.json, P005-E002, smallest_repair), verbatim, second part] “On p. 1517 apply this with $a^c=h-d$ and $b^c=y(1)-\overline{y(1)}$. The same standing assumptions imply $|h_i-d|\le n\kappa$, $n^{-1}\sum_i(h_i-d)^2\le n\kappa d$, and $n^{-1}\sum_i(y_i(1)-\overline{y(1)})^2\le K$. Thus the unscaled variance remains $O(n\kappa^2+\kappa d/n)$ and the scaled variance remains $O(\kappa^2/n+\kappa d/n^3)$.” [Corrected chain, with every hypothesis] Fix $n \ge 4$ and $1 \le n_T \le n-1$, under complete randomization. Suppose $\bar x = 0$ and $\overline{xx'} = I_d$ (Assumption 2 at this $n$) and $\frac1n\sum_iy_i(1)^2 \le K$ (Assumption 1); $h_i = \|x_i\|_2^2$ and $\kappa = \max_ih_i/n$. Write $u_i = h_i - d$ and $v_i = y_i(1) - \overline{y(1)}$. Then (a) $\bar h = d$, $0 \le h_i \le n\kappa$ and $d \le n\kappa$, so $|u_i| \le n\kappa$; (b) $\frac1n\sum_iu_i^2 = \frac1n\sum_ih_i^2 - d^2 \le \frac1n\sum_ih_i^2 \le n\kappa d$; (c) $\frac1n\sum_iv_i^2 = \frac1n\sum_iy_i(1)^2 - \overline{y(1)}^2 \le K$; (d) $\frac1n\sum_i(u_iv_i - \overline{uv})^2 \le \frac1n\sum_iu_i^2v_i^2 \le n^2\kappa^2K$; (e) $$\mathrm{Var}\big[\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T\big] \le \frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_{i=1}^n\big(u_iv_i - \overline{uv}\big)^2 + \frac{2(n-n_T)n}{n_T^3(n-1)}\Big(\frac1n\sum_{i=1}^nu_i^2\Big)\Big(\frac1n\sum_{i=1}^nv_i^2\Big) \le \frac{2(n-n_T)}{n_T(n-1)}n^2\kappa^2K + \frac{2(n-n_T)n}{n_T^3(n-1)}n\kappa dK.$$ (f) [Order statement, as printed] Along a sequence of populations satisfying Assumptions 1–3, the right-hand side of (e) is $O\big(\kappa^2n + \frac{\kappa d}{n}\big)$, and therefore $\mathrm{Var}\big(\frac1n(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T)\big) = O\big(\frac{\kappa^2}{n} + \frac{\kappa d}{n^3}\big)$. Explicitly, if $p_{\min} < n_T/n$ then $\frac{2(n-n_T)}{n_T(n-1)} \le \frac{2}{p_{\min}(n-1)}$ and $\frac{2(n-n_T)n}{n_T^3(n-1)} \le \frac{2}{p_{\min}^3n(n-1)}$. The same statements hold for the control arm, with $y(0)$, $n_C$ and $\widehat{\mathrm{Cov}}_C$ in place of $y(1)$, $n_T$ and $\widehat{\mathrm{Cov}}_T$ (and $1 - p_{\max}$ in place of $p_{\min}$).

: SAP.P005a.T4WholeStatement

Theorem: SAP.P005a.T4

Statement hash: 34c62e507b88f2b44076131f3a4ba1b121b86ea223e0a149c7a2c55bc850d3f4

Statement definition: SAP.P005a.T4WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:15:03Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.Wave1.corrected_leverage_variance_order_proof: observed not used

AronowAudit.P005.centered_leverage_moment_bounds: observed used

AronowAudit.P005.leverage_le_maximum: observed used

AronowAudit.P005.leverage_nonneg: observed used

AronowAudit.P005.leverage_population_mean: observed used

AronowAudit.P005.mean_add: observed used

AronowAudit.P005.mean_const: observed used

AronowAudit.P005.mean_mono: observed used

AronowAudit.P005.mean_mul: observed used

AronowAudit.P005.mean_square_centering: observed used

AronowAudit.P005.mean_sub: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T5

Theorem 4.1, last display on p. 1509 (the third bias component $E[\bar x_C'\nu_{3C}] - E[\bar x_T'\nu_{3T}]$ ), as printed. The witness is the validator's and is not in the paper

p. 1509 (PDF p. 8), Section 4, Theorem 4.1, last display on the page

Theorem 4.1, last display, p. 1509 (PDF p. 8). The printed claim is false: the specified six-unit population satisfies the finite-sample assumptions, has positive variances under all fifteen assignments, and gives 217/2500 for the third bias component while the printed formula gives -217/2500. All individual means, moments, coefficients and expectations are included; reversing both printed signs gives the actual component.

Historical component credits

Counterexample proved AronowAudit.P005.interacted_sign_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The finite third-component identity is contradicted under every required fixed-n condition; insertion at n=6 into a standardized distinct-grid sequence with K=28 also supplies the standing sequence interpretation.
Proved AronowAudit.P005.design_six_two Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.expectation_six_two Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:60484f0e60dc13f45cf1fde1df5fdb3e8d7c9d4bbf30f1cae12da10f09ce33f5

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1509 (PDF p. 8)] Theorem 4.1 is stated “Under Assumptions 1–3” and defines $\nu_{3T} = -(\bar x_T\overline{y^*(1)}_T)$ and $\nu_{3C} = -(\bar x_C\overline{y^*(0)}_C)$. Its last display on the page reads “$$E\big[\bar x_C'\nu_{3C}\big] - E\big[\bar x_T'\nu_{3T}\big] = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}(\overline{hy(0)} - \bar h\times\overline{y(0)}) - \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}(\overline{hy(1)} - \bar h\times\overline{y(1)}).$$” [Counterexample] Take $n = 6$, $n_T = 2$, $n_C = 4$, $d = 1$, $x = (-\frac85, -1, 0, \frac35, \frac45, \frac65)$, $y_i(1) = x_i^2$ and $y_i(0) = 2x_i^2 + 3$, that is, $y(1) = (\frac{64}{25}, 1, 0, \frac{9}{25}, \frac{16}{25}, \frac{36}{25})$ and $y(0) = (\frac{203}{25}, 5, 3, \frac{93}{25}, \frac{107}{25}, \frac{147}{25})$. Then: (i) [The paper's conditions hold at this $n$] $\bar x = 0$ and $\overline{x^2} = 1$ (Assumption 2); $\frac1n\sum_iy_i(1)^2 = \frac{1059}{625}$, $\frac1n\sum_iy_i(0)^2 = \frac{17361}{625}$ and $\frac1n\sum_ix_i^2 = 1$ are at most $K = 28$ (Assumption 1); $p_T = \frac13$ (Assumption 3 with, for example, $p_{\min} = \frac14$, $p_{\max} = \frac34$); the six covariate values are distinct, so $\widehat L_T > 0$ and $\widehat L_C > 0$ under each of the 15 assignments (footnote 4). (ii) $h_i = x_i^2$, $\bar h = 1$, $\overline{y(1)} = 1$, $\overline{y(0)} = 5$, $\overline{hy(1)} - \bar h\times\overline{y(1)} = \frac{434}{625}$ and $\overline{hy(0)} - \bar h\times\overline{y(0)} = \frac{868}{625}$; $N_{\mathrm{TTT}} = \frac{1}{10}$ and $N_{\mathrm{CCC}} = -\frac{1}{80}$. (iii) [Left-hand side, by enumeration of the 15 assignments, from the definitions] $E[\bar x_C'\nu_{3C}] = \frac{217}{12500}$, $E[\bar x_T'\nu_{3T}] = -\frac{217}{3125}$, so $E[\bar x_C'\nu_{3C}] - E[\bar x_T'\nu_{3T}] = \frac{217}{2500}$. (iv) [Printed right-hand side] $\frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}\cdot\frac{868}{625} - \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}\cdot\frac{434}{625} = -\frac{1}{80}\cdot\frac{868}{625} - \frac{1}{10}\cdot\frac{434}{625} = -\frac{217}{2500}$. (v) $\frac{217}{2500} \ne -\frac{217}{2500}$: the printed identity is false for this population; the right-hand side with both signs reversed equals $\frac{217}{2500}$.

: SAP.P005a.T5WholeStatement

Theorem: SAP.P005a.T5

Statement hash: 147e96941977dacbe9ab0f9b79d40dc5373c799d34feb07907a49e34a8811472

Statement definition: SAP.P005a.T5WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:15:03Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.design_six_two: observed used

AronowAudit.P005.expectation_six_two: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T6

Theorem 4.1, last display on p. 1509, as repaired: both signs of the right-hand side reversed (Sol's smallest repair)

p. 1509 (PDF p. 8), Section 4, Theorem 4.1, last display on the page

Theorem 4.1, last display, p. 1509 (PDF p. 8). For centered covariates, both third-moment identities, their signed bias components and the corrected control-minus-treatment formula hold exactly.

Historical component credits

Proved AronowAudit.P005.Wave1.corrected_interacted_third_bias_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The negative arm identities omitted as separate conjuncts are algebraic restatements of the two exported moments, so no independent conclusion is lost. No corrected-sign assertion is mistaken for the printed identity.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:f81f48270dcc67ee96590c34f530409cf9e548487564c90807873dc818e3574a

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] The last display of Theorem 4.1 on p. 1509, as transcribed in P005-T5, with $\nu_{3T} = -(\bar x_T\overline{y^*(1)}_T)$ and $\nu_{3C} = -(\bar x_C\overline{y^*(0)}_C)$. [Sol's corrected text (validated.json, P005-E003, prior_correction.corrected_text), verbatim, with math delimiters added] “$E[\bar x_C'\nu_{3C}]-E[\bar x_T'\nu_{3T}]=-\frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}(\overline{hy(0)}-\bar h\,\overline{y(0)})+\frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}(\overline{hy(1)}-\bar h\,\overline{y(1)}).$” [Corrected statement, with every hypothesis] Let $n \ge 3$, $1 \le n_T \le n-1$ and $n_C = n - n_T$, under complete randomization; let $x_1, \ldots, x_n \in \mathbb{R}^d$ be fixed with $\bar x = 0$ (Assumption 2 (1)) and let $y(1), y(0) \in \mathbb{R}^n$ be fixed; $y_i^*(t) = y_i(t) - \overline{y(t)}$, $h_i = \|x_i\|_2^2$, and, as in Theorem 4.1, $\nu_{3T} = -(\bar x_T\overline{y^*(1)}_T)$, $\nu_{3C} = -(\bar x_C\overline{y^*(0)}_C)$. Write $S_t = \overline{hy(t)} - \bar h\times\overline{y(t)}$. Then (i) $E\big[\bar x_T'\bar x_T\,\overline{y^*(1)}_T\big] = N_{\mathrm{TTT}}S_1$ and $E\big[\bar x_C'\bar x_C\,\overline{y^*(0)}_C\big] = N_{\mathrm{CCC}}S_0$ (Lemma A.2 applied coordinate by coordinate, and its mirror image for the control arm); (ii) $E[\bar x_T'\nu_{3T}] = -N_{\mathrm{TTT}}S_1$ and $E[\bar x_C'\nu_{3C}] = -N_{\mathrm{CCC}}S_0$; (iii) $$E\big[\bar x_C'\nu_{3C}\big] - E\big[\bar x_T'\nu_{3T}\big] = -\frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}\big(\overline{hy(0)} - \bar h\times\overline{y(0)}\big) + \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}\big(\overline{hy(1)} - \bar h\times\overline{y(1)}\big) = -N_{\mathrm{CCC}}S_0 + N_{\mathrm{TTT}}S_1.$$ The printed right-hand side is the negative of (iii). In the theorem the display stands under “Under Assumptions 1–3”; the identity is exact for each $n$ and uses only $\bar x = 0$ and $n \ge 3$ (neither $\overline{xx'} = I_d$, nor invertibility, nor Assumptions 1 and 3).

: T6Statement

Theorem: SAP.P005a.T6

Statement hash: e74ce9ec1140a54778c88863bd65fb6f32086495b9c9fbf7330c99a150134667

Statement definition: SAP.P005a.T6Statement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:01Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T7

Theorem 4.1, final paragraph: the stochastic expansion under “ $\kappa\log d = o(1)$ ”, as printed, at $d = 1$ . The sequence is the validator's and is not in the paper

p. 1510 (PDF p. 9), continuation of Theorem 4.1 (the theorem begins on p. 1509, PDF p. 8), with footnote 8

Theorem 4.1, pp. 1509--1510 (PDF pp. 8--9), footnote 8. The printed claim is false: the explicit one-covariate spike sequence satisfies Assumptions 1--3 and every-assignment regularity, yet its remainder is not of the printed stochastic order. The statement includes all grid identities, pointwise bounds, exact leverage and slopes, fitted-line calculations, sharp bounds, probability-one even-size tails and both exact finite minima.

Historical component credits

Proved AronowAudit.P005.Wave1.univariate_endpoint_counterexample_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Eventual hypotheses and the even-subsequence lower bound give a genuine failure of full-sequence tightness. The fixed-n exact maxima and illustrative enumerations in the target are diagnostics rather than additional requirements on this witness.
Proved AronowAudit.P005.univariate_printed_condition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; A simple auxiliary identity can be faithful even though proving it is elementary; the actual asymptotic counterexample is separately exported by R037-13.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:ced1bdd8e67a9027a75e122bb092cb8fcb6b82a527a44c1860c84fb51a0b4983

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] Theorem 4.1 begins “THEOREM 4.1: Under Assumptions 1–3, …” on p. 1509 (PDF p. 8). [Printed, continuation of Theorem 4.1 on p. 1510 (PDF p. 9)] “If $\kappa\log d = o(1)$,$^{8}$ $$\widehat{\mathrm{ATE}}_I = \mathrm{ATE} + \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta_{I,T}\big) - \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta_{I,C}\big) + O_p\Big(\sqrt{\frac{\kappa d}{n}}\Big).$$” Footnote 8: “The upper bound on the stochastic order of the bias of the interacted case is characterized in equation (15) in Lei and Ding (2021).” [Counterexample: a sequence with one covariate] For every integer $n \ge 6$ let $d = 1$, $n_T = \lfloor n/2\rfloor$, $n_C = n - n_T$, $\epsilon = n^{-2}$, and $v_1 = 0$, $v_{j+2} = -1 + \frac{2j}{n-2}$ for $j = 0, \ldots, n-2$. Put $a_1 = 1$ and $a_i = \epsilon v_i$ for $i \ge 2$; $y_i(1) = y_i(0) = v_i$; $z_i = a_i - \frac1n$; $q = \frac1n\sum_iz_i^2$; and $x_i = z_i/\sqrt q$. Then: (i) [The hypotheses of the theorem hold] $\sum_iv_i = 0$, $\sum_iv_i^2 = \frac{n(n-1)}{3(n-2)}$, $\bar a = \frac1n$ and $q = \frac{n-1}{n^2}\big(1 + \frac{1}{3(n-2)n^2}\big) > 0$. Hence $\bar x = 0$ and $\overline{x^2} = 1$ for every $n \ge 6$ (Assumption 2); $|y_i(1)| = |y_i(0)| \le 1$ and $\frac1n\sum_ix_i^2 = 1$ (Assumption 1 with $K = 1$ for all $n$); $\frac14 < n_T/n < \frac34$ (Assumption 3). Under every assignment each arm has at least three units and positive within-arm variance of $x$, so $\widehat L_T$ and $\widehat L_C$ are invertible (footnote 4). $\mathrm{ATE} = 0$. (ii) [Leverage] $\kappa = \frac{(1 - 1/n)^2}{nq} = \frac{3n(n-1)(n-2)}{3n^2(n-2)+1} \to 1$, while $\kappa\log d = \kappa\log1 = 0$ for every $n$. So the printed condition $\kappa\log d = o(1)$ holds although $\kappa$ does not tend to 0. (iii) [The estimator; even $n \ge 6$, so $n_T = n_C = n/2$] $\widehat{\mathrm{ATE}}_I = (\overline{y(1)}_T - \bar x_T'\widehat\beta_{I,T}) - (\overline{y(0)}_C - \bar x_C'\widehat\beta_{I,C})$ is the difference of the two within-arm fitted lines evaluated at $x = 0$, that is, at $a = \frac1n$. In the arm that does not contain unit 1, $y_i = a_i/\epsilon$ exactly, so its fitted value at $a = \frac1n$ is $\frac{1}{n\epsilon} = n$. In the arm that contains unit 1 the fitted slope $s_H$ in the coordinate $a$ satisfies $|s_H| \le 2$ and the fitted value $p_H$ at $a = \frac1n$ satisfies $|p_H| \le \frac{25}{18}$. Hence $|\widehat{\mathrm{ATE}}_I| \ge n - 3$ under every assignment. (iv) [The leading term] $\beta_{I,T} = \beta_{I,C} = \sqrt q\,b$ with $b = \frac{\sum_iz_iv_i}{\sum_iz_i^2} = \frac{n^2}{3n^2(n-2)+1} \le \frac{1}{12}$, so that $y_i(t) - \overline{y(t)} - x_i'\beta_{I,t} = v_i - bz_i =: e_i$ for $t = 0, 1$, with $|e_i| \le \frac{13}{12}$. The printed leading term is $L_n = \frac{1}{n_T}\sum_{i\in[T]}e_i - \frac{1}{n_C}\sum_{i\in[C]}e_i$, and $|L_n| \le \frac{13}{6} < 3$ under every assignment. (v) [The remainder] $R_n := \widehat{\mathrm{ATE}}_I - \mathrm{ATE} - L_n$ satisfies $|R_n| \ge n - 6$ under every assignment, for every even $n \ge 6$, whereas $\sqrt{\kappa d/n} \le n^{-1/2}$. Hence $|R_n|\big/\sqrt{\kappa d/n} \ge (n-6)\sqrt n \to \infty$ with probability one along the even integers: for every $M$, $\Pr\big(|R_n| > M\sqrt{\kappa d/n}\big) = 1$ for all large even $n$. So $R_n$ is not $O_p\big(\sqrt{\kappa d/n}\big)$, and the printed expansion fails for this sequence. (vi) [Exact values for checking] $n = 6$: $\kappa = \frac{360}{433}$, $b = \frac{36}{433}$, and the minimum of $|R_n|$ over the 20 assignments is $\frac{4357386}{779833} \approx 5.59$. $n = 12$: $\kappa = \frac{3960}{4321}$, $b = \frac{144}{4321}$, and the minimum of $|R_n|$ over the 924 assignments is $\frac{8690712840}{752921287} \approx 11.54$.

