Corrections to: A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage

The problem

The paper distinguishes two results. Proposition 1 concerns the regression whose controls are indicators for bins of width $2k$ of the margin $\mathit{DemVot}_{it}$, the “coarsened control” of Table 1; I call it the coarsened regression. Its OLS coefficient equals the difference in means within the window $(-k,k]$, and every outcome outside the window has weight zero in it. Proposition 2 concerns regressions with a polynomial in the margin, such as the linear and degree-7 controls of Table 1. Appendix B compares the polynomial and shrinking-window RD estimates through the cutoff jump $\beta=f(0)-\lim_{x\uparrow0}g(x)$, rather than through their observation weights.

Read as a whole, the paper separates exact coarsened equivalence from approximate polynomial equivalence in large samples. The phrase “mathematically equivalent” on p. 3 should be read in that context. Nevertheless, the explanation in terms of the mechanics of regression, followed by “rely solely on data” on p. 3 and “only use information” on p. 7, naturally asserts exclusive dependence on close-election outcomes. That assertion is false for the permitted finite polynomial controls. Its failure does not demonstrate failure of either equivalence result. The identified coefficient of a polynomial regression of degree $J$ is a weighted sum $\sum_i w_i\,\mathit{DV}_i$ over all elections. The weight $w_i$ is proportional to the residual of $\mathit{DemWin}$ on the polynomial terms, and this residual is nonzero at all but at most $2J$ distinct margins. Raising the outcome of a distant election with nonzero weight then changes the estimate, even when the polynomial model is exactly correct, while the RD estimate and the coefficient of the coarsened regression do not move.

Two estimators with the same limit need not have the same weights, the same sampling variance or the same bias at a fixed bandwidth. Under shared consistency their difference nevertheless converges to zero in probability; nonzero distant weights do not refute that approximation. For the coarsened regression, equality of the estimators transfers the properties of the specified RD estimate directly. For polynomial controls, Appendix B already supplies the common-target argument, and Section 2 describes assumptions needed for its causal interpretation. Different outcome weights do not remove concerns that arise from that common target and those assumptions. They establish neither different causal bias nor broader external validity in the application.

Evidence an exact counterexample

The evidence is a derivation, two exact counterexamples and a recalculation of the weights on the available House replication data. Let $r$ be the residual of the vector $\mathit{DemWin}$ after least-squares projection on the constant and the powers $\mathit{DemVot},\ldots,\mathit{DemVot}^J$. For an identified coefficient in equation (2), I obtain from the normal equations

which implies that raising one outcome by $\delta$ changes the estimate by $w_i\delta$.

For the linear control ($J=1$), take six elections with margins $-3/4$, $-1/4$, $-1/400$, $1/400$, $1/4$, $3/4$ and outcomes $\mathit{DV}=1/2+\mathit{DemWin}/10+\mathit{DemVot}/20$. Equation (2) then holds exactly with $\beta=1/10$. In the order of the margins, the weights are

With $k=1/200$, only the elections at $\pm1/400$ lie in the window. For the degree-7 control, take sixteen elections with margins $\pm1/8,\pm2/8,\ldots,\pm1$ and outcomes $1/2+\mathit{DemWin}/10+\mathit{DemVot}/20+\mathit{DemVot}^7/100$. Equation (2) with $J=7$ holds exactly, again with $\beta=1/10$, and the sixteen weights, in increasing order of margin, are

none of them zero. With $k=1/5$, only the elections at $\pm1/8$ lie in the window. The table shows what happens when the outcome of the election with the largest margin ($3/4$ in the first example, $1$ in the second) is raised by $1/100$.

In each example the two outcome vectors are realizations of one correctly specified model. Take errors independent of one another and of the margins, with probabilities $1/4$, $1/2$ and $1/4$ at $-1/100$, $0$ and $1/100$. Conditional on the stated margins, the baseline and perturbed error vectors have probabilities $1/64$ and $1/128$ in the six-election example, and $1/65536$ and $1/131072$ in the sixteen-election example. These probabilities establish that the outcomes are possible under the same conditional-mean model; they do not measure the probability of an OLS–RD discrepancy exceeding a fixed positive amount. The finite designs themselves do not test a shrinking-bandwidth asymptotic claim.

