1 Introduction

The regression discontinuity design is one of the most credible tools in the applied econometrician's arsenal. By comparing units just above and just below a known cutoff in a running variable, RDD delivers local causal effects under the mild assumption that potential outcomes are continuous at the cutoff [Imbens and Lemieux, 2008, Calonico et al., 2014].

The classical RDD estimates a scalar treatment effect: the jump in the conditional mean of Y at the cutoff. But many policy-relevant questions concern distributions rather than means. A firm-size tax threshold may shift not just average wages within a firm, but the entire within-firm wage distribution compressing or widening inequality. A school admission cutoff may change not just mean student achievement, but the distribution of outcomes across students with different initial conditions. A health insurance cutoff may shift the utilisation distribution with high users changing behaviour differently from low users.

Recent work, notably Cattaneo et al. [2025] (arXiv:2504.03992), has extended the RDD framework to settings where the outcome of interest is a distribution rather than a scalar. This article explains the motivation, presents the key ideas, discusses estimation and inference, and describes available software.

2 Motivation: When Means Are Not Enough

Consider a concrete example. A government subsidy programme uses a revenue threshold to determine firm eligibility: firms with revenue below the cutoff receive a subsidy, those above do not. The outcome of policy interest is not mean firm revenue or mean employment— it is the distribution of wages paid to workers within each firm.

The allocation unit (firm) differs from the unit at which outcomes are measured (workers within the firm). A standard RDD would estimate the effect of the subsidy on the average wage across firms near the cutoff. But suppose the subsidy primarily raises wages for the lowest-paid workers in each firm compressing the within-firm wage distribution—while leaving mean wages roughly unchanged. The scalar RDD would miss this effect entirely, because the mean is unchanged even as the distribution shifts.

More generally, distributional effects arise whenever:

• The allocation and measurement units differ (firm-level policy, worker-level outcomes).

• Treatment heterogeneity across the marginal distribution of outcomes is the object of interest.

• Policy-makers care about inequality, dispersion, or tail outcomes—not just averages.

3 Framework

3.1 Setup

Maintain the standard RDD setup: a running variable Rᵢ, a cutoff c (normalised to zero without loss of generality), and a sharp treatment indicator Tᵢ = 1[Rᵢ ≥ 0]. For each unit i, define the potential outcome distributions F⁽¹⁾(·|r) and F⁽⁰⁾(·|r), which are the conditional distribution functions of Yᵢ given Rᵢ = r under treatment and control, respectively. These are distribution-valued functions of the running variable.

The standard continuity assumption for distributional RDD is:

Assumption 1 (Distributional Continuity). The potential outcome distributions F⁽¹⁾(·|r) and F⁽⁰⁾(·|r) are continuous in r at r = 0 in some appropriate metric on distributions.

Under this assumption, comparing the distribution of Yᵢ just above and just below the cutoff identifies the distributional treatment effect at the cutoff:‍


for each y in the support of the outcome. The function Δ F(·) is the distributional treatment effect (DTE) at the cutoff. It describes how the cumulative distribution function of Y shifts at the threshold.

3.2 Quantile Treatment Effects

The DTE can be converted to the more interpretable quantile treatment effect (QTE) function. Define Q⁽ᵈ⁾(τ|r) = inf{y : F⁽ᵈ⁾(y|r) ≥ τ} as the τ-quantile of the potential outcome distribution. The QTE at quantile τ is:‍


QTE(τ) > 0 means treatment raises the τth quantile of the outcome distribution at the cutoff. Plotting QTE(τ) as a function of τ ∈ (0,1) gives a complete picture of how the treatment shifts the distribution.

4 Estimation

The estimation approach follows the logic of local polynomial regression, applied to the empirical CDF rather than the conditional mean.

4.1 Local Polynomial CDF Estimation

For a grid of values y₁, ..., yₘ, estimate the conditional CDF F(yₘ|r) on each side of the cutoff using local polynomial regression of the outcome indicator 1[Yᵢ ≤ yₘ] on the running variable Rᵢ:‍


where X⁺ is the design matrix of a pth-order polynomial in Rᵢ for units to the right of the cutoff, W⁺ is a diagonal kernel weight matrix with kernel bandwidth h, and ê₁ extracts the intercept. The DTE estimate is:

where Q̂⁽ᵈ⁾(τ|0) is obtained from F̂(·|0⁽ᵈ⁾) by numerical inversion.

4.2 Bandwidth Selection and Bias Correction

A critical question is the choice of bandwidth h. Standard mean-squared-error-optimal bandwidth selectors developed in Calonico et al. [2014] for the conditional mean case do not directly apply to the CDF. Cattaneo et al. [2025] develop a uniform bandwidth selector that minimises the maximum integrated mean squared error of the DTE estimate across the support of y.

