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 and derive adjustments to the large sample variances of propensity score matching estimators of the average treatment effect (ATE) and the average treatment effect on the treated (ATET). The adjustment for the ATE estimator is negative (or zero in some special cases), implying that matching on the estimated propensity score is more efficient than matching on the true propensity score in large samples. However, for the ATET estimator the sign of the adjustment term depends on the data generating process, and ignoring the estimation error in the propensity score may lead to confidence intervals that are either too large or too small. Finally, we present the results from a simulation exercise that illustrate the implications of our theoretical results. Alberto Abadie, John F. Kennedy School of Government, 79 John F. Kennedy Street, Cambridge MA 02138, USA. E-mail: alberto [email protected], web: http://www.hks.harvard.edu/fs/aabadie/. Guido W. Imbens, Stanford Graduate School of Business, 655 Knight Way Stanford, CA 94305-7298. E-mail: [email protected]. We are grateful to Ben Hansen, James Robins, Paul Rosenbaum, Donald Rubin, and participants in many seminars for comments and discussions. Financial support by the NSF through grants SES 0820361 and 0961707 is gratefully acknowledged. Software implementing these methods is available from the authors.