The targets do agree. For margins uniform on $[-1,1]$ and the same two conditional means, the population coefficient of equation (2) is $1/10$, and the difference in means within bandwidth $k$ is $1/10+k/20$ for $J=1$ and $1/10+k/20+k^7/400$ for $J=7$, which tends to $1/10$ as $k\to0$. With independent observations, the bounded mean-zero errors just specified and, for example, $k_N=N^{-1/3}$, both estimators are consistent for $1/10$. The finite grids are used for conditional sensitivity calculations; no positive probability of drawing those exact grids from a continuous distribution is asserted.

More generally, whenever both estimators converge in probability to the same $\beta$, for each fixed $\epsilon>0$ the triangle inequality gives

This requires no independence between the estimators. It does not assert equality at finite sample sizes, a probability for a particular sample size, or equivalence after division by a shrinking standard error.

Their variance contributions can still differ. With homoskedastic errors, elections whose margins have absolute value at least $1/2$ contribute $1/8$ of the leading error variance of the coefficient of equation (2) for $J=1$ and $572158921/10276044800$ (about 5.6%) for $J=7$. The leading contribution from the errors to the variance is $16\sigma^2/N$ for $J=1$ and $(65536/1225)\sigma^2/N$ for $J=7$, against $800\sigma^2/N$ for the difference in means with $k=1/200$. The polynomial contributions vanish at order $1/N$, so their positive distant shares do not imply a persistent discrepancy between the estimates.

On the available House replication data, the two polynomial specifications of Table 1 also put most of their weight on distant elections. In the 2014 release of the replication data of Caughey and Sekhon (2011), 8,610 elections from 1942–2008 have the outcome (DPctNxt), the margin (DifDPct) and the winner (DemWin) recorded, and a recalculation in exact arithmetic gives the weights $w_i$ for the specifications of rows (b) and (c) on these elections. The available codebook defines this margin as the leading Democrat’s percentage minus the leading non-Democrat’s, rather than a two-party renormalization. The 85 elections within 0.5 points of the threshold form the window of rows (d) and (e). They carry 1.7% of $\sum_i|w_i|$ for the linear control and 4.0% for the degree-7 control. Each of the other 8,525 elections has a nonzero weight in both regressions, and the elections decided by more than 10 points carry 70% and 47% of $\sum_i|w_i|$. In the coarsened specification of row (d), the 85 elections carry all of the weight. The three coefficients are nonetheless close: 11.15, 9.95 and 10.54 points. With the sample restricted to the 8,594 elections flagged Use=1 for Caughey and Sekhon’s (2011) RD analysis, the two shares are 1.8% and 4.0%. These are shares of absolute outcome weights, not probabilities of an OLS–RD discrepancy or causal population weights. These calculations do not establish the authors' exact sample or code, the population fit of either polynomial, or a difference in causal bias or external validity.

What survives

Proposition 1 and Section 5’s statement that OLS “may” rely solely on close-election outcomes are unaffected. When the conditional mean has exactly the form of equation (2), its coefficient and a consistent shrinking-bandwidth RD estimator estimate the same jump; under shared consistency their difference converges to zero in probability. The paper’s distinction between exact and approximate equivalence and Appendix B’s common-target argument are compatible with the weight calculations. Those calculations do not adjudicate the general conditions for polynomial approximation; see also P023-E002.

This clarification does not change any entry of Table 1 (for the entries, see P023-E003 and P023-E004), and the advice to examine regression assumptions stands. The exclusive-data explanation requires qualification for polynomial controls. Concerns about causal identification and external validity can remain shared when they arise from the common cutoff target and the same identifying assumptions; the weight calculation establishes neither different causal bias nor broader causal scope in the application.

The problem

Appendix B writes the conditional mean of the outcome as $\alpha+\beta D+h(X)$, with $X=\mathit{DemVot}$, $D=I[X\ge0]$ and $h$ continuous, approximates $h$ uniformly by a polynomial, and concludes that the coefficient on $D$ is close to the jump $\beta$. The step marked above leaves out a factor. The error of the coefficient is the approximation error weighted by the part of $D$ that the polynomial terms do not explain, divided by the variance of that part. It is bounded by $\kappa_J$ times the uniform approximation error, where the factor $\kappa_J\ge2$ depends on the degree and on the distribution of the margin. For uniform margins it grows without bound in $J$. The continuity assumptions on $f$ and $g$ do not imply Proposition 2. Indeed, for uniform margins there exists a continuous $h$ with $h(0)=0$ such that the coefficient does not converge to $\beta$ as the degree grows.

