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 pre-treatment 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 under a set of much weaker
assumptions and has a smaller asymptotic bias than the existing
estimators. Finally, we evaluate the finite sample performance of
the proposed estimators via simulation and empirical studies. An
open-source software
package is available for implementing the proposed
methods.