Improving Covariate Balancing Propensity Score: A Doubly Robust and Efficient Approach

Inverse probability of treatment weighting (IPTW) is a popular method for estimating causal effects in many disciplines. However, empirical studies show that the IPTW estimators can be sensitive to the misspecification of propensity score model. To address this problem, several researchers have proposed new methods to estimate propensity score by directly optimizing the balance of pre-treatment covariates. While these methods appear to empirically perform well, little is known about their theoretical properties. This paper makes two main contributions. First, we conduct a theoretical investigation of one such methodology, the Covariate Balancing Propensity Score (CBPS) recently proposed by Imai and Ratkovic (2014). We characterize the asymptotic bias and efficiency of the CBPS-based IPTW estimator under both arbitrary and local model misspecification as well as correct specification for general balancing functions. Based on this finding, we address an open problem in the literature on how to optimally choose the covariate balancing function for the CBPS methodology. Second, motivated by the form of the optimal covariate balancing function, we further propose a new IPTW estimator by generalizing the CBPS method. We prove that the proposed estimator is consistent if either the propensity score model or the outcome model is correct. In addition to this double robustness property, we also establish that the proposed estimator is semiparametrically efficient when both the propensity score and outcome models are correctly specified. Unlike the standard doubly robust estimators, however, the proposed methodology does not require the estimation of outcome model. To relax the parametric assumptions on the propensity score model and the outcome model, we further consider a sieve estimation approach to estimate the treatment effect. A new “nonparametric double robustness” phenomenon is observed. Our simulations show that the proposed estimator has better finite sample properties than the standard estimators.