: SAP.P005a.T7WholeStatement

Theorem: SAP.P005a.T7

Statement hash: 2ea70cd2ad0e5fc2de604cb02be40d9e27f3e7a714ecfea261e6e1346da50560

Statement definition: SAP.P005a.T7WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:01Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T8

Theorem 4.1, final paragraph, as repaired: the condition $\kappa\log\max\{d,2\} = o(1)$ (Sol's smallest repair for `P005-E001`)

p. 1510 (PDF p. 9), continuation of Theorem 4.1, with footnote 8; proof B.3 on p. 1517 (PDF p. 16)

No current theorem is supplied for this target.

Historical component credits

Proved, conditional on a published theorem AronowAudit.P005.Wave1.corrected_interacted_expansion_max_conditional Assumes as a hypothesis: J. A. Tropp, "Improved analysis of the subsampled randomized Hadamard transform", Advances in Adaptive Data Analysis 3 (2011), Lemma 3.4, pp. 115-126 Historical support retained; baseline fidelity: ok; manifest: not used; Solution use unverified; The only credited expansion theorem requires the unproved proposition Tropp2011RowSampling. The canonical claim has no such hypothesis, and no unconditional estimator expansion is available from this row. Do not turn the cited proposition into an assumption. The separately targeted logarithm equivalences are retained in T9.

Formal result: omitted

Scope: The available expansion theorem assumes Tropp2011RowSampling instead of proving it. The target claim contains no such hypothesis, so this conditional theorem does not establish its unconditional estimator expansion. The separately targeted logarithm equivalences are retained in T9.

Omission: not formalized

Reason: The available expansion theorem assumes Tropp2011RowSampling instead of proving it. The target claim contains no such hypothesis, so this conditional theorem does not establish its unconditional estimator expansion. The separately targeted logarithm equivalences are retained in T9.

Omission explanation

Assumptions and verification

Claim revision: catalog:73cfad2844215a48f4019f8cb6c0724d153ec400be483e98756af197c7885aae

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] The final paragraph of Theorem 4.1, as transcribed in P005-T7. Its proof, Appendix B.3 (p. 1517, PDF p. 16), reads in full: “The decomposition of the bias is similar to the one in Theorem 3.1. We omit the details. The stochastic order of the bias is derived in Lei and Ding (2021).” [Sol's corrected text (validated.json, P005-E001, prior_correction.corrected_text), verbatim] “Under Assumptions 1–3, if $\kappa\log\max\{d,2\}=o(1)$, $$\widehat{\mathrm{ATE}}_I=\mathrm{ATE}+\frac{1}{n_T}\sum_{i\in[T]}(y_i(1)-\overline{y(1)}-x_i^{\prime}\beta_{I,T})-\frac{1}{n_C}\sum_{i\in[C]}(y_i(0)-\overline{y(0)}-x_i^{\prime}\beta_{I,C})+O_p\!\left(\sqrt{\frac{\kappa d}{n}}\right).$$” [Corrected statement, with every hypothesis] Consider a sequence of finite populations indexed by $n$, with dimension $d = d_n$ ($1 \le d_n \le n$), covariates $x_1, \ldots, x_n \in \mathbb{R}^{d_n}$, potential outcomes $y_i(1), y_i(0)$ and treated-arm size $n_T = n_{T,n}$, under complete randomization at each $n$. Suppose Assumption 1 holds with one constant $K$, Assumption 2 holds for all $n$ large enough, Assumption 3 holds, and $\widehat L_T$, $\widehat L_C$ are invertible (footnote 4). Let $R_n = \widehat{\mathrm{ATE}}_I - \mathrm{ATE} - \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta_{I,T}\big) + \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta_{I,C}\big)$. If $\kappa_n\log\max\{d_n, 2\} \to 0$, then $R_n = O_p\big(\sqrt{\kappa_nd_n/n}\big)$: for every $\varepsilon > 0$ there are $M < \infty$ and $n_0$ such that $\Pr\big(|R_n| > M\sqrt{\kappa_nd_n/n}\big) < \varepsilon$ for all $n \ge n_0$. [Equivalent forms of the repaired condition] (a) the printed condition $\kappa\log d = o(1)$ together with $\kappa = o(1)$ whenever $d = 1$ (for $d \ge 2$, $\max\{d,2\} = d$; for $d = 1$, $\kappa\log2 \to 0$ is $\kappa \to 0$); (b) $\kappa\log(2d) = o(1)$, the form given by the validator of P005-N004 (P005-T9; since $\log d \le \log(2d) \le 2\log d$ for $d \ge 2$). Only the case $d = 1$ changes; estimator, estimand and remainder order are as printed.

Unit certificate (omission has no certified theorem). 20261104T172639Z

Declaration review: omission_ok; 2026-11-04T17:59:36Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Not supplied by this export. Manifest uses are reported separately; actual use is unverified.

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T9

Theorem 4.1, final paragraph, as repaired in the form given by the validator of `P005-N004`: the condition $\kappa\log(2d) = o(1)$ , with its equivalence to the form of `P005-T8`

p. 1510 (PDF p. 9), continuation of Theorem 4.1, with footnote 8; proof B.3 on p. 1517 (PDF p. 16)

Theorem 4.1, p. 1510 (PDF p. 9), and Appendix B.3, p. 1517 (PDF p. 16). Under Assumptions 1--3 and every-assignment invertibility, the repaired condition kappa log(2d) tending to zero implies the full interacted expansion with remainder O_p(sqrt(kappa d/n)). The two repaired logarithms have the stated inequalities and equivalent limits. The repaired condition implies vanishing leverage and the printed condition, and is equivalent to the printed condition plus vanishing leverage on the dimension-one subsequence.

Historical component credits

Proved, conditional on a published theorem AronowAudit.P005.Wave1.corrected_interacted_expansion_two_conditional Assumes as a hypothesis: J. A. Tropp, "Improved analysis of the subsampled randomized Hadamard transform", Advances in Adaptive Data Analysis 3 (2011), Lemma 3.4, pp. 115-126 Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; Requires unproved Tropp2011RowSampling; the stochastic expansion is dropped and only the independently proved canonical log-equivalence clauses are retained.
Proved AronowAudit.P005.repaired_log_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P005.repaired_log_conditions Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Allowing an arbitrary finite prefix is a true generalization, with no loss of target cases.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:9cbe90b8e334ed63f17dd58685a944cd768da9b2ec57d6aa5833cb98c1d89c96

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] The final paragraph of Theorem 4.1 with footnote 8, as transcribed in P005-T7, and its proof B.3, as transcribed in P005-T8. [Sol's smallest repair (validated.json, P005-N004), verbatim] “Replace the expansion’s condition by $\kappa\log(2d)=o(1)$. Equivalently, keep the printed condition and additionally require $\kappa=o(1)$ whenever $d=1$. The repaired remainder bound is proved in artifact/derivation.md.” [Corrected statement, with every hypothesis] Consider a sequence of finite populations indexed by $n$, with dimension $d = d_n$ ($1 \le d_n \le n$), covariates $x_1, \ldots, x_n \in \mathbb{R}^{d_n}$, potential outcomes $y_i(1), y_i(0)$ and treated-arm size $n_T = n_{T,n}$, under complete randomization at each $n$. Suppose Assumption 1 holds with one constant $K$, Assumption 2 holds for all $n$ large enough, Assumption 3 holds, and $\widehat L_T$, $\widehat L_C$ are invertible (footnote 4). Let $R_n = \widehat{\mathrm{ATE}}_I - \mathrm{ATE} - \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta_{I,T}\big) + \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta_{I,C}\big)$. If $\kappa_n\log(2d_n) \to 0$, then $R_n = O_p\big(\sqrt{\kappa_nd_n/n}\big)$. [Equivalence with the form of P005-T8, and with the printed condition for $d \ge 2$] (a) For every integer $d \ge 1$: $\log\max\{d, 2\} \le \log(2d) \le 2\log\max\{d, 2\}$. (b) Hence, for numbers $\kappa_n \ge 0$ and integers $d_n \ge 1$: $\kappa_n\log(2d_n) \to 0$ if and only if $\kappa_n\log\max\{d_n, 2\} \to 0$. (c) $\kappa_n\log(2d_n) \to 0$ implies $\kappa_n \to 0$ (because $\log(2d) \ge \log2 > 0$) and $\kappa_n\log d_n \to 0$; conversely, if $d_n \ge 2$ for all large $n$, then $\kappa_n\log d_n \to 0$ implies $\kappa_n\log(2d_n) \to 0$ (because $\log(2d) \le 2\log d$ for $d \ge 2$). So for $d \ge 2$ the repaired condition is the printed one, and at $d = 1$ it is $\kappa = o(1)$.

: SAP.P005a.T9WholeStatement

Theorem: SAP.P005a.T9

Statement hash: 56b348ebfcd32666db5a1c4e2a6b215e7a5e379143c18ba50f2024d830ef5fd7

Statement definition: SAP.P005a.T9WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:01Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T10

Proof of Theorem 4.2 (Appendix B.4), last display on p. 1518: the weight $\frac{n_T}{(n-2)n_C}$ on the treated-arm covariance, as printed. The witness is the validator's and is not in the paper

p. 1518 (PDF p. 17), Appendix B.4, last display (it ends on p. 1519, PDF p. 18)

Appendix B.4, last display, pp. 1518--1519 (PDF pp. 17--18). The printed claim is false: the six-unit standardized regular population has printed treated expectation 4913/1200 instead of 20/3, while the derived weight is unbiased. All fifteen-assignment positivity, leverage, population covariance, assignmentwise and expected gaps, and rational scaling identities are included.

Historical component credits

Counterexample proved AronowAudit.P005.interacted_weight_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Judged as the finite witness described by this row: it violates unbiasedness of the printed wrong-weight expression and verifies the corrected-weight expectation. No growing-population stochastic-rate refutation is claimed here. The auxiliary S_1 value and pointwise P-Q identity in the target are not necessary to this witness test.
Proved AronowAudit.P005.sampleCov_scale_left Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.sampleCov_scale_both Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.scalar_dot_armN Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.scalar_leverage Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.scalar_armL_det Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.standardized_scalar Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:b20069c624c74af471c465873876c6e40e4c0d074870dab006b5353d4ed71a86

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1518 (PDF p. 17), last display of Appendix B.4] “We can characterize the stochastic expansion for the debiased estimator of the treated group as $$\widehat{\mathrm{ATE}}_{T,I,\mathrm{Debiased}} = \overline{y(1)}_T - \bar x_T\widehat L_T^{-1}\hat N_T - \Big(-\bar x_T'\big(\widehat L_T^{-1}-I_d^{-1}\big)\hat N_T - \frac{n_T}{(n-2)n_C}\big(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T\big)\Big)$$ $$= \overline{y(1)}_T - \bar x_TN_T - \bar x_T(\hat N_T-N_T) + \frac{n_T}{(n-2)n_C}\big(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T\big)$$ $$= \overline{y(1)}_T - \bar x_TN_T + O_p\Big(\sqrt{\frac{\kappa}{n}}\Big),$$” [continued on p. 1519 (PDF p. 18)] “using the stochastic order estimates we derived in the proof of Theorem 4.1 and the fact that $\kappa \in [\frac dn, 1]$.” [Printed, same page, two displays earlier] “Collecting the constants in front of $(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T)$ gives $$\frac{n_C}{n(n_T-1)} - \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2}\times\frac{n_T(n-1)}{(n_T-1)n} = \frac{n_C}{(n-2)n_T}.$$ Thus, an unbiased estimator for $\mathbf{E}[-\bar x_T'(\nu_{1T}+\nu_{2T}+\nu_{3T})]$ is $$-\frac{n_C}{(n-2)n_T}\big(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T\big) - \bar x_T'\big(\widehat L_T^{-1}-I_d^{-1}\big)\hat N_T.$$” [What the printed display asserts] The “debiased estimator of the treated group” is $\overline{y(1)}_T - \bar x_T'\widehat\beta_{I,T}$ minus the estimator of its bias derived two displays earlier, and it is to be unbiased for $\overline{y(1)}$ (Theorem 4.2). After the inverse cancels, the printed first and second lines equal $P := \overline{y(1)}_T - \bar x_T'\hat N_T + \frac{n_T}{(n-2)n_C}\big(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T\big)$ (reading the unprimed $\bar x_T$ as $\bar x_T'$: the missing transposes of P005-E010, restored in P005-T11), whereas the estimator derived two displays earlier gives $Q := \overline{y(1)}_T - \bar x_T'\hat N_T + \frac{n_C}{(n-2)n_T}\big(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T\big)$. [Counterexample] Take $n = 6$, $n_T = 2$, $n_C = 4$, $d = 1$, $a = (-3, -2, -1, 0, 1, 5)$ and $x_i = \sqrt{3/20}\,a_i$, $y_i(1) = a_i^2$, that is, $y(1) = (9, 4, 1, 0, 1, 25)$, and $y(0) = (0, \ldots, 0)$. Then: (i) [The paper's conditions hold at this $n$] $\bar x = 0$ and $\overline{x^2} = \frac{3}{20}\cdot\frac{40}{6} = 1$ (Assumption 2); the six covariate values are distinct, so $\widehat L_T = \frac{3}{20}(\overline{a^2}_T - \bar a_T^2) > 0$ and $\widehat L_C = \frac{3}{20}(\overline{a^2}_C - \bar a_C^2) > 0$ under each of the 15 assignments (footnote 4); $h_i = x_i^2 = \frac{3}{20}a_i^2$ and $\bar h = 1$; $\overline{y(1)} = \frac{20}{3}$; $S_1 := \overline{hy(1)} - \bar h\times\overline{y(1)} = \frac{343}{30}$. (ii) With the printed weight $\frac{n_T}{(n-2)n_C} = \frac18$: $\mathbf{E}[P] = \frac{4913}{1200} \ne \frac{20}{3} = \overline{y(1)}$. (iii) With the derived weight $\frac{n_C}{(n-2)n_T} = \frac12$: $\mathbf{E}[Q] = \frac{20}{3} = \overline{y(1)}$. (iv) $P - Q = \big(\frac{n_T}{(n-2)n_C} - \frac{n_C}{(n-2)n_T}\big)\widehat{\mathrm{Cov}}_T(h, y(1)) = -\frac38\widehat{\mathrm{Cov}}_T(h, y(1))$, and $\mathbf{E}[P] - \mathbf{E}[Q] = -\frac38\cdot\frac{(n_T-1)n}{n_T(n-1)}S_1 = -\frac38\cdot\frac35\cdot\frac{343}{30} = -\frac{1029}{400}$. (In rational terms: $\bar x_T'\hat N_T = \frac{3}{20}\,\bar a_T\big(\overline{ay(1)}_T - \bar a_T\overline{y(1)}_T\big)$ and $\widehat{\mathrm{Cov}}_T(h, y(1)) = \frac{3}{20}\big(\overline{a^2y(1)}_T - \overline{a^2}_T\overline{y(1)}_T\big)$.) So the printed display is not the treated-group part of the debiased estimator of Theorem 4.2, and the expression it prints is not unbiased for $\overline{y(1)}$ in an unbalanced design; it contradicts the unbiased estimator printed two displays above it.

