Large Sample Properties of Matching Estimators for Average Treatment Effects

TLDR

Matching estimators for average treatment effects are widely used, yet their large‑sample properties remain largely unestablished because standard asymptotic expansions fail for fixed‑match estimators, which are highly nonsmooth functionals of the data. The article develops new methods to analyze the large‑sample properties of matching estimators and establishes several new results. The authors focus on matching with replacement using a fixed number of matches, provide a consistent variance estimator that bypasses the need for consistent nonparametric estimation, and offer software implementations in Matlab, Stata, and R. They show that matching estimators are generally not N1/2‑consistent, identify conditions for N1/2‑consistency, demonstrate that even when consistent they fail to reach the semiparametric efficiency bound, and provide a consistent variance estimator.

Abstract

Matching estimators for average treatment effects are widely used in evaluation research despite the fact that their large sample properties have not been established in many cases. The absence of formal results in this area may be partly due to the fact that standard asymptotic expansions do not apply to matching estimators with a fixed number of matches because such estimators are highly nonsmooth functionals of the data. In this article we develop new methods for analyzing the large sample properties of matching estimators and establish a number of new results. We focus on matching with replacement with a fixed number of matches. First, we show that matching estimators are not N1/2-consistent in general and describe conditions under which matching estimators do attain N1/2-consistency. Second, we show that even in settings where matching estimators are N1/2-consistent, simple matching estimators with a fixed number of matches do not attain the semiparametric efficiency bound. Third, we provide a consistent estimator for the large sample variance that does not require consistent nonparametric estimation of unknown functions. Software for implementing these methods is available in Matlab, Stata, and R.