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Estimation of average treatment effects on the treated (ATT) is an importanttopic of causal inference in econometrics and statistics. This problem seems tobe often treated as a simple modification or extension of that of estimatingoverall average treatment effects (ATE). However, the propensity score is nolonger ancillary for estimation of ATT, in contrast with estimation of ATE. Inthis article, we review semiparametric theory for estimation of ATT and the useof efficient influence functionsto derive augmented inverse probabilityweighted (AIPW) estimators that are locally efficient and doubly robust.Moreover, we discuss improved estimation over AIPW by developing calibratedregression and likelihood estimators that are not only locally efficient anddoubly robust, but also intrinsically efficient in achieving smaller variancesthan AIPW estimators when a propensity score model is correctly specified butan outcome regression model may be misspecified. Finally, we present twosimulation studies and an econometric application to demonstrate the advantageof the proposed methods when compared with existing methods.

Improved Estimation of Average Treatment Effects on the Treated: Local Efficiency, Double Robustness, and Beyond