The test of fit in the Remark, which the surrounding text and Appendix B use as a sufficient condition, measures a different quantity from the approximation error of $h$, and it does not certify a given degree. With uniform margins and no jump, the linear and the degree-7 regression can have a coefficient about 10 percentage points from $\beta$ while no further terms could lower the mean squared error by more than $1/12{,}000{,}000$. Rows (b) and (c) of Table 1 use these two degrees. Whether they are adequate for the data of the application cannot be determined from the paper.

Evidence an exact counterexample

The evidence is a derivation and exact counterexamples. Let $X=\mathit{DemVot}$, $D=I[X\ge0]$ and $Y=\mathit{DV}$, with $E[Y\mid X]=\alpha+\beta D+h(X)$ as in Appendix B. Let $r_J$ be the residual of $D$ after projection in $L^2$ on $1,X,\ldots,X^J$, and set $v_J=E[r_J^2]$, which I assume positive. Then $E[r_J\,p(X)]=0$ for every polynomial $p$ of degree at most $J$, and $E[r_JD]=v_J$. Let $b_J$ be the coefficient on $D$ in the population regression (2). Subtracting from the normal equation for $D$ the combination of the other normal equations that defines the projection, I obtain $E[r_J(Y-b_JD)]=0$. Then

By the triangle inequality and the Cauchy–Schwarz inequality, I have

Since $E|r_J|\ge E[r_J(2D-1)]=2v_J$, the factor $\kappa_J$ is at least 2. For fixed $J$ and an i.i.d. sample with finite second moments, $\hat\beta_{OLS2}$ converges in probability to $b_J$, and $b_J-\beta$ is then the difference between the two estimators in large samples.

For margins uniform on $[-1,1]$, I find by a direct computation that $v_J=c_n^2/4$ with $c_n=\binom{2n}{n}4^{-n}$ and $n=\lfloor(J+1)/2\rfloor$, so that $v_J$ is of order $1/J$, and that $\kappa_J=(4/\pi)\log J+O(1)$. The table lists these constants for the two degrees of Table 1, together with the following example. Let $h_\eta(x)=a\,\operatorname{clip}(x/\eta,-1,1)$ with $a=1/20$ and $\eta=1/10000$, and let $Y=1/2+h_\eta(X)+U$, where $U=\pm1/10$ with equal probability, independently of $X$. Then $h_\eta$ is Lipschitz, $h_\eta(0)=0$, the jump is $\beta=0$, and the RD estimate converges to 0. The predictor $1/2+a(2D-1)$ lies in the span of the regressors of (2), and its mean squared error exceeds the noise variance $1/100$ by $a^2\eta/3=1/12{,}000{,}000$. No further terms can lower the mean squared error by more than this amount.

In exact terms, $b_1=99980001/1000000000$. As $\eta\to0$ with $J$ fixed, the bound $a^2\eta/3$ tends to zero and $b_J$ tends to $2a=1/10$. A second example concentrates the margins: all but a fraction $10^{-12}$ of the mass lies in two windows of half-width $1/2000$ around $\pm1/2$, the density is continuous and positive on $[-1,1]$, and $h$ is Lipschitz with $\sup|h|=1/2000$ and vanishes near 0. Then $\beta=0$, further terms can lower the mean squared error by at most $1/4{,}000{,}000$, and yet $b_1=-499999999999503/500000999999500$. With the fraction $10^{-96}$, the degree-7 coefficient lies between $-1001/1000$ and $-999/1000$.

Continuity alone fails even for one fixed design. Since $\kappa_J$ is unbounded for uniform margins, a uniformly convergent series of polynomials defines a continuous $h$ with $h(0)=0$ and $\sup|h|\le1/12$ such that $b_{J_m}\ge m/4$ along a sequence of degrees $J_m\to\infty$, while $\beta=0$.

The corrected statement below rests on two estimates. Comparing with uniform margins on $[-\delta,\delta]$, I have $v_J\ge c\,\delta/(4(J+1))$ under its density condition. Under its Lipschitz condition, convolution with a Jackson kernel yields a polynomial $p_J$ of degree at most $J$ with $p_J(0)=0$ and $\|h-p_J\|_\infty\le4KL/(J+1)$, where $K=3(1+\pi^4)/(4\pi)$. Inserting these into the second bound above, I obtain the rate of the corrected statement. The limit of $\hat\tau_{RD}$ follows from $|E[h(X)\mid0<X\le k]-E[h(X)\mid-k<X<0]|\le2Lk$ and from the growth of the number of observations in the window on each side of the threshold, which has expectation at least $Nck$ on each side.