: SAP.P005a.T10WholeStatement

Theorem: SAP.P005a.T10

Statement hash: ddfdaacefaec314fa62f4b65343aaf0e30200fa7e230e62455df4d3670390a6e

Statement definition: SAP.P005a.T10WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:01Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.sampleCov_scale_both: observed used

AronowAudit.P005.sampleCov_scale_left: observed used

AronowAudit.P005.scalar_armL_det: observed used

AronowAudit.P005.scalar_dot_armN: observed used

AronowAudit.P005.scalar_leverage: observed used

AronowAudit.P005.standardized_scalar: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T11

Proof of Theorem 4.2 (Appendix B.4), last display on p. 1518, as repaired: weight $\frac{n_C}{(n-2)n_T}$ (`P005-E013`), with the transposes of `P005-E010` restored

p. 1518 (PDF p. 17), Appendix B.4, last display, and its last line on p. 1519 (PDF p. 18)

Appendix B.4, corrected last display, pp. 1518--1519 (PDF pp. 17--18). The corrected arm expressions cancel the inverse assignmentwise, have the exact generic expectation, and are unbiased with the derived arm-size weights. The cancelling treated weight equals its unsimplified expression and is unique for arms of size at least two; singleton expectations require no division by s-1. The printed treated weight has the stated bias. Standardization gives d/n <= kappa <= 1 and h_i <= n. Under Assumptions 1--3 both corrected arm remainders are O_p(sqrt(kappa/n)), from any starting size, with no extra rate assumption.

Historical component credits

Proved AronowAudit.P005.treated_arm_cancellation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P005.Wave1.treated_arm_expectation_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P005.Wave1.control_arm_expectation_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The target says the analogous control statements hold, so the nC>=2 restriction is the counterpart of the explicit nT>=2 restriction in T11(b). The iff concerns cancellation of the bias multiplier, not uniqueness of unbiasedness when leverageCov=0.
Proved AronowAudit.P005.Wave1.corrected_arm_debiased_order_proof Historical support retained; baseline fidelity: ok; manifest: not used; not used in Solution; The inverse-free form needs no OLS inverse hypothesis; it certifies precisely the order component assigned to this row. Uniform K and the finite tail convention match the claim reading.
Proved AronowAudit.P005.dot_comm Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.leverage_le_population_size Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.kappa_bounds Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The standalone leverage-range conclusion is not retained; the rate theorem is used directly.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:26d39cc7ca19e84b80075f536556908166fa516c69efcd1eb37a711402c04d65

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] The last display of Appendix B.4 and the collection of constants two displays earlier, as transcribed in P005-T10. [Sol's smallest repair for P005-E013, verbatim] “In both covariance terms replace $n_T/[(n-2)n_C]$ by $n_C/[(n-2)n_T]$. The missing transposes are separately covered by P005-E010.” [Sol's corrected text, verbatim] “$\overline{y(1)}_T-\bar x_T^\prime\widehat N_T+\frac{n_C}{(n-2)n_T}(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T)$. Here $\widehat N_T=\overline{xy(1)}_T-\bar x_T\overline{y(1)}_T$.” [Corrected display] $$\widehat{\mathrm{ATE}}_{T,I,\mathrm{Debiased}} = \overline{y(1)}_T - \bar x_T'\widehat L_T^{-1}\hat N_T - \Big(-\bar x_T'\big(\widehat L_T^{-1}-I_d^{-1}\big)\hat N_T - \frac{n_C}{(n-2)n_T}\big(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T\big)\Big)$$ $$= \overline{y(1)}_T - \bar x_T'N_T - \bar x_T'(\hat N_T-N_T) + \frac{n_C}{(n-2)n_T}\big(\overline{hy(1)}_T-\bar h_T\overline{y(1)}_T\big) = \overline{y(1)}_T - \bar x_T'N_T + O_p\Big(\sqrt{\frac{\kappa}{n}}\Big),$$ using the stochastic order estimates derived in the proof of Theorem 3.2 and the fact that $\kappa \in [\frac dn, 1]$. [Mathematical content, with every hypothesis] (a) [Algebra, assignment by assignment] If $n \ge 3$ and $\widehat L_T$ is invertible, then for every real $c$: $\overline{y(1)}_T - \bar x_T'\widehat L_T^{-1}\hat N_T - \big(-\bar x_T'(\widehat L_T^{-1} - I_d)\hat N_T - c\,\widehat{\mathrm{Cov}}_T(h, y(1))\big) = \overline{y(1)}_T - \bar x_T'\hat N_T + c\,\widehat{\mathrm{Cov}}_T(h, y(1)) = \overline{y(1)}_T - \bar x_T'N_T - \bar x_T'(\hat N_T - N_T) + c\,\widehat{\mathrm{Cov}}_T(h, y(1))$. (b) [Exact expectation] Under complete randomization with $n \ge 3$, $2 \le n_T \le n-1$ and $\bar x = 0$, for every real $c$: $$\mathbf{E}\big[\overline{y(1)}_T - \bar x_T'\hat N_T + c\,\widehat{\mathrm{Cov}}_T(h, y(1))\big] = \overline{y(1)} + \Big(-\frac{n_C}{n_T(n-1)} + N_{\mathrm{TTT}} + c\,\frac{(n_T-1)n}{n_T(n-1)}\Big)S_1,\qquad S_1 = \overline{hy(1)} - \bar h\times\overline{y(1)}.$$ The bracket vanishes exactly when $c = \frac{n_C}{n(n_T-1)} - N_{\mathrm{TTT}}\frac{n_T(n-1)}{(n_T-1)n} = \frac{n_C}{(n-2)n_T}$. So with the corrected weight the treated-group debiased estimator is unbiased for $\overline{y(1)}$, and with the printed weight $\frac{n_T}{(n-2)n_C}$ its expectation is $\overline{y(1)} + \frac{n_T^2 - n_C^2}{(n-2)n_Tn_C}\cdot\frac{(n_T-1)n}{n_T(n-1)}S_1$. (c) [Order] Along a sequence of populations satisfying Assumptions 1–3: $-\bar x_T'(\hat N_T - N_T) + \frac{n_C}{(n-2)n_T}\widehat{\mathrm{Cov}}_T(h, y(1)) = O_p\big(\sqrt{\kappa/n}\big)$. The analogous statements hold for the control group: $\overline{y(0)}_C - \bar x_C'\widehat L_C^{-1}\hat N_C + \bar x_C'(\widehat L_C^{-1} - I_d)\hat N_C + \frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0)) = \overline{y(0)}_C - \bar x_C'\hat N_C + \frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0))$ has expectation $\overline{y(0)}$ and equals $\overline{y(0)}_C - \bar x_C'N_C + O_p(\sqrt{\kappa/n})$.

: SAP.P005a.T11WholeStatement

Theorem: SAP.P005a.T11

Statement hash: b7c94c69260f6dcab4e9e2520d8834ba702e0d89a7982da1023d92c17db49702

Statement definition: SAP.P005a.T11WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:01Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.Wave1.corrected_arm_debiased_order_proof: observed not used

AronowAudit.P005.dot_comm: observed used

AronowAudit.P005.kappa_bounds: observed used

AronowAudit.P005.leverage_le_population_size: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T12

Proof of Theorem 4.2 (Appendix B.4), the display after “Similarly, an unbiased estimator for $\mathbf{E}[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})]$ is”, as printed with $\widehat L_T^{-1}$ . The witness is the validator's and is not in the paper

p. 1518 (PDF p. 17), Appendix B.4

Appendix B.4, display introduced by “Similarly”, p. 1518 (PDF p. 17). The printed claim is false: for the six-unit population with a=(-3,-2,-1,0,1,5), x=sqrt(3/20)a, y(1)=0 and y(0)=a², all 15 assignments have positive treated and control Gram matrices. The population is standardized, has bounded second moments and allocation 1/3, and has h_i=(3/20)a_i² and mean leverage one. The control slope decomposes into its population slope and the three nu terms; the mean control covariate and its population-slope product have expectation zero. The nuisance-product and fitted-slope-product expectations both equal 31085815448171/17062566567774 (approximately 1.822). The printed estimator has expectation 12010469/846720 (approximately 14.18), so is biased; replacing the treated inverse by the control inverse gives exactly the nuisance expectation. The arm sizes, variance scaling, coefficient 1/8 and all rational-coordinate identities hold assignment by assignment.

Historical component credits

Counterexample proved AronowAudit.P005.interacted_inverse_counterexample Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is a witness for T12, not proof that all printed formulas are unbiased. The scalar outcome is used as yC; weightPopulation, which places it in yT for another target, is not used in this statement. Centering gives E[xbar_C]=0, so the slope-product expectation is the nuisance bias specified by the target.
Proved AronowAudit.P005.scalar_slope_product_scale Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.scalar_inverse_product_scale Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:564f8a15fa14744e56844d46a0126fe098908f8d58e7f50c29e4de80b1169e63

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1518 (PDF p. 17), Appendix B.4] “Similarly, an unbiased estimator for $\mathbf{E}[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})]$ is $$\frac{n_T}{(n-2)n_C}\big(\overline{hy(0)}_C-\bar h_C\overline{y(0)}_C\big) + \bar x_C'\big(\widehat L_T^{-1}-I_d^{-1}\big)\hat N_C.$$” [Counterexample] Take $n = 6$, $n_T = 2$, $n_C = 4$, $d = 1$, $a = (-3, -2, -1, 0, 1, 5)$ and $x_i = \sqrt{3/20}\,a_i$, $y_i(0) = a_i^2$, that is, $y(0) = (9, 4, 1, 0, 1, 25)$ (the outcome $y(1)$ does not enter; take $y(1) = 0$). Then: (i) [The paper's conditions hold at this $n$] $\bar x = 0$ and $\overline{x^2} = \frac{3}{20}\cdot\frac{40}{6} = 1$ (Assumption 2); the six covariate values are distinct, so $\widehat L_T = \frac{3}{20}(\overline{a^2}_T - \bar a_T^2) > 0$ and $\widehat L_C = \frac{3}{20}(\overline{a^2}_C - \bar a_C^2) > 0$ under each of the 15 assignments (footnote 4); $h_i = x_i^2 = \frac{3}{20}a_i^2$ and $\bar h = 1$. (ii) [The quantity to be estimated] Since $\widehat\beta_{I,C} = \beta_{I,C} + \nu_{1C} + \nu_{2C} + \nu_{3C}$ and $\mathbf{E}[\bar x_C] = 0$: $\mathbf{E}[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})] = \mathbf{E}[\bar x_C'\widehat\beta_{I,C}] = \frac{31085815448171}{17062566567774}$ ($\approx 1.822$). (iii) [The printed estimator] $\mathbf{E}\big[\frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0)) + \bar x_C'(\widehat L_T^{-1} - I_d^{-1})\hat N_C\big] = \frac{12010469}{846720}$ ($\approx 14.18$), which differs from (ii): the printed expression is not unbiased. (iv) [With the control-arm matrix] $\mathbf{E}\big[\frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0)) + \bar x_C'(\widehat L_C^{-1} - I_d^{-1})\hat N_C\big] = \frac{31085815448171}{17062566567774}$, equal to (ii). (In rational terms, with $\mathrm{cov}_C(a, y) = \overline{ay(0)}_C - \bar a_C\overline{y(0)}_C$, $\mathrm{var}_C(a) = \overline{a^2}_C - \bar a_C^2$ and $\mathrm{var}_T(a)$ likewise: $\bar x_C'\widehat\beta_{I,C} = \bar a_C\,\mathrm{cov}_C(a, y)/\mathrm{var}_C(a)$, $\bar x_C'\hat N_C = \frac{3}{20}\bar a_C\,\mathrm{cov}_C(a, y)$, $\bar x_C'\widehat L_T^{-1}\hat N_C = \bar a_C\,\mathrm{cov}_C(a, y)/\mathrm{var}_T(a)$, $\widehat{\mathrm{Cov}}_C(h, y(0)) = \frac{3}{20}\big(\overline{a^2y(0)}_C - \overline{a^2}_C\overline{y(0)}_C\big)$, and $\frac{n_T}{(n-2)n_C} = \frac18$.)

: SAP.P005a.T12WholeStatement

Theorem: SAP.P005a.T12

Statement hash: 65ab2abaae3e152f65106be06a98c083d46a24fcd6d73093a59253384299ac86

Statement definition: SAP.P005a.T12WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:31Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.scalar_inverse_product_scale: observed used

AronowAudit.P005.scalar_slope_product_scale: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T13

Proof of Theorem 4.2 (Appendix B.4), the control-arm bias estimator, as repaired: $\widehat L_C^{-1}$ in place of $\widehat L_T^{-1}$ (`P005-E012`)

p. 1518 (PDF p. 17), Appendix B.4, the display introduced by “Similarly”

Appendix B.4, corrected display introduced by “Similarly”, p. 1518 (PDF p. 17). The control bias estimator with the control inverse has the required expectation chain, component expectations and coefficient identities.

Historical component credits

Proved AronowAudit.P005.Wave1.corrected_control_bias_estimator_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; All required unbiasedness and component equalities are exported using the control inverse. Positive-dimensional nondegenerate examples exist; any empty-covariate cases are additional harmless cases.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:c30f3ce1d4dcf3c7eadbeb19e0b9bd166578850d7ce76aa92dbe399dd062677e

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed] The display introduced by “Similarly” on p. 1518, as transcribed in P005-T12. [Sol's corrected text (validated.json, P005-E012, prior_correction.corrected_text), verbatim] “$\frac{n_T}{(n-2)n_C}(\overline{hy(0)}_C-\bar h_C\overline{y(0)}_C)+\bar x_C^\prime(\widehat L_C^{-1}-I_d^{-1})\widehat N_C$.” [Corrected statement, with every hypothesis] Let $n \ge 3$, $1 \le n_T \le n-1$, $n_C = n - n_T$, under complete randomization; let $\bar x = 0$ and $\overline{xx'} = I_d$ (Assumption 2 at this $n$); and let $\widehat L_C$ be invertible under every assignment (footnote 4). Then an unbiased estimator for $\mathbf{E}[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})]$ is $$\frac{n_T}{(n-2)n_C}\big(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C\big) + \bar x_C'\big(\widehat L_C^{-1} - I_d^{-1}\big)\hat N_C,$$ that is, $\mathbf{E}\big[\frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0)) + \bar x_C'(\widehat L_C^{-1} - I_d)\hat N_C\big] = \mathbf{E}[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})] = \mathbf{E}[\bar x_C'\widehat\beta_{I,C}]$. [The three components, each exact] With $S_0 = \overline{hy(0)} - \bar h\times\overline{y(0)}$: (a) $\bar x_C'\nu_{2C} = \bar x_C'(\widehat L_C^{-1} - I_d)\hat N_C$ is the second summand itself; (b) $\mathbf{E}[\bar x_C'\nu_{1C}] = \frac{n_T}{(n-1)n_C}S_0$ and $\mathbf{E}[\bar x_C'\nu_{3C}] = -N_{\mathrm{CCC}}S_0$; (c) $\mathbf{E}[\widehat{\mathrm{Cov}}_C(h, y(0))] = \frac{(n_C-1)n}{n_C(n-1)}S_0$; (d) $\Big(\frac{n_T}{(n-1)n_C} - N_{\mathrm{CCC}}\Big) = \frac{n_Tn(n_C-1)}{(n-1)(n-2)n_C^2} = \frac{n_T}{(n-2)n_C}\cdot\frac{(n_C-1)n}{n_C(n-1)}$.

: T13Statement

Theorem: SAP.P005a.T13

Statement hash: 8623e8d7a8ac1ecf47ba48cb5d70531a41f06e31dfd5390cd24b089cca807234

Statement definition: SAP.P005a.T13Statement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:31Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T14

Proof of Theorem 3.2 (Appendix B.2), the expansion of $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}}$ on p. 1516, as repaired: transpose restored and the control sum taken over $[C]$ (`P005-E010`); with the same finding's corrections on pp. 1515 and 1507

p. 1516 (PDF p. 15), Appendix B.2, display after “Now we derive the stochastic expansion for the debiased estimator:”; also p. 1515 (PDF p. 14), first sentence of the proof, and p. 1507 (PDF p. 6), Remark 2

Appendix B.2, pp. 1515–1516 (PDF pp. 14–15), and Remark 2, p. 1507 (PDF p. 6). The corrected debiased expansion, inverse cancellation, arbitrary-vector residual identities, centered arm-mean identities and actual bias decomposition hold. A standardized everywhere regular four-unit example shows that the printed treated-set control sum can differ from the control residual mean. One function of the assignment, covariates and observed outcomes, chosen before the potential outcomes, gives the inverse bias component.

