Regression discontinuity aggregation, with an application to the union effects on inequality

Kirill Borusyak; Matan Kolerman-Shemer

REVIEW 4 major objections 4 minor 29 references

Regression discontinuity aggregation, with an application to the union effects on inequality

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that regression discontinuity designs still identify a local average causal effect when outcomes aggregate many discontinuity events, and that the rate of new unionization reduces within-cell wage inequality in the US.

desk verdict A clean, reusable formalization of aggregated RD designs with a credible equivalence result; the union application is plausible but depends on an untestable exclusion restriction. read the letter →

arxiv 2501.00428 v1 pith:HPSA42J5 submitted 2024-12-31 econ.EM

Kirill Borusyak , Matan Kolerman-Shemer This is my paper · ORCID

classification econ.EM

keywords regressiondiscontinuitydesignRDaggregationshift-shareinstrumentslocalaveragetreatmenteffectfuzzyunionizationandinequalitywageinstrumentalvariables

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the regression discontinuity toolkit from settings where each unit faces one discontinuity to settings where the outcome is measured for units that aggregate many discontinuity events, such as states-by-industry cells aggregating establishment-level union elections. It proposes two estimators, an instrumental-variable regression at the aggregated level and a stacking fuzzy RD regression at the level of individual close elections, and proves that both converge to the same local average causal effect as the bandwidth shrinks. The practical payoff is that researchers can use credible RD variation even when outcomes are only available at a higher level of aggregation, or when spillovers across units matter. Applied to NLRB union elections and census wage data, the method estimates that a higher rate of newly unionized workers in a state-by-industry cell lowers within-cell wage inequality, primarily by compressing the top of the wage distribution. If correct, this provides causal evidence that declining unionization contributed a large share of the rise in within-cell inequality in the US since the 1970s.

What carries the argument

The central object is the weight-share representation of the aggregated treatment and instrument. Each upper-level unit $i$ contains subunits $j$ with known importance weights $s_j$; the treatment is $X_i=\sum_{j\in J_i}s_j z_j$ with $z_j=\mathbf{1}[r_j\ge0]$, and the instrument $Z_i=\sum_{j\in C_i}s_j z_j$ keeps only subunits whose running variable falls within the bandwidth. The load-bearing identity is Proposition 1: the upper-level IV estimator is numerically equivalent to a subunit-level IV regression on the close-elections sample, with $z_j$ as the instrument, $q_j=(1,r_j,r_j^+)$ as controls, and $s_j$ as weights. This equivalence dictates the three RDA controls $Q_i=(\sum_{j\in C_i}s_j,\sum_{j\in C_i}s_j r_j,\sum_{j\in C_i}s_j r_j^+)$ as the correctly aggregated versions of standard local linear controls, and it is what lets the stacking estimator and the upper-level estimator converge to the same $\beta_0$. The lower-level estimator is the same regression without residualization, which is why both inherit the bias behavior of conventional fuzzy RD.

What would settle it

One concrete check is to estimate the reduced-form effect of close-election victories on the change in cell inequality for elections where the union won at the ballot box but certification was later reversed on appeal: under the exclusion restriction these elections should produce no effect, because the unionization share is unchanged, whereas any nonzero effect would signal that the election itself, rather than new unionization, drives the result.

Watch

Extended reading notes

Core claim

It is the paper's central formal claim that aggregated RD still identifies a local average treatment effect. Let $\beta_h^u$ be the estimand of the upper-level IV regression (6) and $\beta_h^\ell$ the estimand of the lower-level stacking regression (9). Proposition 2 states that under continuity, density, monotonicity, and an exclusion restriction (Assumptions 1–5), both estimands converge as $h\to 0$ to the same convexly weighted average of unit-level potential-outcome slopes, $$\beta_0=\frac{\mathbb{E}[s_j\,(Y_{i(j)}(X_{i(j)}(1,z_{i(j)-j}))-Y_{i(j)}(X_{i(j)}(0,z_{i(j)-j})))\mid r_j=0]}{\mathbb{E}[s_j\,(X_{i(j)}(1,z_{i(j)-j})-X_{i(j)}(0,z_{i(j)-j}))\mid r_j=0]},$$ so both procedures recover a causal local parameter rather than a confounded association. The paper further argues, and supports by Monte Carlo simulation, that including the aggregated local linear controls at the upper level—or their standard RD analogues at the lower level—preserves the bias-reduction advantages of local linear RD, which most existing aggregated-RD practice gives up.