What survives

Proposition 1 and Appendix A do not use this step. If $h$ is exactly a polynomial of degree at most $J$, then $b_J=\beta$. Proposition 2 holds as a statement about a sufficiently large degree under the conditions of the corrected statement, which are of the kind its “weak regularity conditions” ordinarily cover. No entry of Table 1 is shown to be wrong, and the similarity of the printed estimates stands as a description. What changes is the proof in Appendix B, the assumptions of Proposition 2, and the Remark. Rows (b) and (c) of Table 1 are instances of Proposition 2 only under a quantitative condition on the degrees 1 and 7, which the paper does not check.

The problem

Row (f) is described as the difference in means $\hat\tau_{RD}$ of row (e) with a triangular kernel in place of the rectangular one, and row (h) as the local linear estimate of row (g) with the same change. I read these descriptions in the standard way. An election with margin $x$ has weight $\max(1-|x|/k,0)$, with $k=0.5$ points and a constant fitted on each side of the threshold in row (f), and with $k=1.0$ point and a line fitted on each side in row (h). On the replication data of Caughey and Sekhon (2011), I find 13.08 and 15.30 points for these two estimators. From the same data and the same elections, I obtain rows (e) and (g) as printed, 10.54 and 13.15, and a coarsened regression equal to row (e), as in row (d). The two printed entries are thus lower by 2.51 and 2.92 points than the values that the described estimators take on the data that reproduce the neighbouring rows.

In the printed table the estimates with the triangular kernel are close to those with the rectangular kernel: the differences are $+0.03$ and $-0.77$ points. Recalculated, the triangular estimates exceed the rectangular ones by 2.54 and 2.16 points, and, with the other entries as printed, the estimates other than the naive one range from 9.97 to 15.30 points, where the printed ones range from 9.97 to 13.15. I do not know the cause of the two differences. A different release of the data, a choice of weights, bandwidth or sample that the paper does not describe, and a slip in computation or reporting are all consistent with the evidence; the paper names no release, and its analysis code was not available.

Evidence a recalculation from the source

The evidence is a recalculation from the data in exact rational arithmetic on the stored values; a second, independent computation agrees with it to $10^{-10}$ points. The data are the release of 10 April 2014, and the sample is the 8,610 elections with the outcome, the margin and the winner recorded. Let $x_i$ be the margin of election $i$ in points and $y_i$ its outcome. Row (f) is the difference between the two sides of the threshold in the weighted mean $\sum_i w_iy_i/\sum_i w_i$, with $w_i=1-|x_i|/0.5$ over the elections with $|x_i|<0.5$. The weighted means are 40.99 for the Democratic losses and 54.08 for the Democratic wins. Row (h) is the difference at $x=0$ between two weighted least-squares lines, one on each side, with $w_i=1-|x_i|$ over the elections with $|x_i|<1$. The two intercepts are 38.86 and 54.16.

No margin equals $0$, $\pm0.5$ or $\pm1.0$ points, so the treatment of ties and of the endpoints of a window cannot account for the differences. If the sample is restricted to the elections that Caughey and Sekhon (2011) flag for their RD analysis, the same elections lie in both windows, and the four rows do not change.

The four outcomes of P023-E004 are one hundredth of their linked next-election percentages at storage precision. All four elections lie within 0.3 points of the threshold, and two of them within 0.06 points, where the triangular weights are largest. With those linked percentages substituted, I obtain 10.36 and 10.91 points for rows (f) and (h). Neither computation returns the printed 10.57 and 12.38; the accuracy of the links and percentages remains unverified.

What survives

Rows (e) and (g) are reproduced as printed, and the coarsened regression of row (d) equals row (e) exactly, as Proposition 1 requires. The propositions and their proofs do not use Table 1. Every estimate other than the naive one remains positive. What changes is two entries of the table and the comparison drawn from them: the sentence below the table that all estimates other than the naive one are “substantively similar” then covers a range from 9.97 to 15.30 points. The paper gives no threshold for this phrase, and I draw no conclusion on whether it still applies.

