Distributional Discontinuity Design

TL;DR

This paper introduces distributional discontinuity designs that estimate causal effects on the entire outcome distribution using Wasserstein distance, capturing heterogeneity and extending to kink designs, with applications to real data.

Contribution

It develops a novel framework for distributional causal inference at discontinuities, utilizing Wasserstein distance and its decomposition, and extends to kink designs with new identification results.

Findings

Wasserstein distance bounds average treatment effect under certain conditions.

Decomposition into L-moments quantifies effects on distribution shape components.

Application to real data demonstrates the method's practical utility.

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.