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Distributional Discontinuity Design

Kyle Schindl, Larry Wasserman

Abstract

Regression discontinuity and kink designs are typically analyzed through mean effects, even when treatment changes the shape of the entire outcome distribution. To address this, we introduce distributional discontinuity designs, a framework for estimating causal effects for a scalar outcome at the boundary of a discontinuity in treatment assignment. Our estimand is the Wasserstein distance between limiting conditional outcome distributions; a single scale-interpretable measure of distribution shift. We show that this weakly bounds the average treatment effect, where equality holds if and only if the treatment effect is purely additive; thus, departure from equality measures effect heterogeneity. To further encode effect heterogeneity we show that the Wasserstein distance admits an orthogonal decomposition into squared differences in $L$-moments, thereby quantifying the contribution from location, scale, skewness, and higher-order shape components to the overall distributional distance. Next, we extend this framework to distributional kink designs by evaluating the Wasserstein derivative at a policy kink; this describes the flow of probability mass through the kink. In the case of fuzzy kink designs, we derive new identification results. Finally, we apply our methods on real data by re-analyzing two natural experiments to compare our distributional effects to traditional causal estimands.

Distributional Discontinuity Design

Abstract

Regression discontinuity and kink designs are typically analyzed through mean effects, even when treatment changes the shape of the entire outcome distribution. To address this, we introduce distributional discontinuity designs, a framework for estimating causal effects for a scalar outcome at the boundary of a discontinuity in treatment assignment. Our estimand is the Wasserstein distance between limiting conditional outcome distributions; a single scale-interpretable measure of distribution shift. We show that this weakly bounds the average treatment effect, where equality holds if and only if the treatment effect is purely additive; thus, departure from equality measures effect heterogeneity. To further encode effect heterogeneity we show that the Wasserstein distance admits an orthogonal decomposition into squared differences in -moments, thereby quantifying the contribution from location, scale, skewness, and higher-order shape components to the overall distributional distance. Next, we extend this framework to distributional kink designs by evaluating the Wasserstein derivative at a policy kink; this describes the flow of probability mass through the kink. In the case of fuzzy kink designs, we derive new identification results. Finally, we apply our methods on real data by re-analyzing two natural experiments to compare our distributional effects to traditional causal estimands.
Paper Structure (34 sections, 10 theorems, 185 equations, 5 figures, 3 tables)

This paper contains 34 sections, 10 theorems, 185 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setup & Notation
  3. Distributional Discontinuity Design
  4. Identification
  5. Interpretation
  6. Relation to the Average Treatment Effect
  7. Visualizing Quantile Effect Curves
  8. Direction of the Treatment Effect
  9. Decomposition into $L$-Moments
  10. Estimation and Asymptotics
  11. Statistical Inference
  12. Testing the Null Hypothesis
  13. Constructing Confidence Intervals
  14. Fuzzy Distributional Discontinuity Design
  15. Estimation of the Fuzzy Wasserstein Effect
  16. ...and 19 more sections

Key Result

Lemma 1

Under assumptions $(i)$-$(iv)$ by FRANDSEN2012382 it follows that where $Q_{1}(u) = \text{inf} \, \{ y : \text{lim}_{x \downarrow x_0} F_{Y \mid X} (y \! \mid \! x) \geq u\}$ and $Q_{0}(u) = \text{inf} \, \{ y : \text{lim}_{x \uparrow x_0} F_{Y \mid X} (y \! \mid \! x) \geq u\}$ are the limiting conditional quantiles of $Y \mid X = x$ above and below the cutoff.

Figures (5)

  • Figure 1: Optimal transport maps between counterfactual distributions.
  • Figure 2: Counterfactual distributions above and below a treatment discontinuity
  • Figure 3: Quantile effect curves (left panel) and contribution curves (right panel) for a hypothetical null effect curve (solid) and a skewed effect curve (dashed). Both effect curves are defined such that the average treatment effect $\tau = 0$.
  • Figure 4: Example of a hypothetical regression kink design.
  • Figure 5: Monte Carlo confidence interval widths and coverage for bootstrap intervals (dashed) and simple intervals (solid) across trimming levels $\gamma$ and sample sizes $n$.

Theorems & Definitions (21)

  • Lemma 1: Identification
  • Theorem 1: Effect Inequality
  • Theorem 2: $L$-Moment Decomposition
  • Remark 1: Conditioning on covariates
  • Theorem 3: Eigenvalue Test
  • Corollary 1: Eigenvalue Decay
  • Proposition 1: Conservative Test
  • Lemma 2: Conservative Interval
  • Remark 2: Counterfactual Definition
  • Lemma 3: Fuzzy Structural Representation
  • ...and 11 more