The problem

In the replication data of Caughey and Sekhon (2011) the outcome is the variable DPctNxt, which the codebook defines as the “Democratic percentage of the vote in the next election”, and the variable NxtElcOb gives the observation number of that election. For four elections of 2000 the recorded outcome is one hundredth of the Democratic percentage at the linked observation, to the precision of the stored numbers. The linked observation is the election of 2002 with the same state and district number. I call this percentage the linked percentage. No other positive outcome shows this ratio.

The four elections were decided by margins of $-0.28$, $-0.05$, $-0.05$ and $0.22$ points. They are therefore among the 85 elections in the 0.5-point window of rows (d) to (f) and among the 168 elections in the 1.0-point window of rows (g) and (h). In the smaller window, the three proportions recorded for Democratic losses lower the mean outcome of the losing side by 2.62 points, and the proportion recorded for the Democratic win lowers the mean of the winning side by 1.41 points. The difference in means of rows (d) and (e) is then higher by 1.22 points than with the linked percentages. With the four outcomes replaced by the linked percentages and nothing else changed, rows (d) and (e) fall from 10.54 to 9.32 points and row (g) falls from 13.15 to 10.40. The printed entries of these three rows agree to the second decimal with the values computed from the four outcomes as recorded.

Two points are not established. The first is that the linked percentages are the right outcomes for the four elections. The codebook codes the outcome as missing when the seat was redistricted between the two elections, and the linked elections of 2002 are not among the elections that Caughey and Sekhon (2011) flag for their RD analysis and carry no link back to a previous election. The data alone do not show that the two elections are in the same district. The second is that the published analysis read these four values. The paper names no release of the data and its analysis code was not available. Agreement of the printed rows (d), (e) and (g) with calculations from the release of 10 April 2014 does not establish historical use.

Evidence a recalculation from the source

I recalculated the estimates from the data (the release of 10 April 2014, with 16,968 observations, of which 8,980 have an outcome and a linked percentage). The four observations listed in the first table below are the only ones with a positive outcome at which 100 times the outcome equals the linked percentage, to the precision of the stored numbers. Their recorded outcomes also equal, to storage precision, the Democratic shares at the linked observations: $83265/217115$, $70920/230597$, $121121/286860$ and $104806/171713$.

The eight Table 1 specifications were translated using DifDPct as the margin and a common sample of 8,610 elections with the outcome, margin and winner recorded. Each was computed twice with the stored numbers treated as exact binary fractions: once with the outcomes as recorded, and once with the four outcomes replaced by the linked percentages. The second table reports both computations in percentage points; the changes are computed before rounding.

With the sample restricted to the 8,594 elections that Caughey and Sekhon (2011) flag for their RD analysis, rows (a) to (c) are 34.40, 11.14 and 9.97 with the outcomes as recorded and 34.38, 11.12 and 9.89 with the four replaced, and rows (d) to (h) are as in the table. Neither sample reproduces all eight printed entries. Row (d) equals row (e) exactly in every computation. For rows (f) and (h), whose printed entries neither computation returns, see P023-E003.

What survives

The equality of rows (d) and (e) holds for every vector of outcomes and is unaffected, and the propositions do not use the table. Rows (a) to (c) move by less than 0.1 point. In this calculation, with the four outcomes replaced, rows (b) to (h) lie between 9.32 and 11.13 points, and their numerical agreement stands. What changes is the size of the close-election estimates: rows (d), (e) and (g) fall by 1.22, 1.22 and 2.75 points. The four values are in the data as Caughey and Sekhon released them, and nothing shows that the authors of the paper introduced them.

The problem

The paper gives no numerator, denominator or counting rule for this figure. The replication data of Caughey and Sekhon (2011), which Section 4 uses for the application, contain 13,923 observations for 1944–2006, one for each seat and election. In 103 of them the recorded margin between the leading Democrat and the leading non-Democrat is below 0.5 points in absolute value, that is, in about 0.74% of them. A share that rounds to 0.6% would require between 77 and 90 such elections.