Historical component credits

Proved AronowAudit.P005.noninteracted_debiased_expansion Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P005.control_minus_treated_mean_identities Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Fidelity sweep: ok
Proved AronowAudit.P005.inverse_component_observable Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The function is chosen before the population. Although the proof can ignore some hypotheses, they remain in the theorem statement; unused hypotheses do not themselves export a stronger theorem.
Proved AronowAudit.P005.Wave1.noninteracted_exact_decomposition_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The registry shares the Theorem 3.1 exact decomposition with T14; the exported bias identity supplies T14(d). The sole public description also mentions the constant-effect and balance conclusions, both of which are exported.
Proved AronowAudit.P005.mean_dot Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.
Proved AronowAudit.P005.mean_congr Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Not invoked separately: its relevant content is already incorporated in the selected primary proof; no additional conclusion is asserted from this auxiliary.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:97c70ee075ece1bd9256459b2b27758a4222a5ee7daf9da705b967352704cd5e

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1516 (PDF p. 15)] “Now we derive the stochastic expansion for the debiased estimator: $$\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \big(\overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T - \bar x_C)\widehat L^{-1}\hat N\big) - (\bar x_C - \bar x_T)'\big(\widehat L^{-1} - I_d^{-1}\big)\hat N + \frac{1}{n-2}(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T) - \frac{1}{n-2}(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C)$$ $$= \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - x_i'N\big) - \frac{1}{n_C}\sum_{i\in[T]}\big(y_i(0) - x_i'N\big)\ \underbrace{-\ (\bar x_T - \bar x_C)'(\hat N - N) + \frac{1}{n-2}(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T) - \frac{1}{n-2}(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C)}_{(*)}.$$” [Printed, p. 1515 (PDF p. 14)] “PROOF: We first propose estimators for the bias $\mathbf{E}[(\bar x_C - \bar x_T)(\nu_1 + \nu_2 + \nu_3)]$. Note, by Assumption 2, $(\bar x_C - \bar x_T) = \frac{1}{p_T}\bar x_C = -\frac{1}{p_C}\bar x_T$.” [Printed, p. 1507 (PDF p. 6)] “REMARK 2: The bias in $E[(\bar x_C - \bar x_T)'v_2]$ has been discussed previously. $(\bar x_C - \bar x_T)'v_2$ contains no unknowns and hence can be subtracted directly for debiasing purposes.” [Sol's corrected text (validated.json, P005-E010, prior_correction.corrected_text), verbatim] “PDF 15: $(\bar x_T-\bar x_C)^\prime\widehat L^{-1}\widehat N$ and $n_C^{-1}\sum_{i\in[C]}(y_i(0)-x_i^\prime N)$. PDF 17: $\bar x_T^\prime$ before $\widehat L_T^{-1}\widehat N_T$, both $N_T$ occurrences, and $(\widehat N_T-N_T)$. PDF 14: $\mathbb E[(\bar x_C-\bar x_T)^\prime(\nu_1+\nu_2+\nu_3)]$. PDF 6 Remark 2: $\nu_2$.” [Corrected display, p. 1516] $$\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \big(\overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T - \bar x_C)'\widehat L^{-1}\hat N\big) - (\bar x_C - \bar x_T)'\big(\widehat L^{-1} - I_d^{-1}\big)\hat N + \frac{1}{n-2}(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T) - \frac{1}{n-2}(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C)$$ $$= \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - x_i'N\big) - \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - x_i'N\big) - (\bar x_T - \bar x_C)'(\hat N - N) + \frac{1}{n-2}(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T) - \frac{1}{n-2}(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C).$$ [Mathematical content, with every hypothesis] For every assignment with $1 \le n_T \le n-1$, $n \ge 3$ and $\widehat L$ invertible, with $\widehat{\mathrm{ATE}}$ as in (3), $\widehat\beta = \widehat L^{-1}\widehat N$ and $\widehat{\mathrm{Bias}}$ as in (7)–(8): (a) $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \widehat{\mathrm{ATE}} - \widehat{\mathrm{Bias}}$ equals the first line of the corrected display; (b) $-(\bar x_T - \bar x_C)'\widehat L^{-1}\widehat N - (\bar x_C - \bar x_T)'(\widehat L^{-1} - I_d)\widehat N = -(\bar x_T - \bar x_C)'\widehat N$, so that $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T - \bar x_C)'\widehat N + \frac{1}{n-2}\widehat{\mathrm{Cov}}_T(h, y(1)) - \frac{1}{n-2}\widehat{\mathrm{Cov}}_C(h, y(0))$ (the inverse cancels); (c) $\overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T - \bar x_C)'\widehat N = \frac{1}{n_T}\sum_{i\in[T]}(y_i(1) - x_i'N) - \frac{1}{n_C}\sum_{i\in[C]}(y_i(0) - x_i'N) - (\bar x_T - \bar x_C)'(\widehat N - N)$ for every vector $N \in \mathbb{R}^d$, in particular for $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$. These identities are algebraic and use no assumption on the population. The printed sum $\frac{1}{n_C}\sum_{i\in[T]}(y_i(0) - x_i'N)$ is not equal to $\overline{y(0)}_C - \bar x_C'N$ in general. (d) [p. 1515, corrected] “the bias $\mathbf{E}[(\bar x_C - \bar x_T)'(\nu_1 + \nu_2 + \nu_3)]$”; and if $\bar x = 0$ then $\bar x_C - \bar x_T = \frac{1}{p_T}\bar x_C = -\frac{1}{p_C}\bar x_T$ for every assignment. (e) [Remark 2, corrected] “The bias in $E[(\bar x_C - \bar x_T)'\nu_2]$ has been discussed previously. $(\bar x_C - \bar x_T)'\nu_2$ contains no unknowns and hence can be subtracted directly for debiasing purposes”, with $\nu_2 = (\widehat L^{-1} - I_d^{-1})\widehat N$ as defined in Theorem 3.1: $(\bar x_C - \bar x_T)'\nu_2$ is a function of the assignment, the covariates and the observed outcomes $Y_i$ only.

: SAP.P005a.T14WholeStatement

Theorem: SAP.P005a.T14

Statement hash: 1cdbc76f3014ddbb3b961e425c68301232a3f6fb4fff527aa488f14c086d0aa1

Statement definition: SAP.P005a.T14WholeStatement

Certificate. 20261104T172639Z

Declaration review: faithful; 2026-11-04T18:12:31Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.mean_congr: observed used

AronowAudit.P005.mean_dot: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T15

Proof of Theorem 3.2 (Appendix B.2), the second sentence “Collecting the constants in front of …” on p. 1516, as repaired: the control-arm covariance $\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C$ (`P005-E009`)

p. 1516 (PDF p. 15), Appendix B.2, top of the page; equations (11) and (12) are on p. 1515 (PDF p. 14)

Appendix B.2, p. 1516 (PDF p. 15), using (11)–(12) on p. 1515. The corrected control covariance and the treatment covariance have coefficients 1/(n-2) and -1/(n-2), including the printed intermediate fraction and their combined expression.

Historical component credits

Proved AronowAudit.P005.collected_covariance_coefficients Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The stronger algebraic statement covers the target by substitution; it does not independently assert that equations (11) and (12) are unbiased, which is not this row's component.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:6ccf2dbf20e8541ee79c7af7c490a261215221cc74b08446b583e8399a84c790

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1516 (PDF p. 15)] “Collecting the constants in front of $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T$ gives $$-\frac{1}{n-1}\frac{n_T(n-1)}{(n_T-1)n} + \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}\frac{n_T^2(n-1)}{n_C(n_T-1)n} = \frac{n - n_Tn}{(n-2)(n_T-1)n} = -\frac{1}{n-2}.$$ Collecting the constants in front of $\overline{hy(0)}_C - \overline{hy(0)y(0)}_C$ gives $$\frac{1}{n-1}\frac{n_C(n-1)}{(n_C-1)n} - \frac{(n_T-n_C)}{(n-1)(n-2)n_T}\frac{n_T(n-1)}{(n_C-1)n} = \frac{1}{n-2}.$$ Collecting terms gives the expression in the main text.” [Sol's corrected text (validated.json, P005-E009, prior_correction.corrected_text), verbatim] “$\overline{hy(0)}_C-\bar h_C\overline{y(0)}_C$; coefficient $1/(n-2)$.” [Corrected sentence] “Collecting the constants in front of $\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C$ gives $\frac{1}{n-1}\frac{n_C(n-1)}{(n_C-1)n} - \frac{(n_T-n_C)}{(n-1)(n-2)n_T}\frac{n_T(n-1)}{(n_C-1)n} = \frac{1}{n-2}$.” [Mathematical content, with every hypothesis] Let $n_T \ge 2$, $n_C \ge 2$ be integers and $n = n_T + n_C$ (so $n \ge 4$). Then (a) $\frac{1}{n-1}\cdot\frac{n_C(n-1)}{(n_C-1)n} - \frac{n_T-n_C}{(n-1)(n-2)n_T}\cdot\frac{n_T(n-1)}{(n_C-1)n} = \frac{1}{n-2}$; (b) $-\frac{1}{n-1}\cdot\frac{n_T(n-1)}{(n_T-1)n} + \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}\cdot\frac{n_T^2(n-1)}{n_C(n_T-1)n} = \frac{n - n_Tn}{(n-2)(n_T-1)n} = -\frac{1}{n-2}$ (as printed); (c) consequently, for every assignment, the sum of the estimators (11) and (12) of p. 1515, $$\frac{1}{n-1}\Big(\frac{n_C(n-1)}{(n_C-1)n}\widehat{\mathrm{Cov}}_C(h, y(0)) - \frac{n_T(n-1)}{(n_T-1)n}\widehat{\mathrm{Cov}}_T(h, y(1))\Big) + \frac{N_{\mathrm{TTT}}n_T^2(n-1)}{n_C(n_T-1)n}\widehat{\mathrm{Cov}}_T(h, y(1)) - \frac{N_{\mathrm{TTC}}n_T(n-1)}{(n_C-1)n}\widehat{\mathrm{Cov}}_C(h, y(0)),$$ equals $\frac{1}{n-2}\widehat{\mathrm{Cov}}_C(h, y(0)) - \frac{1}{n-2}\widehat{\mathrm{Cov}}_T(h, y(1))$, which is line (7) of Theorem 3.2. The expression whose coefficient is collected in the second sentence is the control-arm sample covariance of leverage and outcome, $\widehat{\mathrm{Cov}}_C(h, y(0)) = \overline{hy(0)}_C - \bar h_C\overline{y(0)}_C$, not the printed $\overline{hy(0)}_C - \overline{hy(0)y(0)}_C$.

: SAP.P005b.T15Statement

Theorem: SAP.P005b.T15

Statement hash: 23e1d09ced9dbcd6945aad8bd350df1d2ad17476b606feaa06095a459fbc397e

Statement definition: SAP.P005b.T15Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:36:59Z; binding v2

Declaration review: faithful; 2026-11-04T16:54:44Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T16

Proof of Lemma A.2, as repaired: $\mathbf{E}[D_iD_j]$ in the first display (p. 1511) and upper limit $n$ in the last display (p. 1512) (`P005-E005`)

p. 1511 (PDF p. 10), proof of Lemma A.2, first display; p. 1512 (PDF p. 11), last display of the proof

proof of Lemma A.2, pp. 1511–1512 (PDF pp. 10–11). The corrected finite indicator calculations, coefficient simplification, and complement relation give both the treated third moment and the mixed treated-control third moment.

Historical component credits

Proved AronowAudit.P005.Wave1.corrected_third_moment_calculation_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the corrected calculation, not a proof of the malformed printed displays. The standalone coefficient equality is a separately credited statement in R040-08; the current statement exports both the raw and simplified moment formulas, without cancelling a possibly zero sum of triple products.
Proved AronowAudit.P005.third_moment_coefficient Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is algebraic coefficient simplification, so the larger m domain is meaningful without pretending there is a randomization law for m>n.
Proved AronowAudit.P005.inclusion_count Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting sampling or linearity fact already incorporated in the aggregate third-moment row; not separately invoked.
Proved AronowAudit.P005.indicator_expectation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting sampling or linearity fact already incorporated in the aggregate third-moment row; not separately invoked.
Proved AronowAudit.P005.expectation_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting sampling or linearity fact already incorporated in the aggregate third-moment row; not separately invoked.
Proved AronowAudit.P005.expectation_mul Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting sampling or linearity fact already incorporated in the aggregate third-moment row; not separately invoked.
Proved AronowAudit.P005.expectation_congr Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting sampling or linearity fact already incorporated in the aggregate third-moment row; not separately invoked.
Proved AronowAudit.P005.indicator_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting sampling or linearity fact already incorporated in the aggregate third-moment row; not separately invoked.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:cf6ca375a180998bc300cd3a01f9043a010e2dc67408478dbd8a955807c3128b

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1511 (PDF p. 10)] “Let $x_i$, $y_i$, and $z_i$ be three possibly identical vectors such that $\bar x = \bar y = \bar z = 0$. LEMMA A.2: For $n \ge 3$,$^{10}$ $$\mathbf{E}[\bar x_T\bar y_T\bar z_T] = N_{\mathrm{TTT}}\frac1n\sum_{i=1}^nx_iy_iz_i\quad\text{and}\quad\mathbf{E}[\bar x_T\bar y_T\bar z_C] = N_{\mathrm{TTC}}\frac1n\sum_{i=1}^nx_iy_iz_i.$$ PROOF: We only prove the first equality. The second one can be proved analogously. First, notice two useful equalities: $$\mathbf{E}\Big[\sum_{i=1}^nD_ix_iy_i\sum_{j\ne i}D_jz_j\Big] = \sum_{i=1}^n\sum_{j\ne i}[D_iD_j]x_iy_iz_j = \frac{n_T(n_T-1)}{n(n-1)}\sum_{i=1}^n\sum_{j\ne i}x_iy_iz_j = \frac{n_T(n_T-1)}{n(n-1)}\Big(\sum_{i=1}^n\sum_{j=1}^nx_iy_iz_j - \sum_{i=1}^nx_iy_iz_i\Big)$$ [p. 1512 (PDF p. 11)] $$= -\frac{n_T(n_T-1)}{n(n-1)}\sum_{i=1}^nx_iy_iz_i,$$ where the fourth equality uses the fact that $\sum_{i=1}^nz_i = 0$. Also, $$\mathbf{E}\Big[\sum_{i=1}^nD_ix_i\sum_{j\ne i}D_jy_j\sum_{s\notin\{i,j\}}D_sz_s\Big] = \sum_{i=1}^n\sum_{j\ne i}\sum_{s\notin\{i,j\}}\mathbf{E}[D_iD_jD_s]x_iy_jz_s = \frac{n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\sum_{i=1}^n\sum_{j\ne i}\sum_{s\notin\{i,j\}}x_iy_jz_s$$ $$= \frac{n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\Big(\sum_{i=1}^n\sum_{j\ne i}\sum_{s=1}^nx_iy_jz_s - \sum_{i=1}^n\sum_{j\ne i}x_iy_j(z_i+z_j)\Big) = \frac{n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\Big(-\sum_{i=1}^n\sum_{j=1}^nx_iy_j(z_i+z_j) + 2\sum_{i=1}^nx_iy_iz_i\Big) = \frac{2n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\sum_{i=1}^nx_iy_iz_i,$$ where the fourth and fifth equality use $\sum_{i=1}^nx_i = \sum_{i=1}^ny_i = \sum_{i=1}^nz_i = 0$. Finally, $$\mathbf{E}[\bar x_T\bar y_T\bar z_T] = \frac{1}{n_T^3}\Big(\mathbf{E}\Big[\sum_iD_ix_iy_iz_i\Big] + \mathbf{E}\Big[\sum_{i=1}^n\sum_{j\ne i}D_iD_j(x_iy_iz_j + x_iy_jz_i + x_jy_iz_i)\Big] + \mathbf{E}\Big[\sum_{i=1}^{N}D_ix_i\sum_{j\ne i}D_jy_j\sum_{s\notin\{i,j\}}D_sz_s\Big]\Big)$$ $$= \frac{1}{n_T^3}\Big(\frac{n_T}{n} - \frac{3n_T(n_T-1)}{n(n-1)} + \frac{2n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\Big)\sum_{i=1}^nx_iy_iz_i,$$ where, for the last equality, we apply the previous two equalities. Simplifying the coefficients gives $\frac1nN_{\mathrm{TTT}}$.” Footnote 10: “We note the following equalities can also be derived using results in Finucan, Galbraith, and Stone (1974).” [Sol's corrected text (validated.json, P005-E005, prior_correction.corrected_text), verbatim] “$\mathbb E[D_iD_j]$ in the double sum; $\sum_{i=1}^{n}$ in the triple-distinct contribution.” [Corrected statements, with every hypothesis] Let $n \ge 3$, $1 \le n_T \le n-1$, under complete randomization with indicators $D_1, \ldots, D_n$; let $x, y, z \in \mathbb{R}^n$ be fixed vectors of scalars with $\sum_ix_i = \sum_iy_i = \sum_iz_i = 0$. Then (a) [first display, corrected] $\mathbf{E}\big[\sum_{i=1}^nD_ix_iy_i\sum_{j\ne i}D_jz_j\big] = \sum_{i=1}^n\sum_{j\ne i}\mathbf{E}[D_iD_j]x_iy_iz_j = \frac{n_T(n_T-1)}{n(n-1)}\sum_{i=1}^n\sum_{j\ne i}x_iy_iz_j = \frac{n_T(n_T-1)}{n(n-1)}\big(\sum_{i=1}^n\sum_{j=1}^nx_iy_iz_j - \sum_{i=1}^nx_iy_iz_i\big) = -\frac{n_T(n_T-1)}{n(n-1)}\sum_{i=1}^nx_iy_iz_i$, where $\mathbf{E}[D_iD_j] = \frac{n_T(n_T-1)}{n(n-1)}$ for $i \ne j$; (b) [second display, as printed] $\mathbf{E}\big[\sum_{i=1}^nD_ix_i\sum_{j\ne i}D_jy_j\sum_{s\notin\{i,j\}}D_sz_s\big] = \frac{2n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\sum_{i=1}^nx_iy_iz_i$, where $\mathbf{E}[D_iD_jD_s] = \frac{n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}$ for distinct $i, j, s$; (c) [last display, corrected] $\mathbf{E}[\bar x_T\bar y_T\bar z_T] = \frac{1}{n_T^3}\Big(\mathbf{E}\big[\sum_iD_ix_iy_iz_i\big] + \mathbf{E}\big[\sum_{i=1}^n\sum_{j\ne i}D_iD_j(x_iy_iz_j + x_iy_jz_i + x_jy_iz_i)\big] + \mathbf{E}\big[\sum_{i=1}^{n}D_ix_i\sum_{j\ne i}D_jy_j\sum_{s\notin\{i,j\}}D_sz_s\big]\Big) = \frac{1}{n_T^3}\Big(\frac{n_T}{n} - \frac{3n_T(n_T-1)}{n(n-1)} + \frac{2n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\Big)\sum_{i=1}^nx_iy_iz_i$; (d) [the coefficient] $\frac{1}{n_T^3}\Big(\frac{n_T}{n} - \frac{3n_T(n_T-1)}{n(n-1)} + \frac{2n_T(n_T-1)(n_T-2)}{n(n-1)(n-2)}\Big) = \frac{(n-n_T)(n-2n_T)}{n\,n_T^2(n-1)(n-2)} = \frac1nN_{\mathrm{TTT}}$; (e) [Lemma A.2, which these displays prove; unchanged] $\mathbf{E}[\bar x_T\bar y_T\bar z_T] = N_{\mathrm{TTT}}\frac1n\sum_{i=1}^nx_iy_iz_i$ and $\mathbf{E}[\bar x_T\bar y_T\bar z_C] = N_{\mathrm{TTC}}\frac1n\sum_{i=1}^nx_iy_iz_i$ (the second because $\bar z_C = -\frac{n_T}{n_C}\bar z_T$ for centred $z$, and $N_{\mathrm{TTC}} = -\frac{n_T}{n_C}N_{\mathrm{TTT}}$).