Load-bearing premise

The causal interpretation of the inequality estimates rests on the exclusion restriction that a close union election affects a cell's inequality only through the share of newly unionized workers, not through election campaigns, organizing threats, or other channels.

Editorial extensions

If this is right

In any design where treatment is a weighted sum of RD shocks, the upper-level IV with RDA controls and the lower-level stacking estimator converge to the same local average treatment effect, so aggregated RD studies can keep the bias-reducing controls of local linear RD.
For the union application, a 1 percentage point increase in the rate of new unionization lowers the Gini coefficient by about 0.018, the top-10% income share by about 0.14 percentage points, and the variance of log wages by about 0.0025, which is roughly 1% of its mean.
The inequality reduction comes mostly from lower wages at the top of the distribution (college graduates, the 90th percentile, and managers), not from higher wages at the bottom.
If the estimated effects persist, the decline in new unionization since the 1960s explains about 34–38% of the growth in within-cell inequality from 1970 to 2010.
The same upper-level or stacking template applies to legislature seat shares, firm political connections, school bond referenda, and spillover designs, where the paper argues current practice typically omits the necessary aggregated controls.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves formal bias expansions to future work; a natural extension would prove that the RDA controls reduce bias at a quadratic rate as $h\to0$ under smooth potential outcomes, turning the Monte Carlo evidence into a theorem.
Because the weights in $\beta_0$ are $s_j$ times the first-stage effect, the target parameter is implicitly election-weighted; researchers wanting unit-level rather than subunit-level weights should reweight by the inverse of $\sum_{j\in C_i}s_j^2$, an adjustment the paper mentions only briefly.
A testable consequence of the exclusion restriction is that close elections whose certification was later reversed should show zero reduced-form effect on cell inequality; the appeal cases the paper cites could serve as a placebo sample.
The control for the total weight of close elections is equivalent to a recentering adjustment under a local randomization view of RD, suggesting that a local-randomization variant of RDA could sharpen inference in small samples.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

This paper is a solid, reusable methodological contribution. It formalizes settings where the treatment is an aggregate of many RD shocks—e.g., a state-level outcome with district-level elections—and shows two estimators are equivalent: an IV regression at the outcome level with properly aggregated local linear controls, and a stacked fuzzy RD at the subunit level. The equivalence rests on the published Borusyak–Hull–Jaravel shift-share result, and the identification proof in the appendix is short but credible. The Monte Carlo supports the bias-reduction claims, and the upper-level estimator looks somewhat more efficient. This will be cited.

The empirical application to unions and inequality is careful and well documented. The data construction is thorough, the balance tests are encouraging, and the counterfactual is clearly labeled as back-of-the-envelope. The survey of more than fifty papers that fit the RDA template is a real service to the field.

The soft spots are real but manageable. The main one is the exclusion restriction in the union application: the instrument is the share of workers unionized through close elections, and the causal interpretation requires that close wins affect cell-level inequality only through new unionization. Threat effects, employer responses to organizing campaigns, or political spillovers could break that. The balance tests rule out correlation with observables but not this kind of unobserved channel. The paper states the assumption explicitly, so this is a limitation rather than a hidden flaw, but it should be discussed more directly.

Second, the main inequality table omits first-stage coefficients and F-statistics. The balance table has partial F's for the instrument, but readers need the first stage for the IV specification to assess strength. This is an easy fix. Third, the identification proof assumes the control coefficients converge as the bandwidth shrinks without fully stating conditions. It is plausible but should be tightened. Fourth, the abstract claims effects on all inequality measures, but only three of the five are statistically significant; a softer phrasing would be more accurate.

These are all repairs, not deep flaws. The methodological core holds up, and the application, while dependent on the exclusion restriction, is honest about its assumptions. I would send this to referees. If I were the editor, I'd ask for first-stage diagnostics, a more direct discussion of the exclusion restriction, and a modest softening of the abstract. For an applied or methodological reader, this paper is worth the time.

