Matching on the Estimated Propensity Score

Propensity score matching estimators (Rosenbaum and Rubin, 1983) are widely used in evaluation research to estimate average treatment effects. In this article, we derive the large sample distribution of propensity score matching estimators. Our derivations take into account that the propensity score is itself estimated in a first step, prior to matching. We prove that first step estimation of the propensity score affects the large sample distribution of propensity score matching estimators. Moreover, we derive an adjustment to the large sample variance of propensity score matching estimators that corrects for first step estimation of the propensity score. In spite of the great popularity of propensity score matching estimators, these results were previously unavailable in the literature.