Evidence a recalculation from the source

I recalculated the share from the data (the release of 10 April 2014; the paper names no release). The margin is missing for 587 of the 13,923 elections, and the missing margins cannot restore the printed figure. By the codebook’s definition of the margin, each of the 103 elections was decided by less than 0.5 points between the winner and the runner-up. The share of all elections is then at least $103/13{,}923$. Among the elections with a recorded margin the share is $103/13{,}336\approx0.77\%$. The ratio $85/13{,}923\approx0.61\%$ rounds to the printed figure, but its numerator counts only the close elections that Caughey and Sekhon (2011) flag for their RD analysis and omits 18 others, while its denominator counts all elections. The paper does not say how its figure was computed.

What survives

Each of the recorded close-election shares above is below 1%. The figure enters no proposition, no proof and no entry of Table 1. The data do not determine the exact share of all elections decided by less than 0.5 points, since the margin between winner and runner-up is not recorded for every election; this share rounds to 0.7% if at most one further close election lies outside the 103.

The problem

The surname of the first author of the cited study is misspelled in both footnotes.

Evidence a check of the printed page

I checked the pages: the reference list on p. 11 and the text on pp. 6, 7 and 9 have “Caughey”.

What survives

Nothing depends on the spelling, and the reference list identifies the work.

The problem

The codebook of the replication data of Caughey and Sekhon (2011) defines a flag, Use, for a narrower “RD incumbency advantage sample”. Its margin variable, DifDPct, is the percentage of the leading Democrat minus that of the leading non-Democrat, while the paper defines $\mathit{DemVot}_{it}$ as the difference between Democratic and Republican vote share. Section 4 does not say which release of the data was used, which variables stand for the margin and the outcome, whether the sample was restricted by the flag, or how missing values were treated. Table 1 reports no sample sizes and no standard errors. The table cannot be reproduced exactly from this description. This is a gap in the description, separate from the numerical and data issues in P023-E003 and P023-E004.

Evidence a check of the printed page

I checked pp. 7–8 against the codebook of the data (the copy posted by the first author of the source study, dated 22 December 2025), which defines Use on its p. 1 and DifDPct on its p. 2. Independent recalculations from the corrected release of 10 April 2014 used 8,610 elections complete on DPctNxt, DifDPct and DemWin, or 8,594 when also requiring Use=1. Neither sample reproduces every row to the second decimal. The small differences in rows (a) and (c) do not change the comparison that the table serves.

What survives

The broad comparison between the naive estimator and the others survives, with the numerical qualifications in P023-E003 and P023-E004. The two definitions of the margin agree under the two-party supposition of p. 5. The propositions do not use the table.

The problem

The set $U_b$ consists of pairs $it$, and the summands depend on $t$, so a sum over $i$ alone is not defined.

Evidence a check of the printed page

I checked the page: $U_b=\{it:-k<\mathit{DemVot}_{it}\le k\}$ is defined on p. 6, and the display on p. 6 and the next display on p. 9 sum over $it\in U_b$.

What survives

The proof of Proposition 1 is unaffected; the intended sums are over the pairs in the central bin.

Intended meaning

The RD estimate displayed on p. 6 is a difference in means over $U_b=\{it:-k<\mathit{DemVot}_{it}\le k\}$. It exists as soon as $U_b$ contains one Democratic win and one Democratic loss, so the strict inequalities of Condition 1 are sufficient for it and not necessary. For example, with $k=1/2$, margins $-3/4,-1/4,1/2,3/4$ and outcomes $1/10,1/5,4/5,9/10$, Condition 1 fails and the RD and coarsened OLS estimates both equal $3/5$. Conditions 1–3 allow a bin to be empty, and $|\tilde U_j|^{-1}$ is then undefined. I read the intended convention as dropping empty bins; the coefficient on $\mathit{DemWin}$ is then unique, and Proposition 1 holds. The 50% threshold is that of a two-party election, which the paper supposes from p. 5 on. With three parties a plurality can be below 50% (shares of 40%, 35% and 25%, say).

Intended meaning

In Section 3.1 and Appendix B the margin $\mathit{DemVot}_{it}$ lies in $[-1,1]$, and the formula $J=2\lceil(1-k)/(2k)\rceil$ of Condition 2 takes $k$ on the same scale. The bandwidths of the application are stated in percentage points. I convert 0.5 points to $k=0.005$, for which $J=200$ in Condition 2, that is, 201 bins one percentage point wide, and 1.0 point to $k=0.01$. Read literally, $k=0.5$ would give $J=2$ and a window of 50 points on each side of the threshold. No result changes.