Referee Report

4 major / 4 minor

Summary. The paper extends regression discontinuity (RD) methods to settings where the outcome is measured at a higher level of aggregation than the running variable, proposing two estimators: an upper-level IV estimator that instruments an aggregated treatment with a weighted sum of close-election RD shocks while controlling for aggregated local-linear controls, and a lower-level 'stacking' estimator that runs a fuzzy RD on the sample of close elections with repeated cell outcomes. The central theoretical claim (Proposition 2) is that both estimators converge, as the bandwidth h goes to zero, to the same convexly weighted average of potential-outcome slopes, under continuity, monotonicity, and an exclusion restriction. The paper also provides Monte Carlo evidence on bias reduction, a survey of existing practice, and an application to the effect of unionization on within-state-industry wage inequality, reporting significant negative effects on the Gini coefficient, top-10% share, and variance of log wages, with a back-of-the-envelope calculation attributing roughly 35% of the 1970-2010 rise in within-cell inequality to the decline in new unionization.

Significance. If Proposition 2 is correct, the paper contributes a practical and credible approach to a class of designs (legislative composition, spillovers, temporal aggregation) that currently rely on ad hoc or less transparent methods. The connection to shift-share equivalence (Borusyak, Hull, and Jaravel 2022) is elegant and likely useful for inference and bandwidth choice. The empirical application is policy-relevant, and the paper is careful in data construction, balance checks, placebo tests, and robustness analysis. The Monte Carlo evidence is honestly presented, and the authors explicitly flag the lack of formal bias theory. The main weakness is that the causal interpretation of the application hinges on an exclusion restriction that is plausible but not directly tested, and the empirical tables omit first-stage diagnostics.

major comments (4)

[§2.4 and §5.1] The causal interpretation of the union application rests on the exclusion restriction stated in §2.4: that a close election's outcome affects cell-level inequality only through the share of newly unionized workers. In this setting, narrowly lost elections may affect inequality through union threat effects, employer responses to organizing drives, or changes in worker composition, all of which would violate the restriction. The balance tests in §5.3 and Appendix Table A5 establish exogeneity of the instrument with respect to observables but cannot address exclusion. The paper should either provide additional evidence against threat effects (e.g., reduced-form effects of close losses on outcomes in cells where unionization did not occur) or explicitly discuss the plausibility of the restriction in light of the literature on union threat effects (e.g., Fortin et al. 2022). This is load-bearing for the empirical claim.
[§5.4, Table 3] The main empirical table reports only the IV coefficients, not the first-stage coefficients, F-statistics, or reduced-form coefficients. Given that the instrument is a constructed employment-share weighted sum of close-election outcomes, the first stage is not mechanical, and the strength of the instrument is important for interpreting the IV estimates. The paper should report the first-stage coefficient on Z_sit and the associated F-statistic (or effective F-statistic) for each specification in Table 3, and ideally the reduced-form coefficients corresponding to the main outcomes. This omission weakens the empirical contribution as currently presented.
[§2.5 and Appendix A] The proof of Proposition 2 in Appendix A is only sketched. It relies on two unproven steps: (i) that the population coefficients on r_j and r_j^+ in the lower-level specification have limits as h goes to zero, and (ii) that the coefficients on the RDA controls in the upper-level specification converge so that residualization on Q_i is asymptotically irrelevant. The latter step is asserted with 'assuming again that the population coefficients ... converge' rather than established. Since Proposition 2 is the central theoretical result, the proof should be made complete. Relatedly, the paper explicitly states in §2.5 that 'We leave theoretical results establishing this property to future drafts' for the bias-reduction claims; the abstract and introduction do not overstate this, but the Monte Carlo evidence alone does not establish that the estimators 'inherit' the bias properties of local-linear RD. The authors should either provide a formal bias statement or temper the corresponding language.
[§5.4, Panel B; §5.4 inference discussion] The lower-level stacking estimator repeats the same cell outcome and treatment for multiple elections in the same state-industry-decade, creating mechanical within-cell correlation in the error term. The paper states that clustering does not substantially change the standard errors but does not report the clustered estimates. Given that the stacking estimator is one of the two main proposals, the manuscript should show cluster-robust (by cell) standard errors for Panel B, or at least provide a quantitative comparison to heteroskedasticity-robust errors. As it stands, the reader cannot verify whether the reported significance of the lower-level estimates is robust to this clustering concern.