: SAP.P005b.T16Statement

Theorem: SAP.P005b.T16

Statement hash: cfa62f64cb1db476a7ada0af24e0009f5ba26f0aa72f46034cc94984658bb4d0

Statement definition: SAP.P005b.T16Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:36:59Z; binding v2

Declaration review: faithful; 2026-11-04T16:54:44Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.expectation_congr: observed used

AronowAudit.P005.expectation_mul: observed used

AronowAudit.P005.expectation_sum: observed used

AronowAudit.P005.inclusion_count: observed used

AronowAudit.P005.indicator_expectation: observed used

AronowAudit.P005.indicator_sum: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T17

Appendix B.1.1, the bound on $\mathrm{Var}[(*)]$ on p. 1514, third line, as repaired: matching summation index (`P005-E007`)

p. 1514 (PDF p. 13), Appendix B.1.1, the display bounding $\mathrm{Var}[(*)]$ and the sentence after it

Appendix B.1.1, p. 1514 (PDF p. 13), corrected matching summation index. The variance of the actual common-residual term (*) is bounded by the centered double sum, then by the raw contraction. The exact Gram contraction and leverage bound give deterministic order O(kappa/n), with bounded residual-difference second moments, and (*) is O_p(sqrt(kappa*d/n)), including dimension zero.

Historical component credits

Proved AronowAudit.P005.corrected_contraction_chain Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The row exports the deterministic part of the chain only. R040-12 separately gives the unscaled contraction identity (b), and R040-09 supplies the rate (e); no variance inequality is inferred merely from this deterministic statement.
Proved AronowAudit.P005.Wave1.corrected_contraction_order Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; No positive lower bound on d_n, inverse regularity, or assumed order bound is hidden in the hypotheses. The target concerns the corrected deterministic part (e); the printed variance bound preceding it is contextual.
Proved AronowAudit.P005.covariate_contraction Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Removing the n>=2 restriction is a harmless generalization; the leverage and the squared residual coefficient carry the same summation index.
Proved AronowAudit.P005.dot_population_slope Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The contraction-chain row already includes the needed dot/average contraction; this helper is not separately invoked.
Proved AronowAudit.P005.common_residual_difference_second_moment Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The same fitted slope appears in both residuals and cancels, leaving a centered outcome difference; the 4K bound is not a field or hypothesis in disguise.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:aab7dfd5db24385098eaf8b850dcff916a501202fa3ce88c37d60e69c9e5a418

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1514 (PDF p. 13)] “By Lemma A.5 in Lei and Ding (2021), the variance can be upper-bounded as $$\mathrm{Var}\big[(*)\big] \le \Big(\frac{1}{p_Cn_Tn}\Big)^2\frac{n_Tn_C}{n(n-1)}\sum_{i,j}\Big(x_i'\Big(x_j\big(e_j(1) - e_j(0)\big) - \frac1n\sum_{i=1}^nx_i\big(e_i(1) - e_i(0)\big)\Big)\Big)^2$$ $$\le \Big(\frac{1}{p_Cn_Tn}\Big)^2\frac{n_Tn_C}{n(n-1)}\sum_{i,j}\big(x_i'x_j\big(e_j(1) - e_j(0)\big)\big)^2$$ $$= \Big(\frac{1}{p_Cn_Tn}\Big)^2n^2\frac{n_Tn_C}{n(n-1)}\sum_in^{-1}\|x_i\|_2^2\big(e_j(1) - e_j(0)\big)^2 = O\Big(\frac{\kappa}{n}\Big),$$ where, for the second inequality, we use the fact that, for each $x_i$, $\frac1n\sum_j(x_i'x_j(e_j(1) - e_j(0)))^2 \ge \frac1n\sum_j(x_i'(x_j(e_j(1) - e_j(0)) - \frac1n\sum_{j=1}^nx_j(e_j(1) - e_j(0))))^2$. For the second to last equality, we use the fact $\sum_{i,j}(x_i'x_ja_j)^2 = \sum_ja_j^2x_j'(\sum_ix_ix_i')x_j = n\sum_ja_j^2\|x_j\|_2^2$ by Assumption 2. Thus, the term $(*)$ is of order $O_p(\sqrt{\frac{\kappa d}{n}})$.” [Sol's corrected text (validated.json, P005-E007, prior_correction.corrected_text), verbatim] “$\sum_j n^{-1}\|x_j\|_2^2(e_j(1)-e_j(0))^2$.” [Corrected third line] $$= \Big(\frac{1}{p_Cn_Tn}\Big)^2n^2\frac{n_Tn_C}{n(n-1)}\sum_jn^{-1}\|x_j\|_2^2\big(e_j(1) - e_j(0)\big)^2 = O\Big(\frac{\kappa}{n}\Big).$$ [Mathematical content, with every hypothesis] Fix $n \ge 2$ and $1 \le n_T \le n-1$; let $x_1, \ldots, x_n \in \mathbb{R}^d$ with $\sum_ix_ix_i' = nI_d$ (Assumption 2 (2)); let $\delta_j = e_j(1) - e_j(0)$ (any real numbers $\delta_1, \ldots, \delta_n$ will do in (a)–(d)), and $\mu = \frac1n\sum_kx_k\delta_k$. Then (a) for each $i$: $\sum_j\big(x_i'(x_j\delta_j - \mu)\big)^2 \le \sum_j(x_i'x_j\delta_j)^2$ (the second inequality of the display); (b) $\sum_{i,j}(x_i'x_ja_j)^2 = \sum_ja_j^2x_j'\big(\sum_ix_ix_i'\big)x_j = n\sum_ja_j^2\|x_j\|_2^2$ for all real $a_1, \ldots, a_n$; (c) hence $\Big(\frac{1}{p_Cn_Tn}\Big)^2\frac{n_Tn_C}{n(n-1)}\sum_{i,j}(x_i'x_j\delta_j)^2 = \Big(\frac{1}{p_Cn_Tn}\Big)^2n^2\frac{n_Tn_C}{n(n-1)}\sum_jn^{-1}\|x_j\|_2^2\delta_j^2 = \frac{1}{p_Tp_Cn(n-1)}\cdot\frac1n\sum_jh_j\delta_j^2$; (d) and, since $h_j \le n\kappa$, this is at most $\frac{\kappa}{p_Tp_C(n-1)}\cdot\frac1n\sum_j\delta_j^2$; (e) [order] along a sequence of populations satisfying Assumptions 1–3, $\frac1n\sum_j\delta_j^2 = O(1)$ (because $\frac1n\sum_ie_i^2(1) = O(1)$ and $\frac1n\sum_ie_i^2(0) = O(1)$, p. 1513), so the quantity in (c) is $O(\frac{\kappa}{n})$, as printed. As printed, the third line sums over $i$ while the residual difference carries the free index $j$; the leverage and the residual must carry the same index.

: SAP.P005b.T17WholeStatement

Theorem: SAP.P005b.T17

Statement hash: ce385b30949ee76c6d8092d31e700f7864e503f146a6fd607e0685cd9ef4a509

Statement definition: SAP.P005b.T17WholeStatement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:36:59Z; binding v2

Declaration review: faithful; 2026-11-04T16:54:44Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.dot_population_slope: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T18

Appendix B.1.1, the three-line equality for the term $(*)$ at the top of p. 1514, as repaired: the unmatched closing parenthesis of the second line deleted (the part of `P005-N005` that is established)

p. 1514 (PDF p. 13), Appendix B.1.1, first display and the sentence after it

Appendix B.1.1, top of p. 1514 (PDF p. 13), corrected brackets. The weighted normal equation gives all three bracketed vectors and identifies their first expression with the actual common-residual term (*).

Historical component credits

Proved AronowAudit.P005.corrected_residual_brackets Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Covers T18(a)-(b). The normal equation is the explicit premise of the generic residual-vector claim, and R041-04 derives it for the actual paper residuals; it is not an unproved extra premise for the combined target.
Proved AronowAudit.P005.corrected_residual_term_identification Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Covers T18(c). Under the stated normalization the paper minimizer has alpha*=bar yC, tau*=ATE, beta*=N, so commonResidual is the actual residual, not an arbitrary substitute.
Proved AronowAudit.P005.popMean_sum Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.
Proved AronowAudit.P005.populationSlope_centered Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.
Proved AronowAudit.P005.populationSlope_projection Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.
Proved AronowAudit.P005.populationSlope_common_residual Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.
Proved AronowAudit.P005.common_residual_normal_equation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.
Proved AronowAudit.P005.complement_x_mean_identity Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.
Proved AronowAudit.P005.observed_residual_slope Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Supporting residual algebra already incorporated in the two selected aggregate rows; not separately invoked.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:5ce863b566a115420d917b3676915466ea2073e8fe574a04dbfe3c943f04fef9

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1514 (PDF p. 13)] “Note $\bar x_T - \bar x_C = \frac{1}{p_C}\bar x_T$. We bound the stochastic order of $(*)$: $$\frac{1}{p_C}\bar x_T'\Big(\frac1n\sum_{i\in[T]}x_i\big(e_i(1) - e_i(0)\big) + \frac1n\sum_{i=1}^nx_ie_i(0)\Big)$$ $$= \frac{1}{p_C}\bar x_T'\Big(\frac1n\sum_{i\in[T]}\Big(x_i\big(e_i(1) - e_i(0)\big) + \frac{1}{p_T}\frac1n\sum_{i=1}^nx_ie_i(0)\Big)\Big)\big)$$ $$= \frac{1}{p_C}\bar x_T'\Big(\frac1n\sum_{i\in[T]}\Big(x_i\big(e_i(1) - e_i(0)\big) - \frac1n\sum_{i=1}^nx_i\big(e_i(1) - e_i(0)\big)\Big)\Big),$$ where, for the second equality, we use the fact that $p_T\sum_{i=1}^nx_ie_i(1) + p_C\sum_{i=1}^nx_ie_i(0) = 0$ implies $\sum_{i=1}^nx_i(e_i(1) - e_i(0)) = -\frac{1}{p_T}\sum_{i=1}^nx_ie_i(0)$.” (The second line closes one parenthesis more than it opens.) [Sol's smallest repair (validated.json, P005-N005), verbatim] “Delete the final unmatched small closing parenthesis on the second line; retain all other brackets and factors.” [Corrected display] $$\frac{1}{p_C}\bar x_T'\Big(\frac1n\sum_{i\in[T]}x_i\big(e_i(1) - e_i(0)\big) + \frac1n\sum_{i=1}^nx_ie_i(0)\Big) = \frac{1}{p_C}\bar x_T'\Big(\frac1n\sum_{i\in[T]}\Big(x_i\big(e_i(1) - e_i(0)\big) + \frac{1}{p_T}\frac1n\sum_{k=1}^nx_ke_k(0)\Big)\Big) = \frac{1}{p_C}\bar x_T'\Big(\frac1n\sum_{i\in[T]}\Big(x_i\big(e_i(1) - e_i(0)\big) - \frac1n\sum_{k=1}^nx_k\big(e_k(1) - e_k(0)\big)\Big)\Big)$$ (the inner sums are written with the dummy index $k$ here; the paper reuses $i$). [Mathematical content, with every hypothesis] Fix $n$ and an assignment with $1 \le n_T \le n-1$, $p_T = n_T/n$, $p_C = n_C/n$; let $x_i \in \mathbb{R}^d$ and let $e_i(1), e_i(0)$ be real numbers with $p_T\sum_{i=1}^nx_ie_i(1) + p_C\sum_{i=1}^nx_ie_i(0) = 0$. Write $\delta_i = e_i(1) - e_i(0)$, $A = \frac1n\sum_{i=1}^nx_i\delta_i$, $B = \frac1n\sum_{i=1}^nx_ie_i(0)$ and $U = \frac1n\sum_{i\in[T]}x_i\delta_i$. Then (a) $B = -p_TA$, that is, $\sum_{i=1}^nx_i\delta_i = -\frac{1}{p_T}\sum_{i=1}^nx_ie_i(0)$; (b) the three bracketed vectors are equal: $U + B = \frac1n\sum_{i\in[T]}\big(x_i\delta_i + \frac{1}{p_T}B\big) = \frac1n\sum_{i\in[T]}\big(x_i\delta_i - A\big)$, because a constant vector summed over $[T]$ and divided by $n$ is multiplied by $n_T/n = p_T$; (c) [the first line is the term $(*)$] if moreover $\bar x = 0$ and $e_i(1), e_i(0)$ are the residuals of p. 1513, then $\bar x_T - \bar x_C = \frac{1}{p_C}\bar x_T$, the normal equation $p_T\sum_ix_ie_i(1) + p_C\sum_ix_ie_i(0) = 0$ holds, and $(*) = (\bar x_T - \bar x_C)'\frac1n\sum_{i=1}^nx_ie_i(D_i) = \frac{1}{p_C}\bar x_T'(U + B)$. No factor $p_T$ is missing from the third line: in both the second and the third line the population average stands inside the parentheses governed by the treated-arm sum.

: SAP.P005b.T18Statement

Theorem: SAP.P005b.T18

Statement hash: e0da6d789c90e26f5f660517a6bc352e1a3e77644194b4ef78ff165a49ea33bd

Statement definition: SAP.P005b.T18Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:36:59Z; binding v2

Declaration review: faithful; 2026-11-04T16:54:44Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.common_residual_normal_equation: observed used

AronowAudit.P005.complement_x_mean_identity: observed used

AronowAudit.P005.observed_residual_slope: observed used

AronowAudit.P005.popMean_sum: observed used

AronowAudit.P005.populationSlope_centered: observed used

AronowAudit.P005.populationSlope_common_residual: observed used

AronowAudit.P005.populationSlope_projection: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T19

Lemma A.1, as printed

p. 1511 (PDF p. 10), Appendix A.2; definitions on p. 1510 (PDF p. 9)

The printed claim is false: the universal conjunction of the printed expectation identity and the uncentered covariance variance bound fails under complete randomization. Lemma A.1, pp. 1510–1511 (PDF pp. 9–10).

