Optimal Covariate Balancing Conditions in Propensity Score Estimation

Inverse probability of treatment weighting (IPTW) is a popular method for estimating the average treatment effect (ATE). However, empirical studies show that the IPTW estimators can be sensitive to the misspecification of the propensity score model. To address this problem, researchers have proposed to estimate propensity score by directly optimizing the balance of pretreatment covariates. While these methods appear to empirically perform well, little is known about how the choice of balancing conditions affects their theoretical properties. To fill this gap, we first characterize the asymptotic bias and efficiency of the IPTW estimator based on the Covariate Balancing Propensity Score (CBPS) methodology under local model misspecification. Based on this analysis, we show how to optimally choose the covariate balancing functions and propose an optimal CBPS-based IPTW estimator. This estimator is doubly robust; it is consistent for the ATE if either the propensity score model or the outcome model is correct. In addition, the proposed estimator is locally semiparametric efficient when both models are correctly specified. To further relax the parametric assumptions, we extend our method by using a sieve estimation approach. We show that the resulting estimator is globally efficient ∗Supported by NSF grants DMS-1854637, DMS-1662139 and DMS-1712591 and NIH grant R01-GM072611-12. An earlier version of this paper is entitled, “Improving Covariate Balancing Propensity Score: A Doubly Robust and Efficient Approach.” †Department of Operations Research and Financial Engineering, Princeton University ‡Department of Government and Department of Statistics, Harvard University §Department of Electrical Engineering and Computer Science, Northwestern University. ¶Department of Statistics and Data Science, Cornell University ‖Amazon

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