On the consistency of bootstrap for matching estimators

TL;DR

This paper demonstrates that the naive bootstrap becomes consistent for matching estimators when the number of matches increases, resolving previous concerns about its inconsistency with fixed matches.

Contribution

It proves that the naive bootstrap is consistent for matching estimators when the number of matches diverges, clarifying the source of earlier inconsistency issues.

Findings

Naive bootstrap is consistent with diverging matches

Inconsistency arises only with fixed number of matches

Provides theoretical clarification on bootstrap behavior

Abstract

In a landmark paper, Abadie and Imbens (2008) showed that the naive bootstrap is inconsistent when applied to nearest neighbor matching estimators of the average treatment effect with a fixed number of matches. Since then, this finding has inspired numerous efforts to address the inconsistency issue, typically by employing alternative bootstrap methods. In contrast, this paper shows that the naive bootstrap is provably consistent for the original matching estimator, provided that the number of matches, $M$, diverges. The bootstrap inconsistency identified by Abadie and Imbens (2008) thus arises solely from the use of a fixed $M$.