Historical component credits

Counterexample proved AronowAudit.P005.ConfidenceFollowup02.printed_covariance_lemma_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This negates the conjunction by its variance clause; it neither negates the valid expectation clause separately nor proves the corrected centered covariance bound.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:5225f4deba6ccbd26bfdca8b7ebf173b84badddf2621b063f3aece7c6ced6de2

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1510 (PDF p. 9), Appendix A.2] “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$. The following lemma characterizes the mean and variance of these covariance estimators.” [Printed, p. 1511 (PDF p. 10)] “LEMMA A.1: Let $t \in \{T, C\}$ and $n \ge 4$. Then, $$\mathbf{E}\big[\widehat{\mathrm{Cov}}_t(a,b)\big] = \frac{(n_t-1)n}{n_t(n-1)}(\overline{ab} - \bar a\bar b),$$ $$\mathrm{Var}\big(\widehat{\mathrm{Cov}}_t(a,b)\big) \le \frac{2(n-n_t)}{n_t(n-1)}\frac1n\sum_{i=1}^n(a_ib_i-\overline{ab})^2 + \frac{2(n-n_t)n}{n_t^3(n-1)}\frac1n\sum_{i=1}^n a_i^2 \times \frac1n\sum_{i=1}^n b_i^2.$$” [Printed, proof of Lemma A.1, p. 1511] “To upper-bound the variance, we first note that the covariance estimator is location invariant and we can assume, without loss of generality, $\bar a = 0$ and $\bar b = 0$.$^{9}$ We then have $$\mathrm{Var}(\overline{ab}_T - \bar a_T\bar b_T) \le 2\,\mathrm{Var}(\overline{ab}_T) + 2\,\mathrm{Var}(\bar a_T\bar b_T) = 2\,\mathrm{Var}(\overline{ab}_T) + \frac{2}{n_T^4}\mathrm{Var}\Big(\sum_{i,j\in[T]}a_ib_j\Big)$$ $$\le \frac{2(n-n_T)}{n_T(n-1)}\frac1n\sum_{i=1}^n(a_ib_i-\overline{ab})^2 + \frac{2(n-n_T)n}{n_T^3(n-1)}\frac1n\sum_{i=1}^n a_i^2\,\frac1n\sum_{i=1}^n b_i^2,$$ where the last inequality follows from the variance calculation of simple random samplings (first term) and Lemma A.5 in Lei and Ding (2021) (second term).” Footnote 9: “This is required by the conditions of Lemma A.5 in Lei and Ding (2021).” [The statement as printed, with its hypotheses made explicit] For all $n \ge 4$, all $n_T$ with $1 \le n_T \le n-1$, each $t \in \{T, C\}$ and all fixed vectors $a, b \in \mathbb{R}^n$ (not required to be centred), under complete randomization: (i) $\mathbf{E}[\widehat{\mathrm{Cov}}_t(a,b)] = \frac{(n_t-1)n}{n_t(n-1)}(\overline{ab} - \bar a\bar b)$; (ii) $\mathrm{Var}(\widehat{\mathrm{Cov}}_t(a,b)) \le \frac{2(n-n_t)}{n_t(n-1)}\frac1n\sum_{i=1}^n(a_ib_i - \overline{ab})^2 + \frac{2(n-n_t)n}{n_t^3(n-1)}\frac1n\sum_{i=1}^na_i^2\times\frac1n\sum_{i=1}^nb_i^2$. Part (i) is true. Part (ii) is false as printed (P005-T1); it is true when $\bar a = \bar b = 0$ and, in general, with $a$ and $b$ centred on the right-hand side (P005-T2).

: SAP.P005b.T19Statement

Theorem: SAP.P005b.T19

Statement hash: 00bb619d0f018f7b40b74f6d8a74d9b0399c336d674e9b9188950ed20f655467

Statement definition: SAP.P005b.T19Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:36:59Z; binding v2

Declaration review: faithful; 2026-11-04T16:54:44Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T20

Theorem 4.1, as printed

pp. 1509–1510 (PDF pp. 8–9), Section 4; proof B.3 on p. 1517 (PDF p. 16)

The printed claim is false: both the universal third-component sign identity and the stochastic expansion under the literal logarithm condition fail; the stated exact slope decompositions, bias expression, and first-component moments hold. Theorem 4.1, pp. 1509–1510 (PDF pp. 8–9).

Historical component credits

Proved AronowAudit.P005.Wave1.interacted_exact_decomposition_completion Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Only the valid exact finite-sample portion of T20 is certified here. The n>=2 formulation and the admitted d=0 case exclude no required instance; the nonvacuity example has positive dimension.
Counterexample proved AronowAudit.P005.ConfidenceFollowup02.printed_interacted_third_bias_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The finite sign-identity component is refuted with the printed negative definition of armNu3. Other valid components of T20 receive no refutation credit from this row.
Counterexample proved AronowAudit.P005.Wave1.printed_interacted_expansion_refuted Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Metadata inconsistency: claims.json marks T20 asymptotic=false although its statement includes the expansion and this row is explicitly credited to refute it. The record target note calls T20 finite-sample-only and assigns the expansion to T7/T8, conflicting with the packet statement and row attribution. This scope issue does not change the fidelity of the displayed negation. No claim-stage verdict changed.

Formal result: refuted

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:5e4541e5319b1b2a14e738482a3aa167ba6eef808b369b96e424045926d2bd5d

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1509 (PDF p. 8)] “THEOREM 4.1: Under Assumptions 1–3, the OLS coefficient vectors for the covariates of the interacted ATE estimator can be written as $\widehat\beta_{I,T} = \beta_{I,T} + \nu_{1T} + \nu_{2T} + \nu_{3T}$, and $\widehat\beta_{I,C} = \beta_{I,C} + \nu_{1C} + \nu_{2C} + \nu_{3C}$, with $$\nu_{1T} = \overline{y^*(1)x}_T - \overline{y^*(1)x}, \qquad \nu_{2T} = \big(\widehat L_T^{-1} - I_d^{-1}\big)\hat N_T, \qquad \nu_{3T} = -(\bar x_T\overline{y^*(1)}_T),$$ $$\nu_{1C} = \overline{y^*(0)x}_C - \overline{y^*(0)x}, \qquad \nu_{2C} = \big(\widehat L_C^{-1} - I_d^{-1}\big)\hat N_C, \qquad \nu_{3C} = -(\bar x_C\overline{y^*(0)}_C).$$ The bias of the $\widehat{\mathrm{ATE}}_I$ estimator is $E[\widehat{\mathrm{ATE}}_I - \mathrm{ATE}] = E[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})] - E[\bar x_T'(\nu_{1T}+\nu_{2T}+\nu_{3T})]$. We have $$E\big[\bar x_C'\nu_{1C}\big] - E\big[\bar x_T'\nu_{1T}\big] = \frac{n_T}{(n-1)n_C}(\overline{hy(0)} - \bar h\times\overline{y(0)}) - \frac{n_C}{(n-1)n_T}(\overline{hy(1)} - \bar h\times\overline{y(1)}),$$ $$E\big[\bar x_C'\nu_{3C}\big] - E\big[\bar x_T'\nu_{3T}\big] = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}(\overline{hy(0)} - \bar h\times\overline{y(0)}) - \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}(\overline{hy(1)} - \bar h\times\overline{y(1)}).$$” [Printed, continuation of Theorem 4.1 on p. 1510 (PDF p. 9)] “If $\kappa\log d = o(1)$,$^{8}$ $$\widehat{\mathrm{ATE}}_I = \mathrm{ATE} + \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta_{I,T}\big) - \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta_{I,C}\big) + O_p\Big(\sqrt{\frac{\kappa d}{n}}\Big).$$” Footnote 8: “The upper bound on the stochastic order of the bias of the interacted case is characterized in equation (15) in Lei and Ding (2021).” [Printed proof, Appendix B.3, p. 1517 (PDF p. 16), in full] “The decomposition of the bias is similar to the one in Theorem 3.1. We omit the details. The stochastic order of the bias is derived in Lei and Ding (2021).” [Status of the parts as printed] (1) The two decompositions of $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$, the expression of the bias and the first displayed identity (the $\nu_1$ components) hold (P005-T24). (2) The second displayed identity (the $\nu_3$ components) is false as printed whenever the component is nonzero: its right-hand side has both signs reversed (counterexample P005-T5; corrected statement P005-T6; finding P005-E003, Moderate). (3) The final expansion is false as printed for $d = 1$ with non-vanishing maximum leverage (counterexample P005-T7; corrected condition P005-T8; finding P005-E001, Minor).

: SAP.P005b.T20Statement

Theorem: SAP.P005b.T20

Statement hash: 03f92f70baf1697ffea13bb076319335f1907048075196081d7995acdf08c4ce

Statement definition: SAP.P005b.T20Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:35:28Z; binding v2

Declaration review: faithful; 2026-11-04T17:07:45Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T21

Theorem 3.2, first part: exact unbiasedness of $\widehat{\mathrm{Bias}}$ and of $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}}$ (noninteracted case)

pp. 1508–1509 (PDF pp. 7–8), Section 3, equations (7)–(8); proof in Appendix B.2, pp. 1515–1516 (PDF pp. 14–15)

Theorem 3.2, pp. 1508–1509 (PDF pp. 7–8); proof in Appendix B.2. The printed noninteracted bias estimator is exactly unbiased for the bias, and its correction is exactly unbiased for the ATE. The closed-form coefficients are the unique OLS minimizer. The inverse-free formula, bias decomposition, inverse-component identity and both population component expectations hold for all nonempty arms. Expressions (11) and (12), their expectations and their collection into (7) require only their presupposed nonzero arm-size denominators.

Historical component credits

Proved AronowAudit.P005.Wave1.noninteracted_exact_unbiasedness_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The target explicitly incorporates n>=3, nonempty arms, fixed-n Assumption 2 and footnote-4 all-assignment invertibility; none is an added restriction. The separate generalized-inverse extension in the footnote is outside the nonsingular component credited to this row.
Proved AronowAudit.P005.mean_unbiased Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The exact-unbiasedness row already incorporates the sample-mean identity; not separately invoked.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:afd7a73facb0408e4462811202449af1aceb00bfdc3bbaac5fc696cfd5ee11da

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1508 (PDF p. 7)] “THEOREM 3.2: Under Assumption 2, an unbiased estimator for the bias of the noninteracted ATE estimator is $$\widehat{\mathrm{Bias}} = \frac{1}{n-2}(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C) - \frac{1}{n-2}(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T) \qquad (7)$$ $$\qquad + (\bar x_C - \bar x_T)'\big(\widehat L^{-1} - I_d^{-1}\big)\widehat N. \qquad (8)$$” [Printed, continuation on p. 1509 (PDF p. 8)] “The estimator $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \widehat{\mathrm{ATE}} - \widehat{\mathrm{Bias}}$ is unbiased for estimating the ATE.” (The theorem continues with the asymptotic expansion, which is P005-T26.) [The statement with its hypotheses made explicit] Let $n \ge 3$ and $1 \le n_T \le n-1$, under complete randomization; let the covariates satisfy $\bar x = 0$ and $\overline{xx'} = I_d$ (Assumption 2 at this $n$); let $\widehat L$ be invertible under every assignment (footnote 4); the potential outcomes are arbitrary fixed numbers. With $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$ and $\widehat\beta = \widehat L^{-1}\widehat N$ as in (3) (by Lin (2013) and the Frisch–Waugh–Lovell theorem, as the paper cites them, $\widehat{\mathrm{ATE}}$ is the OLS coefficient $\widehat\tau$ of regression (1)), and $\widehat{\mathrm{Bias}}$ as in (7)–(8): (i) $\mathbf{E}[\widehat{\mathrm{Bias}}] = \mathbf{E}[\widehat{\mathrm{ATE}}] - \mathrm{ATE}$; (ii) $\mathbf{E}[\widehat{\mathrm{ATE}}_{\mathrm{Debiased}}] = \mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, where $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \widehat{\mathrm{ATE}} - \widehat{\mathrm{Bias}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T - \bar x_C)'\widehat N + \frac{1}{n-2}\widehat{\mathrm{Cov}}_T(h, y(1)) - \frac{1}{n-2}\widehat{\mathrm{Cov}}_C(h, y(0))$. [The steps of the printed proof, B.2, each exact] (a) $\mathbf{E}[\widehat{\mathrm{ATE}}] - \mathrm{ATE} = \mathbf{E}[(\bar x_C - \bar x_T)'(\nu_1 + \nu_2 + \nu_3)]$ (Theorem 3.1); (b) $(\bar x_C - \bar x_T)'\nu_2$ is line (8) itself; (c) the expectation of (11) is the right-hand side of (5), by the expectation part of Lemma A.1; (d) the expectation of (12) is the right-hand side of (6), likewise, with $\frac{p_T}{p_C}N_{\mathrm{TTT}}$ and $\frac{p_T}{p_C}N_{\mathrm{TTC}}$ as coefficients of $S_1$ and $S_0$; (e) (11) $+$ (12) $=$ line (7) (P005-T15).

: SAP.P005b.T21WholeStatement

Theorem: SAP.P005b.T21

Statement hash: fe3fae5eb16ef5238b4f476524b3508624f70fb2697db4908b844abb46d3b48c

Statement definition: SAP.P005b.T21WholeStatement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:35:29Z; binding v2

Declaration review: faithful; 2026-11-04T17:07:45Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.mean_unbiased: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T22

Theorem 4.2, first part: exact unbiasedness of $\widehat{\mathrm{Bias}}_I$ and of $\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}}$ (interacted case)

p. 1510 (PDF p. 9), Section 4; proof in Appendix B.4, p. 1518 (PDF p. 17)

Theorem 4.2, p. 1510 (PDF p. 9). The interacted bias correction and corrected ATE are exactly unbiased, with the inverse-free identity and both armwise expectations, including the stated weaker centered-covariate hypotheses.

Historical component credits

Proved AronowAudit.P005.Wave1.interacted_exact_unbiasedness_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Both armwise conclusions are exported, in addition to the two estimator-unbiasedness conclusions, under the target common normalized nonsingular regime. No asymptotic sequel is credited here.
Proved AronowAudit.P005.interacted_debiased_cancellation Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; This is the cancellation component alone, not a positive unbiasedness theorem. The extended algebraic domain loses no target cases.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:520f9aa2afd09ae071a9fe0a81af2f0109bc34e7344d3cbba114be09c5b625f6

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1510 (PDF p. 9)] “THEOREM 4.2: Under Assumption 2, an unbiased estimator for the bias of the interacted ATE estimator is $$\widehat{\mathrm{Bias}}_I = \frac{n_T}{(n-2)n_C}(\overline{hy(0)}_C - \bar h_C\overline{y(0)}_C) + \bar x_C'\big(\widehat L_C^{-1} - I_d^{-1}\big)\hat N_C - \frac{n_C}{(n-2)n_T}(\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T) - \bar x_T'\big(\widehat L_T^{-1} - I_d^{-1}\big)\hat N_T.$$ The estimator $\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}} = \widehat{\mathrm{ATE}}_I - \widehat{\mathrm{Bias}}_I$ is unbiased for estimating the ATE.” (The theorem continues with the asymptotic expansion, which is P005-T27.) [The statement with its hypotheses made explicit] Let $n \ge 3$ and $1 \le n_T \le n-1$, under complete randomization; let $\bar x = 0$ and $\overline{xx'} = I_d$ (Assumption 2 at this $n$); let $\widehat L_T$ and $\widehat L_C$ be invertible under every assignment (footnote 4); the potential outcomes are arbitrary fixed numbers. With $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$, $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$, $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$ as in Section 4 (by Lin (2013), as the paper cites it, $\widehat{\mathrm{ATE}}_I$ is the OLS coefficient $\widehat\tau_I$ of the interacted regression (2)), and $\widehat{\mathrm{Bias}}_I$ as displayed: (i) $\mathbf{E}[\widehat{\mathrm{Bias}}_I] = \mathbf{E}[\widehat{\mathrm{ATE}}_I] - \mathrm{ATE}$; (ii) $\mathbf{E}[\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}}] = \mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, where $\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}} = \widehat{\mathrm{ATE}}_I - \widehat{\mathrm{Bias}}_I = \Big(\overline{y(1)}_T - \bar x_T'\widehat N_T + \frac{n_C}{(n-2)n_T}\widehat{\mathrm{Cov}}_T(h, y(1))\Big) - \Big(\overline{y(0)}_C - \bar x_C'\widehat N_C + \frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0))\Big)$; (iii) arm by arm: $\mathbf{E}\big[\overline{y(1)}_T - \bar x_T'\widehat N_T + \frac{n_C}{(n-2)n_T}\widehat{\mathrm{Cov}}_T(h, y(1))\big] = \overline{y(1)}$ and $\mathbf{E}\big[\overline{y(0)}_C - \bar x_C'\widehat N_C + \frac{n_T}{(n-2)n_C}\widehat{\mathrm{Cov}}_C(h, y(0))\big] = \overline{y(0)}$ (these two need only $\bar x = 0$, $n \ge 3$ and $n_T \ge 2$, respectively $n_C \ge 2$).

: SAP.P005b.T22Statement

Theorem: SAP.P005b.T22

Statement hash: 68271e8dae26cf37204856e5c24e3134dcf909c80b6a8eccf3462d4fee32243d

Statement definition: SAP.P005b.T22Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:35:29Z; binding v2

Declaration review: faithful; 2026-11-04T17:07:45Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T23

Theorem 3.1, exact part: the decomposition of $\widehat\beta$ and the bias components (5) and (6); with the two exact consequences stated in Remark 1

p. 1507 (PDF p. 6), Section 3; proof in Appendix B.1, pp. 1512–1513 (PDF pp. 11–12), and B.2, p. 1515 (PDF p. 14)

Theorem 3.1 and Remark 1, p. 1507 (PDF p. 6). The exact noninteracted slope and bias decompositions give (5), (6), the covariance interpretation, and the two stated vanishing consequences of Remark 1.