minor comments (4)

[§5.4, paragraph after Table 3] The numerical magnitudes reported in the text are inconsistent with the coefficients in Table 3. If NewUnions is measured as a share (0.01 = 1 percentage point), then a 1 pp increase changes the college premium by approximately 0.003 log points, the 90/10 ratio by 0.0046 log points, and the Gini by 0.0018, not 0.31, 0.46, and 0.018 as stated. The top-10 share and variance statements are consistent with the table. Please correct these magnitudes or clarify the units of the treatment variable.
[§2.2, equation (6) and text after it] The sentence in §3.2 describing the recentering procedure says 'recenter the instrument Z_i by subtracting 0.5 times the total weight of narrow elections' — should say 'the total weight of narrow elections' rather than omitting the word 'the' before 'narrow'; also, for consistency with the rest of the paper, 'narrow' is used interchangeably with 'close' in a few places. A unified terminology would improve readability.
[§2.5, Figure 1 notes] Panel (c) of Figure 1 shows that the bias of the proposed estimators has slope approaching zero for smaller h, but the vertical axis range differs across panels; it would help to report the numeric bias values in the text or table, as the confidence intervals are stated to be hard to see on some panels.
[§4.2, paragraph on Cellini et al.] The conjecture at the end of the paragraph ('we cautiously conjecture that the upper-level solutions also continues to be valid') deserves more justification or a formal statement; as written, it is an untested claim in a methodological paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivalence result is a parameter-free algebraic theorem from prior published work, and the empirical estimates are not fitted versions of their own outputs.

full rationale

The paper's central claim (Proposition 2) is that the upper-level IV estimand (6) and lower-level stacking estimand (9) converge to the same convex-weighted average beta_0 in (10) under Assumptions 1-5. The proof is self-contained except for Proposition 1, which is imported from Borusyak, Hull, and Jaravel (2022, Prop. 1) and used to demonstrate a numerical equivalence between the upper-level IV regression and a shock-level IV regression. This is a parameter-free algebraic equivalence with stated assumptions that do not include the target result; it is independent support even though one author co-authored the cited paper. The empirical application estimates inequality effects using close-election RD variation; the treatment, instrument, and RDA controls are constructed from the data, not fitted to the outcomes, and the bandwidth is chosen from prior practice and optimal-bandwidth calculations. The exclusion restriction for the union application is an untested identifying assumption, but an untested assumption is not circularity. No equation in the paper reduces to its own inputs by construction, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central method does not introduce new physical or structural entities. The application relies on standard RD assumptions plus an exclusion restriction, and on several hand-selected sample thresholds. The only notable reliance on prior work is the Borusyak et al. shift-share equivalence theorem, which is published and not fitted to the target results.

free parameters (3)

Close-election bandwidth = 10 percentage points (50% plus or minus 10pp)
Chosen based on the smallest bandwidth in recent union RD papers and aligned with Calonico et al. optimal bandwidths of 9.5-13.7%; robustness to 15% is reported. This is a hand-selected tuning parameter for the application.
Instrument sample restrictions = Minimum 20 votes; exclude ties and one-vote margins
These rules determine which elections enter the instrument and follow prior union RD literature. They are justified by manipulation evidence but are still sample choices that could affect estimates.
Cell size threshold = At least 1,000 workers after projection weights
The paper drops 24.9% of cells but only 0.8% of employment and 1.3% of close elections. This ad hoc threshold stabilizes inequality measures but is a modeling choice.

assumptions (6)

domain assumption Assumption 3(a): expected reweighted potential outcomes and treatments are continuous in the running variable at the cutoff
Used in Proposition 2 to equate one-sided limits of outcomes and treatments at r_j = 0. This is the standard RD continuity assumption, extended to reweighted unit-level objects.
domain assumption Assumption 2: subunit labels within a unit are exchangeable
Required to induce a distribution over subunits and to give the lower-level estimator a meaningful interpretation as a fuzzy RD in the sample of elections.
domain assumption Assumption 4: the density of the running variable is positive and continuous at zero
Necessary for the RD interpretation and for conditioning on r_j = 0 to be well-defined. In the application this means no strategic manipulation of union vote shares except in the excluded ties and one-vote margins.
domain assumption Assumption 5: monotonicity and positive first stage for the treatment
Needed for the IV estimand to be a convexly weighted average of causal effects; trivially satisfied when the treatment is the share of unionized workers.
domain assumption Exclusion restriction: close election outcomes affect inequality only through the new unionization rate
Stated in Section 2.4 and applied in Section 5. It is the key identifying assumption for the empirical application and is not directly testable.
standard math Borusyak, Hull, and Jaravel (2022) Proposition 1 on shift-share equivalence
Used to prove Proposition 1, the numerical equivalence between the upper-level IV specification and a subunit-level fuzzy RD specification. This is a published external result.

