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Regression Discontinuity Design with Distribution-Valued Outcomes

David Van Dijcke

TL;DR

This paper develops Regression Discontinuity Design for distribution-valued outcomes (R3D), defining the Local Average Quantile Treatment Effects (LAQTE) to capture shifts in entire outcome distributions around a treatment cutoff. It introduces two estimators—local polynomial regression on random quantiles and local Fréchet regression in 2-Wasserstein space—along with uniform, debiased confidence bands and data-driven bandwidth selection, with theoretical guarantees and a multiplier bootstrap for inference. Simulations show R3D estimators are less biased and provide valid uniform bands, outperforming quantile RD in this setting. The empirical application using a close-election RD to study gubernatorial control reveals an equality–efficiency trade-off: Democratic governorships tend to reduce high-end income and compress the distribution, while effects at the bottom are milder and often not significant. Overall, R3D extends causal inference to distributional outcomes, enabling nuanced, policy-relevant distribution-centered insights.

Abstract

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fréchet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

Regression Discontinuity Design with Distribution-Valued Outcomes

TL;DR

This paper develops Regression Discontinuity Design for distribution-valued outcomes (R3D), defining the Local Average Quantile Treatment Effects (LAQTE) to capture shifts in entire outcome distributions around a treatment cutoff. It introduces two estimators—local polynomial regression on random quantiles and local Fréchet regression in 2-Wasserstein space—along with uniform, debiased confidence bands and data-driven bandwidth selection, with theoretical guarantees and a multiplier bootstrap for inference. Simulations show R3D estimators are less biased and provide valid uniform bands, outperforming quantile RD in this setting. The empirical application using a close-election RD to study gubernatorial control reveals an equality–efficiency trade-off: Democratic governorships tend to reduce high-end income and compress the distribution, while effects at the bottom are milder and often not significant. Overall, R3D extends causal inference to distributional outcomes, enabling nuanced, policy-relevant distribution-centered insights.

Abstract

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fréchet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

Paper Structure

This paper contains 62 sections, 14 theorems, 135 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Regression Discontinuity with Distribution-Valued Outcomes
  3. Setting
  4. Motivating Examples
  5. Local Average Quantile Treatment Effects
  6. Definition
  7. Identification
  8. Discontinuities in Average Distributions
  9. Comparison to Quantile RDD
  10. Estimators
  11. Local Polynomial Regression on Quantiles
  12. Local Fréchet Regression
  13. Overview of Inference
  14. Extensions
  15. Fuzzy R3D
  16. ...and 47 more sections

Key Result

Lemma 1

Under Assumptions asspt:i_cont and asspt:i_dens, the unobserved $\tau^{\mathrm{R3D}}$ is identified from the joint distribution of the observed $(X,Y)$ as,

Figures (14)

  • Figure 1: Example of a Distribution-Valued RDD
  • Figure 2: Local Polynomial Estimator: Illustration
  • Figure 3: Simulated Bias of R3D and Q-RD Estimators
  • Figure 4: Simulated Variance of R3D Estimators
  • Figure 5: R3D Plot: Average Income Quantiles vs. Democrat Vote Share
  • ...and 9 more figures

Theorems & Definitions (35)

  • Example 1: Administrative units
  • Example 2: Institutions
  • Example 3: Establishments
  • Definition 1: Local Average Quantile Treatment Effects (LAQTE)
  • Example 4
  • proof
  • Lemma 1: Identification
  • Definition 2: Fuzzy LAQTE
  • Lemma 2: Fuzzy Identification
  • Theorem 1: Convergence: Conditional Means
  • ...and 25 more