Historical component credits

Proved AronowAudit.P005.Wave1.noninteracted_exact_decomposition_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The exact fixed-n target and both vanishing cases are exported; the separate stochastic expansion is covered by R040-20 rather than asserted here.
Proved AronowAudit.P005.noninteracted_slope_decomposition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected decomposition row already contains the slope decomposition.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:ad200e42f434f249ba1d79ca9f62f190a05913e854c781e77a5845bd9e9f5dee

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1507 (PDF p. 6)] “THEOREM 3.1: Under Assumptions 1–3, the OLS coefficient estimators for the covariates, $\widehat\beta$, can be decomposed as $\widehat\beta = \beta^* + \nu_1 + \nu_2 + \nu_3$ with $$\nu_1 = p_T(\overline{y^*(1)x}_T - \overline{y^*(1)x}) + p_C(\overline{y^*(0)x}_C - \overline{y^*(0)x}),$$ $$\nu_2 = \big(\widehat L^{-1} - I_d^{-1}\big)\widehat N,$$ $$\nu_3 = -(p_T\overline{y^*(1)}_T\bar x_T + p_C\overline{y^*(0)}_C\bar x_C).$$ The bias of the $\widehat{\mathrm{ATE}}$ estimator is $\mathbf{E}[(\bar x_C - \bar x_T)'(\nu_1+\nu_2+\nu_3)]$, where $$\mathbf{E}\big[(\bar x_C - \bar x_T)'\nu_1\big] = \frac{1}{n-1}(\overline{hy(0)} - \bar h\times\overline{y(0)}) - \frac{1}{n-1}(\overline{hy(1)} - \bar h\times\overline{y(1)}) \qquad (5)$$ and $$\mathbf{E}\big[(\bar x_C - \bar x_T)'\nu_3\big] = \frac{n_C-n_T}{(n-1)(n-2)n_T}(\overline{hy(1)} - \bar h\times\overline{y(1)}) - \frac{n_T-n_C}{(n-1)(n-2)n_C}(\overline{hy(0)} - \bar h\times\overline{y(0)}). \qquad (6)$$” (The theorem continues with the stochastic expansion, which is P005-T25.) [Printed, Remark 1, p. 1507] “REMARK 1: The first bias term (5) is the main component of the bias. It is the scaled covariance between the leverage $h_i$ and individual effects $y_i(1) - y_i(0)$. Note that the formula suggests that the first bias term is 0 when the treatment effect is additive, for example, if there is no treatment effect on all subjects. However, a large bias may result from the presence of highly heterogeneous effects. The third term $\mathbf{E}[(\bar x_C - \bar x_T)'\nu_3]$ is 0 when $n_T = n_C = \frac12n$. See also Freedman (2008b) and Lin (2013).” [The statement with its hypotheses made explicit] Let $n \ge 3$ and $1 \le n_T \le n-1$, under complete randomization; let $\bar x = 0$ and $\overline{xx'} = I_d$ (Assumption 2 at this $n$); let $\widehat L$ be invertible under every assignment (footnote 4). Then (i) for every assignment, $\widehat\beta = \widehat L^{-1}\widehat N = \beta^* + \nu_1 + \nu_2 + \nu_3$, with $\beta^* = N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$ and $\nu_1, \nu_2, \nu_3$ as displayed; (ii) $\mathbf{E}[\widehat{\mathrm{ATE}}] - \mathrm{ATE} = \mathbf{E}[(\bar x_C - \bar x_T)'(\nu_1 + \nu_2 + \nu_3)]$; (iii) equation (5): $\mathbf{E}[(\bar x_C - \bar x_T)'\nu_1] = \frac{1}{n-1}(S_0 - S_1) = -\frac{1}{n-1}\cdot\frac1n\sum_{i=1}^n(h_i - \bar h)\big(y_i(1) - y_i(0)\big)$, where $S_t = \overline{hy(t)} - \bar h\times\overline{y(t)}$; (iv) equation (6): $\mathbf{E}[(\bar x_C - \bar x_T)'\nu_3] = \frac{n_C-n_T}{(n-1)(n-2)n_T}S_1 - \frac{n_T-n_C}{(n-1)(n-2)n_C}S_0$; (v) [Remark 1] if $y_i(1) - y_i(0)$ is the same for all $i$, the right-hand side of (5) is 0; if $n_T = n_C$, the right-hand side of (6) is 0. The theorem is stated “Under Assumptions 1–3”; (i)–(v) are exact for each $n$ and use only Assumption 2, the invertibility of $\widehat L$ (for (i), (ii)) and $n \ge 3$ (for (6)).

: SAP.P005b.T23Statement

Theorem: SAP.P005b.T23

Statement hash: 2feaabc5525de02ca73d5aa69908e652a5bf06609335b3d9c21187006d5e7b08

Statement definition: SAP.P005b.T23Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:35:29Z; binding v2

Declaration review: faithful; 2026-11-04T17:07:45Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.noninteracted_slope_decomposition: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T24

Theorem 4.1, exact part that stands as printed: the decompositions of $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ , the expression of the bias, and the first bias component

p. 1509 (PDF p. 8), Section 4; proof B.3 on p. 1517 (PDF p. 16) omits the details

Theorem 4.1, p. 1509 (PDF p. 8), its valid exact part. Both interacted slopes decompose as printed, and the bias expression and separate first-component expectations give their stated difference.

Historical component credits

Proved AronowAudit.P005.Wave1.interacted_exact_decomposition_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; Covers all of T24, which expressly excludes the separately targeted third-component formula and stochastic expansion. The finite target itself supplies normalization and all-assignment invertibility, so these are not added restrictions.
Proved AronowAudit.P005.armN_decomposition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected decomposition row already contains the two arm slope and numerator identities.
Proved AronowAudit.P005.arm_slope_decomposition Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The selected decomposition row already contains the two arm slope and numerator identities.

Formal result: whole

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:43328987c47c5d6f08d8faeb6c298bbc8436cede57b4e1bc5e7e8590d80d90a9

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1509 (PDF p. 8)] “THEOREM 4.1: Under Assumptions 1–3, the OLS coefficient vectors for the covariates of the interacted ATE estimator can be written as $\widehat\beta_{I,T} = \beta_{I,T} + \nu_{1T} + \nu_{2T} + \nu_{3T}$, and $\widehat\beta_{I,C} = \beta_{I,C} + \nu_{1C} + \nu_{2C} + \nu_{3C}$, with $$\nu_{1T} = \overline{y^*(1)x}_T - \overline{y^*(1)x}, \qquad \nu_{2T} = \big(\widehat L_T^{-1} - I_d^{-1}\big)\hat N_T, \qquad \nu_{3T} = -(\bar x_T\overline{y^*(1)}_T),$$ $$\nu_{1C} = \overline{y^*(0)x}_C - \overline{y^*(0)x}, \qquad \nu_{2C} = \big(\widehat L_C^{-1} - I_d^{-1}\big)\hat N_C, \qquad \nu_{3C} = -(\bar x_C\overline{y^*(0)}_C).$$ The bias of the $\widehat{\mathrm{ATE}}_I$ estimator is $E[\widehat{\mathrm{ATE}}_I - \mathrm{ATE}] = E[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})] - E[\bar x_T'(\nu_{1T}+\nu_{2T}+\nu_{3T})]$. We have $$E\big[\bar x_C'\nu_{1C}\big] - E\big[\bar x_T'\nu_{1T}\big] = \frac{n_T}{(n-1)n_C}(\overline{hy(0)} - \bar h\times\overline{y(0)}) - \frac{n_C}{(n-1)n_T}(\overline{hy(1)} - \bar h\times\overline{y(1)}),$$” (The theorem continues with the display for the third component, which is false as printed and is P005-T5, P005-T6 and P005-T20, and with the stochastic expansion, which is P005-T7 and P005-T8.) [The statement with its hypotheses made explicit] Let $n \ge 2$ and $1 \le n_T \le n-1$, under complete randomization; let $\bar x = 0$ and $\overline{xx'} = I_d$ (Assumption 2 at this $n$); let $\widehat L_T$ and $\widehat L_C$ be invertible under every assignment (footnote 4). Then (i) for every assignment, $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T = \beta_{I,T} + \nu_{1T} + \nu_{2T} + \nu_{3T}$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C = \beta_{I,C} + \nu_{1C} + \nu_{2C} + \nu_{3C}$, with $\beta_{I,T} = \overline{y(1)x}$, $\beta_{I,C} = \overline{y(0)x}$ and the six vectors $\nu$ as displayed; (ii) $E[\widehat{\mathrm{ATE}}_I - \mathrm{ATE}] = E[\bar x_C'(\nu_{1C}+\nu_{2C}+\nu_{3C})] - E[\bar x_T'(\nu_{1T}+\nu_{2T}+\nu_{3T})]$; (iii) $E[\bar x_C'\nu_{1C}] - E[\bar x_T'\nu_{1T}] = \frac{n_T}{(n-1)n_C}\big(\overline{hy(0)} - \bar h\times\overline{y(0)}\big) - \frac{n_C}{(n-1)n_T}\big(\overline{hy(1)} - \bar h\times\overline{y(1)}\big)$; separately, $E[\bar x_T'\nu_{1T}] = \frac{n_C}{(n-1)n_T}S_1$ and $E[\bar x_C'\nu_{1C}] = \frac{n_T}{(n-1)n_C}S_0$. The theorem is stated “Under Assumptions 1–3”; (i)–(iii) are exact for each $n$ and use only $\bar x = 0$ (and, for (i), (ii), the invertibility of $\widehat L_T$, $\widehat L_C$).

: SAP.P005b.T24Statement

Theorem: SAP.P005b.T24

Statement hash: 0c56602ef7c3078d9cd57b34fbcaa5ddc31f5d675a7a4b37688192d34a1ce37c

Statement definition: SAP.P005b.T24Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:35:29Z; binding v2

Declaration review: faithful; 2026-11-04T17:07:45Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Manifest annotation reconciliation: Effective use follows the source-bound dependency observation; original annotations are retained in the crosswalk.

AronowAudit.P005.armN_decomposition: observed used

AronowAudit.P005.arm_slope_decomposition: observed used

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T25

Theorem 3.1, last part: the stochastic expansion of $\widehat{\mathrm{ATE}}$ (noninteracted case)

p. 1507 (PDF p. 6), Section 3; proof in Appendix B.1.1, pp. 1513–1515 (PDF pp. 12–14)

last part of Theorem 3.1, p. 1507 (PDF p. 6). The noninteracted regression remainder is O_p(sqrt(kappa*d/n)) when d/n tends to zero. The dimension convention 1 <= d_n <= n agrees with the canonical order-symbol definitions. No conclusion is omitted.

Historical component credits

Proved AronowAudit.P005.Wave1.noninteracted_stochastic_expansion_proof Historical support retained; baseline fidelity: ok; manifest: used; used in Solution; The target definitions explicitly require 1<=d_n, so the positive-dimension field is not an added assumption. Uniform K, eventual standardization and eventual regularity have the required all-sequence meaning; finitely many initial sizes do not affect O_p.

Formal result: whole

Scope: The asymptotic result assumes invertibility of the noninteracted sample matrix for every treatment assignment at all sufficiently large sample sizes.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:7e44f8b6f0c09ea443e0e50e7a6584574378c215b25dff1f9aa00640a7c5ee19

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1507 (PDF p. 6)] Theorem 3.1 is stated “Under Assumptions 1–3” (its exact part is P005-T23) and ends: “If $\frac dn = o(1)$, we have the stochastic expansion $$\widehat{\mathrm{ATE}} = \mathrm{ATE} + \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta^*\big) - \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta^*\big) + O_p\Big(\sqrt{\frac{\kappa d}{n}}\Big).$$” [The statement with the sequence made explicit] Consider a sequence of finite populations indexed by $n$ (dimension $d = d_n \le n$, covariates, potential outcomes and $n_T = n_{T,n}$ depending on $n$), under complete randomization at each $n$, satisfying Assumptions 1–3, with $\widehat L$ invertible (footnote 4). If $d_n/n \to 0$, then $$R_n := \widehat{\mathrm{ATE}} - \mathrm{ATE} - \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta^*\big) + \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta^*\big) = O_p\Big(\sqrt{\frac{\kappa_nd_n}{n}}\Big).$$

: SAP.P005b.T25Statement

Theorem: SAP.P005b.T25

Statement hash: e0c39e29049c4a56a77260c97db9a2987b3d48cba60e6575cd12f8a04c5b1c05

Statement definition: SAP.P005b.T25Statement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:33:57Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T26

Theorem 3.2, second part: the asymptotic linear expansion of $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}}$

p. 1509 (PDF p. 8), Section 3 (the theorem begins on p. 1508, PDF p. 7); proof in Appendix B.2, pp. 1516–1517 (PDF pp. 15–16)

Theorem 3.2, p. 1509 (PDF p. 8); Appendix B.2, pp. 1516–1517. The debiased noninteracted remainder equals the p. 1516 term (*) exactly and is O_p(sqrt(kappa/n)) under Assumptions 1–3, without leverage decay. If kappa tends to zero, this gives the printed common-slope expansion with o_p(n^(-1/2)) error.

Historical component credits

Proved AronowAudit.P005.Wave1.noninteracted_debiased_linear_expansion_proof Limited support; baseline fidelity: narrower; manifest: used; used in Solution; {'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/DebiasedLinearTailC.lean:130, the declaration assumes hκ (κ_n → 0) and concludes only that R_n = debiasedATE − ate − commonLeading is o_p(n^−1/2). The final sentence of the supplied P005-T26 claim additionally asserts R_n = O_p(√(κ_n/n)) without that assumption, and identifies R_n exactly with the printed term (*) on p. 1516. Neither the stronger rate without hκ nor the exact identity is included in this declaration’s conclusion. The printed Theorem 3.2 little-o formula is matched, but the supplied target and the row’s full-coverage description assert more.', 'suggested_fix': 'Extend the statement to export stochasticO m R (fun n => Real.sqrt (kappa (P n) / n)) under standingAssumptions and eventual noninteractedRegular without hκ, with the existing stochasticLittleO conclusion conditional on hκ. Also export the fixed-n identity R = −dot (xMean P S − xMean P (control S)) (Nhat P m S − N P m) + (sampleCov S (leverage P) P.yT − sampleCov (control S) (leverage P) P.yC) / (n−2) for n ≥ 3, standardized P, valid arm sizes, S in design n m and nonzero determinant of Lhat.', 'second_review': "P005/Wave1/DebiasedLinearTailC.lean:130 assumes κ_n → 0 and exports only stochasticLittleO at scale sqrt(1/n), with the residual and randomization interpretation matching the printed expansion through P005/P005_Formalization.lean:310 and P005/P005_Formalization.lean:325. P005-T26's claim text additionally asserts O_p(sqrt(κ_n/n)) without κ_n → 0 and an exact identification with (*); neither conclusion is exported. The little-o conclusion cannot recover that finer rate or cover nonvanishing-leverage sequences, so this is a major statement omission, even though the printed Theorem 3.2 formula itself is represented correctly.", 'severity': 'major'}

Formal result: whole

Scope: The asymptotic result assumes invertibility of the noninteracted sample matrix for every treatment assignment at all sufficiently large sample sizes.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:67ec9f0dfa9acc19273241cedede466b72a9ba4038ce231fcbe9fce372866b82

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1509 (PDF p. 8)] Theorem 3.2 (its first part is P005-T21) ends: “Under Assumptions 1–3 and if $\kappa = o(1)$, the estimator $\widehat{\mathrm{ATE}}_{\mathrm{Debiased}}$ has the asymptotic linear expansion $$\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} = \mathrm{ATE} + \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta^*\big) - \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta^*\big) + o_p\Big(\sqrt{\frac1n}\Big).$$” [The statement with the sequence made explicit] Consider a sequence of finite populations indexed by $n$ (dimension $d = d_n \le n$, covariates, potential outcomes and $n_T = n_{T,n}$ depending on $n$), under complete randomization at each $n$, satisfying Assumptions 1–3, with $\widehat L$ invertible (footnote 4). If $\kappa_n \to 0$, then $$\widehat{\mathrm{ATE}}_{\mathrm{Debiased}} - \mathrm{ATE} - \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta^*\big) + \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta^*\big) = o_p\big(n^{-1/2}\big).$$ The printed proof shows more: without the condition $\kappa = o(1)$, the left-hand side is the term $(*)$ of p. 1516 and $(*) = O_p\big(\sqrt{\kappa/n}\big)$.

