Causal Inference with Groupwise Matching

TL;DR

This paper develops a unified framework for causal inference using groupwise matching, connecting methods like difference-in-differences and synthetic control through generalized matching conditions, and introduces a new inference procedure.

Contribution

It generalizes the parallel trend assumption, unifies existing methods under a common framework, and proposes a new statistical inference approach for synthetic control with differencing.

Findings

Difference-in-differences and synthetic control are complementary methods.

Synthetic control with differencing is equivalent to difference-in-differences under parallel trends.

The new inference procedure demonstrates practical usefulness in empirical applications.

Abstract

This paper examines methods of causal inference based on groupwise matching when we observe multiple large groups of individuals over several periods. We formulate causal inference validity through a generalized matching condition, generalizing the parallel trend assumption in difference-in-differences designs. We show that difference-in-differences, synthetic control, and synthetic difference-in-differences designs are distinguished by the specific matching conditions that they invoke. Through regret analysis, we demonstrate that difference-in-differences and synthetic control with differencing are complementary; the former dominates the latter if and only if the latter's extrapolation error exceeds the former's matching error up to a term vanishing at the parametric rate. The analysis also reveals that synthetic control with differencing is equivalent to difference-in-differences when the parallel trend assumption holds for both the pre-treatment and post-treatment periods. We develop a statistical inference procedure based on synthetic control with differencing and present an empirical application demonstrating its usefulness.