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Pith. "Pith review of Regression discontinuity aggregation, with an application to the union effects on inequality." pith.science (2026). https://pith.science/paper/HPSA42J5

@misc{pith2026250100428,
  author       = {Pith},
  title        = {Pith review of: Regression discontinuity aggregation, with an application to the union effects on inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HPSA42J5}},
  note         = {Machine review of arXiv:2501.00428}
}

read the original abstract

We extend the regression discontinuity (RD) design to settings where each unit's treatment status is an average or aggregate across multiple discontinuity events. Such situations arise in many studies where the outcome is measured at a higher level of spatial or temporal aggregation (e.g., by state with district-level discontinuities) or when spillovers from discontinuity events are of interest. We propose two novel estimation procedures - one at the level at which the outcome is measured and the other in the sample of discontinuities - and show that both identify a local average causal effect under continuity assumptions similar to those of standard RD designs. We apply these ideas to study the effect of unionization on inequality in the United States. Using credible variation from close unionization elections at the establishment level, we show that a higher rate of newly unionized workers in a state-by-industry cell reduces wage inequality within the cell.

Figures

Figures reproduced from arXiv: 2501.00428 by the authors.

**Figure 1.** Monte Carlo Simulations (a) Bias: Outcome #1 (b) Bias: Outcome #2 (c) Bias: Outcome #3 0 .5 1 1.5 Bias .25 .5 .75 1 1.25 Bandwidth Upper-level Lower-level Upper-level, fewer controls -.03 -.02 -.01 0 .01 Bias .25 .5 .75 1 1.25 Bandwidth Upper-level Lower-level Upper-level, fewer controls -.2 0 .2 .4 .6 .8 Bias .25 .5 .75 1 1.25 Bandwidth Upper-level Lower-level Upper-level, fewer controls (d) Standard deviation: Out… view at source ↗

**Figure 2.** Lower-Level Specifications: RD Plots Notes: The first plot visualizes the first-stage specification (19) with no fixed effects. The other plots visualize the reduced-form version of (18) with no fixed effects for the outcomes included in [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** The Counterfactual with Unionization Rates Remaining at the 1960s Level [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 10, 2026 · model on record in the stance chip above.

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Chao Fu and Jesse Gregory

doi: 10.1086/711852. Chao Fu and Jesse Gregory. Estimation of an equilibrium model with externalities: Post-disaster neighbor- hood rebuilding. Econometrica, 87(2):387–421,

work page doi:10.1086/711852

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Algorithm is experiment: Machine learning, market design, and policy eligibility rules

Yusuke Narita and Kohei Yata. Algorithm is experiment: Machine learning, market design, and policy eligibility rules. arXiv preprint arXiv:2104.12909 ,

arXiv

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Regression discontinuity designs under interfer- ence

Elena Dal Torrione, Tiziano Arduini, and Laura Forastiere. Regression discontinuity designs under interfer- ence. arXiv preprint arXiv:2410.02727 ,

work page arXiv

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state” samples. For 1970, we combined the 2% “state

Invoking Assumption 3b again and assuming again that the population coefficients on Qi in the first-stage and reduced-form of (6) converge, this implies that residualizing on Qi makes no difference in (8), provided an intercept is included in that subunit-level specification. This concludes the proof. B Data Appendix Our analysis is at the level of state-...

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Executive, Administrative, and Managerial

to identify the county composition of each metropolitan area and county group, and obtained the 1970 county populations from IPUMS NHGIS. 39 For each worker, we observe their pre-tax annual labor income (variable INCWAGE). Top-coded income values are multiplied by 1.5 [Autor et al., 2008]. We then compute the hourly wage as the income dividing by the week...

work page 1970