A Test for Treatment Heterogeneity under a Distributional Difference-in-Difference Framework

2026-06-19

AI Summary

This study addresses a key limitation of traditional difference-in-differences (DiD) methods, which focus solely on mean effects and fail to capture heterogeneous treatment impacts across the entire outcome distribution—including shifts in location, scale, shape, and tails. To overcome this, the authors propose a distributional DiD framework that leverages optimal transport to estimate the distributional shift in the control group before and after treatment and then transports this shift to the treated group’s baseline distribution to construct a counterfactual. Building on this, they develop the first nonparametric omnibus test capable of jointly detecting multidimensional distributional changes by employing Maximum Mean Discrepancy (MMD) in a reproducing kernel Hilbert space. Theoretical results establish the asymptotic null distribution and Pitman local power under alternatives. Simulations and an empirical application to the Card–Krueger minimum wage study demonstrate the method’s ability to uncover significant distributional effects missed by conventional DiD.

Abstract

We develop a novel distributional Difference-in-Differences (DiD) framework to capture treatment heterogeneity across outcome distributions. By leveraging optimal transport, we use the control group to estimate the untreated distributional drift from the pre- to post-treatment period and apply it to the treated group's pre-treatment baseline, constructing a counterfactual distribution under the assumption of no treatment effect. We frame the null hypothesis as a distributional equality between the transported counterfactual distribution and the observed treated post-treatment distribution, and test it using a maximum mean discrepancy statistic in a reproducing kernel Hilbert space (RKHS). The resulting nonparametric omnibus test is sensitive to changes in location, scale, shape, and tail behavior. Under the null, we derive the asymptotic Gaussian quadratic-form limit of the test statistic, while under local alternatives, we provide a unified characterization of power that establishes its Pitman local power and moderate-deviation consistency. Our theory reveals how detectability is shaped by the interaction between transport-induced drift and RKHS geometry. Simulations and an application to the Card--Krueger minimum-wage data demonstrate that the proposed method identifies key distributional treatment effects missed by classical mean-based DiD.

Problem

• treatment heterogeneity
• distributional difference-in-differences
• counterfactual distribution
• nonparametric test
• optimal transport

Innovation

• distributional Difference-in-Differences
• optimal transport
• maximum mean discrepancy
• reproducing kernel Hilbert space
• treatment heterogeneity