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Inverse probability of treatment weighting (IPTW) is a popular method forestimating the average treatment effect (ATE). However, empirical studies showthat the IPTW estimators can be sensitive to the misspecification of thepropensity score model. To address this problem, researchers have proposed toestimate propensity score by directly optimizing the balance of pre-treatmentcovariates. While these methods appear to empirically perform well, little isknown about how the choice of balancing conditions affects their theoreticalproperties. To fill this gap, we first characterize the asymptotic bias andefficiency of the IPTW estimator based on the Covariate Balancing PropensityScore (CBPS) methodology under local model misspecification. Based on thisanalysis, we show how to optimally choose the covariate balancing functions andpropose an optimal CBPS-based IPTW estimator. This estimator is doubly robust;it is consistent for the ATE if either the propensity score model or theoutcome model is correct. In addition, the proposed estimator is locallysemiparametric efficient when both models are correctly specified. To furtherrelax the parametric assumptions, we extend our method by using a sieveestimation approach. We show that the resulting estimator is globally efficientunder a set of much weaker assumptions and has a smaller asymptotic bias thanthe existing estimators. Finally, we evaluate the finite sample performance ofthe proposed estimators via simulation and empirical studies. An open-sourcesoftware package is available for implementing the proposed methods.

Optimal Covariate Balancing Conditions in Propensity Score Estimation