: SAP.P005b.T26WholeStatement

Theorem: SAP.P005b.T26

Statement hash: 8b948c460094e10cd7e8662689812227bb47eebf750f2a066aa07abe3d0e6fa2

Statement definition: SAP.P005b.T26WholeStatement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:33:57Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

P005-T27

Theorem 4.2, second part: the asymptotic linear expansion of $\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}}$

p. 1510 (PDF p. 9), Section 4; proof in Appendix B.4, pp. 1518–1519 (PDF pp. 17–18)

Theorem 4.2, p. 1510 (PDF p. 9); Appendix B.4, pp. 1518–1519. The debiased interacted remainder is O_p(sqrt(kappa/n)) under Assumptions 1–3, without leverage decay. If kappa tends to zero, the printed arm-specific linear expansion has o_p(n^(-1/2)) error.

Historical component credits

Proved AronowAudit.P005.Wave1.interacted_debiased_linear_expansion_proof Limited support; baseline fidelity: narrower; manifest: used; used in Solution; {'source': 'adjudication+second review', 'adjudicator': 'At P005/Wave1/DebiasedLinearTailC.lean:195, the declaration assumes hκ (κ_n → 0) and concludes only that R_n = interactedDebiasedATE − ate − armLeading is o_p(n^−1/2). The final sentence of the supplied P005-T27 claim also asserts R_n = O_p(√(κ_n/n)) without assuming κ_n → 0. That stronger rate and its wider sequence domain are absent from the declaration, although the printed Theorem 4.2 little-o formula is represented correctly.', 'suggested_fix': 'Extend the statement to export stochasticO m R (fun n => Real.sqrt (kappa (P n) / n)) under standingAssumptions and eventual interactedRegular without hκ, retaining the existing stochasticLittleO conclusion as conditional on hκ.', 'second_review': "P005/Wave1/DebiasedLinearTailC.lean:195 assumes κ_n → 0 and concludes only stochasticLittleO for interactedDebiasedATE − ate − armLeading at scale sqrt(1/n); armLeading uses the target's two population arm slopes (P005/P005_Formalization.lean:302 and P005/P005_Formalization.lean:313). P005-T27's claim text also requires O_p(sqrt(κ_n/n)) without the decay assumption. That finer rate on the wider sequence domain does not follow from the exported little-o statement, so the target needs a substantive additional statement rather than a description edit.", 'severity': 'major'}

Formal result: whole

Scope: The asymptotic result assumes invertibility of both arm sample matrices for every treatment assignment at all sufficiently large sample sizes.

Formal statement · Proof · Definitions

Assumptions and verification

Claim revision: catalog:88011ec1b109ee929518f754f0f34f6f8a2a152edc2046b72f99a6a1944a14cd

Standing assumptions: Setting, notation and standing assumptions, as printed. Every passage was read from the rendered page images (staged/pages/p-NN.png, and 220 dpi renderings of the same PDF pages for subscripts and primes). Pages are those of Econometrica, Vol. 92, No. 5 (September, 2024), 1503–1519, with the PDF page in parentheses (PDF p. 1 is the journal's cover sheet, so printed page = PDF page + 1501). Appendix A (pp. 1510–1512) and Appendix B (pp. 1512–1519) are inside the article; there is no supplement. [Setting: Section 2, p. 1504 (PDF p. 3)] “We follow the setting of Freedman (2008a), Lin (2013), Abadie, Athey, Imbens, and Wooldridge (2020), and Lei and Ding (2021), which assume a Neyman model with covariates (Splawa-Neyman, Dabrowska, and Speed (1923)). There are $n$ subjects indexed by $i = 1, \ldots, n$. For each subject, we observe an outcome $Y_i$ and a column vector of covariates $x_i = (x_{i1}, x_{i2}, \ldots, x_{id})' \in \mathbb{R}^d$. Each subject has two potential outcomes $y_i(1)$ and $y_i(0)$. We observe $Y_i = y_i(1)$ if subject $i$ is assigned to the treatment arm $T$, and $Y_i = y_i(0)$ if subject $i$ is assigned to the control arm $C$. Let $D_i$ be a binary variable, where $D_i = 1$ indicates that subject $i$ is assigned to the treatment arm.” [Design: p. 1505 (PDF p. 4)] “The experiment is assumed to be completely randomized, where $n_T$ out of $n$ subjects are randomly assigned to the treatment arm $T$ and the remaining $n_C = n - n_T$ subjects are assigned to the control arm $C$. Random assignment is the sole source of randomness in our statistical analysis. The potential outcomes and covariates are considered fixed, and the bias of an estimator is assessed relative to the randomization distribution of the estimator. We do not assume the existence of a superpopulation: the $n$ subjects are the population of interest.” [Notation: p. 1505 (PDF p. 4)] “We define $[T]$ as the set of subjects chosen for the treatment arm, given by $\{i \mid D_i = 1\}$, and similarly $[C]$ as the set of subjects chosen for the control arm, given by $\{i \mid D_i = 0\}$. For a possibly matrix-valued variable $a_i$, we use the notation $\bar a = \frac1n\sum_{i=1}^n a_i$ to represent the population average, $\bar a_T = \frac{1}{n_T}\sum_{i\in[T]}a_i$ to denote the treatment group average, and $\bar a_C = \frac{1}{n_C}\sum_{i\in[C]}a_i$ to denote the control group average. The average treatment effect (ATE) can be expressed in this notation as $\mathrm{ATE} = \overline{y(1)} - \overline{y(0)}$, and the difference-in-means estimator is given by $\overline{Y}_T - \overline{Y}_C$. Similarly, we can write $\frac1n\sum_{i=1}^n x_ix_i' = \overline{xx'}$ for $x_i \in \mathbb{R}^d$ and $\frac1n\sum_{i=1}^n y_i(1)x_i = \overline{y(1)x}$ for $y_i(1) \in \mathbb{R}$ and $x_i \in \mathbb{R}^d$.” [Assumptions: p. 1505 (PDF p. 4)] “We make the following assumptions throughout the paper, which are standard in the literature.” ASSUMPTION 1: “For all $n$, there exists a finite constant $K$ such that $\frac1n\sum_{i=1}^n y_i(1)^2 \le K$, $\frac1n\sum_{i=1}^n y_i(0)^2 \le K$, $\frac1n\sum_{i=1}^n x_{ik}^2 \le K$, for all $k = 1, \ldots, d$ and $d \le n$.” ASSUMPTION 2: “For all $n$ large enough, (1) $\bar x = 0$, and (2) $\overline{xx'} = I_d$, the $d \times d$ identity matrix.” ASSUMPTION 3: “Let $p_{T,n} = \frac{n_T}{n}$ and $p_{C,n} = \frac{n-n_T}{n}$ denote the inclusion probabilities into the treatment arm $T$ and control arm $C$, respectively. There exist positive constants $p_{\min}$ and $p_{\max}$ such that $0 < p_{\min} < p_{\max} < 1$ and $p_{\min} < p_{T,n} < p_{\max}$ for all $n$.” Then: “We shall hereafter omit the $n$ subscript in $p_{T,n}$ and $p_{C,n}$, and write $p_T$ and $p_C$ unless otherwise noted.” [The two estimators: pp. 1505–1506 (PDF pp. 4–5)] “The first estimator results from a OLS regression: $Y_i \sim \alpha + \tau D_i + x_i'\beta$, (1) where one regresses observed outcome $Y_i$ on the treatment indicator $D_i$ and covariates $x_i$. We denote the OLS estimators for this case as $(\widehat\alpha, \widehat\tau, \widehat\beta)$. The estimator $\widehat\tau$ is hereafter referred to as the noninteracted ATE estimator. The second estimator results from an interacted OLS regression where researchers expand the covariates by including terms interacting the treatment indicator and covariates: $Y_i \sim \alpha_I + \tau_ID_i + x_i'\beta_{I,C} + D_ix_i'\beta_{I,T}$. (2) We denote the OLS estimators for the interacted case as $(\widehat\alpha_I, \widehat\tau_I, \widehat\beta_{I,C}, \widehat\beta_{I,T})$. The estimator $\widehat\tau_I$ is hereafter referred to as the interacted ATE estimator.” [Section 3 objects: p. 1506 (PDF p. 5)] “As shown in Lin (2013), the noninteracted ATE estimator can be written as $\widehat{\mathrm{ATE}} = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta - \bar x_C'\widehat\beta)$, (3) where $\widehat\beta$ is the OLS coefficient estimators for the covariates.” “The coefficient estimators $\widehat\beta$ can be algebraically written as $\widehat\beta = \widehat L^{-1}\widehat N$, where $\widehat L = I_d - p_T\bar x_T\bar x_T' - p_C\bar x_C\bar x_C'$ and $\widehat N = p_T(\overline{y(1)x}_T - \overline{y(1)}_T\bar x_T) + p_C(\overline{y(0)x}_C - \overline{y(0)}_C\bar x_C)$ by an application of the Frisch–Waugh–Lovell theorem.” “The OLS coefficient estimators can be thought of as estimating the (finite) population coefficients $\beta^* = L^{-1}N$, where $L = I_d$ and $N = p_T\overline{y(1)x} + p_C\overline{y(0)x}$.” Footnote 4: “Note we shall hereafter assume for simplicity that all design matrices, $\widehat L$, are invertible. In the case of noninvertible design matrices, our debiased procedure will still work after choosing an arbitrary generalized inverse matrix and computing the ATE estimators accordingly.” “Let $y_i^*(1)$ and $y_i^*(0)$ be the centered potential outcomes, that is, $y_i^*(1) = y_i(1) - \overline{y(1)}$ and $y_i^*(0) = y_i(0) - \overline{y(0)}$. Denote the (rescaled) leverage of the $i$th subject as $h_i = \|x_i\|_2^2$. As in Lei and Ding (2021), we define the maximum leverage as $\kappa = \max_{i=1,\ldots,n}\frac{h_i}{n} = \max_{i=1,\ldots,n}\frac{\|x_i\|_2^2}{n}$. (4)” [Section 4 objects: p. 1509 (PDF p. 8)] “We write the regression coefficient estimators of the pretreatment covariates in the interacted case as $\widehat\beta_{I,T} = \widehat L_T^{-1}\widehat N_T$ and $\widehat\beta_{I,C} = \widehat L_C^{-1}\widehat N_C$, and their (finite) population counterparts as $\beta_{I,T} = I_d^{-1}N_T$ and $\beta_{I,C} = I_d^{-1}N_C$, with $\widehat L_T = \overline{xx'}_T - \bar x_T\bar x_T'$, $\widehat N_T = \overline{y(1)x}_T - \overline{y(1)}_T\bar x_T$, $\widehat L_C = \overline{xx'}_C - \bar x_C\bar x_C'$, $\widehat N_C = \overline{y(0)x}_C - \overline{y(0)}_C\bar x_C$, $N_T = \overline{y(1)x}$, and $N_C = \overline{y(0)x}$. As shown in Lin (2013), the OLS regression-adjusted ATE estimator can be written as $\widehat{\mathrm{ATE}}_I = \overline{y(1)}_T - \overline{y(0)}_C - (\bar x_T'\widehat\beta_{I,T} - \bar x_C'\widehat\beta_{I,C})$ for the interacted case, where $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the OLS coefficients on covariates $D_ix_i$ and $x_i$, repsectively.” [Appendix A objects: p. 1510 (PDF p. 9)] “We define the following three constants: $N_{\mathrm{TTT}} = \frac{(n-n_T)(n-2n_T)}{(n-1)(n-2)n_T^2} = \frac{n_C(n_C-n_T)}{(n-1)(n-2)n_T^2}$, $N_{\mathrm{CCC}} = \frac{(n-n_C)(n-2n_C)}{(n-1)(n-2)n_C^2} = \frac{n_T(n_T-n_C)}{(n-1)(n-2)n_C^2}$, $N_{\mathrm{TTC}} = -\frac{(n-2n_T)}{(n-1)(n-2)n_T} = \frac{n_T-n_C}{(n-1)(n-2)n_T}$.” “Let $a = (a_1, \ldots, a_n) \in \mathbb{R}^n$ and $b = (b_1, \ldots, b_n) \in \mathbb{R}^n$ be arbitrary $n$-vectors with fixed elements. We define the covariance estimator for the treated group as $\widehat{\mathrm{Cov}}_T(a,b) = \overline{ab}_T - \bar a_T\bar b_T$, and similarly for the control group as $\widehat{\mathrm{Cov}}_C(a,b) = \overline{ab}_C - \bar a_C\bar b_C$.” [Readings fixed by the printed text and used in every target] (a) $\mathbf{E}$ (also printed $E$), $\mathrm{Var}$ and all probability statements refer to the randomization distribution: $[T]$ is uniformly distributed over the $\binom{n}{n_T}$ subsets of $\{1,\ldots,n\}$ of size $n_T$, $[C]$ is its complement, both arms are non-empty ($1 \le n_T \le n-1$), and everything else is fixed. (b) Vectors are column vectors and a prime is a transpose; for $n$-vectors $a$, $b$ of scalars, $\overline{ab}$ is the average of the products $a_ib_i$, so $\overline{hy(1)}_T - \bar h_T\overline{y(1)}_T = \widehat{\mathrm{Cov}}_T(h, y(1))$, and $\overline{hy(1)} - \bar h\times\overline{y(1)}$ is the finite-population covariance of $h$ and $y(1)$ with divisor $n$. (c) $I_d^{-1}$ is printed for $I_d$ throughout, and the paper prints both $\hat N$ and $\widehat N$ (also $\hat N_T$, $\widehat N_T$) for the same objects. (d) In Assumption 1, $K$ is one constant, the same for every $n$: this is the reading the proofs use (the printed order of quantifiers is the clarity note P005-N002, not an error). (e) Assumption 2 is printed “for all $n$ large enough”; an exact finite-sample statement is read at a fixed $n$ at which (1) and (2) hold. Under Assumption 2, $\bar h = \frac1n\sum_i\|x_i\|_2^2 = \mathrm{tr}(\overline{xx'}) = d$, $h_i \le n$, and $\frac dn \le \kappa \le 1$ (the paper uses “$\kappa \in [\frac dn, 1]$” on p. 1516). (f) In Section 4, $\widehat\beta_{I,T}$ and $\widehat\beta_{I,C}$ are the within-arm slope vectors $\widehat L_T^{-1}\widehat N_T$ and $\widehat L_C^{-1}\widehat N_C$ defined there; their description as the coefficients on $D_ix_i$ and $x_i$ does not match parameterization (2), under which the coefficient on $D_ix_i$ is the difference of the two arm slopes (clarity note P005-E004, not an error). Every target uses the Section 4 definitions. (g) “Design matrices … are invertible” (footnote 4) is read as: $\widehat L$ (Section 3), respectively $\widehat L_T$ and $\widehat L_C$ (Section 4), is invertible under every assignment. (h) $O_p$, $o_p$, $O$, $o$ refer to a sequence of finite populations indexed by $n$, in which $d = d_n$, $n_T = n_{T,n}$, the covariates and the potential outcomes may all change with $n$; $\log$ is the natural logarithm.

Effective claim (source TeX): [Printed, p. 1510 (PDF p. 9)] Theorem 4.2 (its first part is P005-T22) ends: “Under Assumptions 1–3 and $\kappa = o(1)$, the estimator $\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}}$ has the asymptotic linear expansion $$\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}} = \overline{y(1)} - \overline{y(0)} + \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta_{I,T}\big) - \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta_{I,C}\big) + o_p\Big(\sqrt{\frac1n}\Big).$$” [The statement with the sequence made explicit] Consider a sequence of finite populations indexed by $n$ (dimension $d = d_n \le n$, covariates, potential outcomes and $n_T = n_{T,n}$ depending on $n$), under complete randomization at each $n$, satisfying Assumptions 1–3, with $\widehat L_T$ and $\widehat L_C$ invertible (footnote 4). If $\kappa_n \to 0$, then $$\widehat{\mathrm{ATE}}_{I,\mathrm{Debiased}} - \big(\overline{y(1)} - \overline{y(0)}\big) - \frac{1}{n_T}\sum_{i\in[T]}\big(y_i(1) - \overline{y(1)} - x_i'\beta_{I,T}\big) + \frac{1}{n_C}\sum_{i\in[C]}\big(y_i(0) - \overline{y(0)} - x_i'\beta_{I,C}\big) = o_p\big(n^{-1/2}\big).$$ The proof shows more: without the condition $\kappa = o(1)$, the left-hand side is $O_p\big(\sqrt{\kappa/n}\big)$.

: SAP.P005b.T27WholeStatement

Theorem: SAP.P005b.T27

Statement hash: 7efefb0865546b975d54b0106457c6dad6d5d82d68bd43f7c3f97e3a7337a9fd

Statement definition: SAP.P005b.T27WholeStatement

Certificate. 20261104T161519Z

Declaration review: faithful; 2026-11-04T16:33:57Z; binding v2

Evidence scope: Certification checks formal proof objects. Statement fidelity is a model judgment. Author review is separate.

Proof dependencies: Independently extracted

Source and evidence metadata · sap-2026-11-08-assured-release

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