Bias-corrected confidence bands for the QTE function QTE(τ) are constructed by estimating the asymptotic bias from higher-order local polynomial fits. and applying the bias correction, then constructing simultaneous confidencebands using the theory of empirical processes.

5 Testing for Distributional Effects

Beyond point estimates, researchers may want to test:

1. Whether any distributional effect exists. The null H₀ : Δ F(y) = 0 for all y can be tested with a Kolmogorov-Smirnov-type statistic.‍
2. Whether the effect is uniform across quantiles. The null H₀ : QTE(τ) = δ for all τ (constant QTE) can be tested against a heterogeneous alternative.‍
3. Stochastic dominance. Whether the treated distribution first-order stochastically dominates the control distribution: Δ F(y) ≤ 0 for all y.

These tests are implemented using bootstrap or analytical critical values that accountfor the functional nature of the object being estimated.

6 An Application: Firm Subsidy and Within-Firm Wage Inequality

To illustrate, consider a stylised application. A firm subsidy programme uses a revenue threshold to determine eligibility. The researcher observes individual wages for all workers in firms near the threshold. Using distributional RDD:

• Estimate Q̂TE(τ) for τ ∈ {0.1, 0.2, ..., 0.9}.

• If the subsidy raises wages uniformly, Q̂TE(τ) ≈ c for all τ.

• If the subsidy compresses the within-firm wage distribution, Q̂TE(τ) is large and positive at low τ (raises the bottom) and near zero or negative at high τ (does not raise or lowers top earners) .

• Testing H₀ : QTE(τ) = QTE(0.5) for all τ formally tests whether the effect is heterogeneous across the distribution.

The standard mean-focused RDD might conclude the subsidy has “no effect” if the average wage is unchanged, while the distributional analysis reveals that the subsidy redistributedwithin the firm.

7 Connections to the Literature

Distributional RDD builds on several strands of earlier work:

• IV quantile treatment effects. Chernozhukov and Hansen [2005] develop quantile treatment effects under IV. using a local-to-the-instrument framework analogous to thelocal-to-the-cutoff framework of RDD.

• Distributional DiD. Callaway and Li [2019] extend the difference-in-differences estimator to quantile treatment effects. identifying the quantile analogue of the ATT underconditional parallel trends.

• Local randomisation RDD. In the local randomisation framework [Cattaneo et al., 2015], units near the cutoff are treated as if they were randomly assigned. Under localrandomisation, Fisher’s exact test and randomisation inference for distributional effectsfollow naturally.

8 Software

The rdrobust package for R and Stata [Calonico et al., 2014] provides the infrastructure for standard RDD estimation. Extensions for distributional RDD are available in development versions linked to the work of Cattaneo and collaborators; the rddensity and rdlocrand packages provide supporting tools. for density tests and local randomisation inference. Researchers implementing distributional RDD currently need to combine local polynomial CD Festimation (feasible with base R or locpol) with the robust inference procedures describedabove.

9 Conclusion

Regression discontinuity with distribution-valued outcomes extends one of applied economics' most credible designs to questions about inequality, distributional effects, and tail outcomes. The key insight is simple: replace the conditional mean with the conditional CDF in the local polynomial estimation framework, and a rich set of distributional treatment effects follows. Bandwidth selection, bias correction, and uniform confidence bands require additional methodological care, but the framework is theoretically coherent and practically implementable. As policy-makers increasingly focus on distributional consequencesof interventions—not just average effects—this extension of the RDD toolkit addresses a genuine gap in the applied econometrics literature.

References

1. Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica, 82(6):2295-2326.
2. Callaway, B. and Li, T. (2019). Quantile treatment effects in difference in differences models under dependence restrictions and with only two time periods. Journal of Econometrics, 206(2):395-413.
3. Cattaneo, M. D., Frandsen, B. R., and Titiunik, R. (2015). Randomization inference in the regression discontinuity design: An application to party advantages in the U.S. Senate. Journal of Causal Inference, 3(1):1-24.
4. Cattaneo, M. D., Titiunik, R., and Vazquez-Bare, G. (2025). Regression discontinuity design with distribution-valued outcomes. arXiv: 2504.03992.
5. Chernozhukov, V. and Hansen, C. (2005). An IV model of quantile treatment effects. Econometrica, 73(1):245-261.
6. Imbens, G. W. and Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. Journal of Econometrics, 142(2):615-635.
7. Imbens, G. W. and Rubin, D. B. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences. Cambridge University Press.