Matched Triple-Differences: A Framework for Covariate Adjustment

2026-10-03

AI Summary

This study addresses the inconsistency in average treatment effect estimation caused by covariate adjustment in conventional difference-in-differences-in-differences (DDD) models. To resolve this, we propose a matching-based triple differences framework that eliminates bias inherent in three-way fixed effects regression through a generalized matching procedure. The core contributions include constructing a consistent matching estimator with established asymptotic normality and developing variance estimation that accounts for matching variability. Furthermore, this work integrates nearest-neighbor and kernel matching approaches to refine econometric inference theory. Empirical applications demonstrate that both the estimates and qualitative conclusions derived from the proposed method differ substantially from those obtained via traditional DDD, thereby providing a more robust analytical tool for causal inference.

Abstract

A common empirical strategy in triple-differences (DDD) is to include either covariate trends or covariate levels to a three-way fixed effects (3WFE) regression. This strategy is typically motivated by the conditional parallel gaps assumption which assumes that deviations from parallel trends are similar among units with comparable observed covariates. We formally study both 3WFE specifications and show that, in general, neither consistently estimates the average treatment effect on the treated (ATT). Our diagnosis has two parts. First, we show that the OLS estimands of both specifications fail to satisfy the covariate balancing condition that is sufficient for them to equal the ATT. Second, we characterize the additional assumptions for each OLS estimand to equal the ATT. To address these limitations, we propose a matched triple-differences framework that accommodates a general class of matching procedures, including nearest-neighbor matching and kernel matching. Within this framework, we construct a class of consistent matching estimators, establish their asymptotic normality, and provide a consistent variance estimator that accounts for the variability introduced by the matching step. Our empirical application shows that the proposed matched triple-differences estimator can yield estimates and qualitative conclusions that differ from those obtained using the 3WFE regression.

Problem

• Triple-Differences
• Covariate Adjustment
• Three-Way Fixed Effects
• Average Treatment Effect on the Treated

Innovation

• Triple-Differences
• Covariate Adjustment
• Matching Estimator
• Asymptotic Normality
• Treatment Effect