  In this paper, we propose a robust method to estimate the average treatment
effects in observational studies when the number of potential confounders is
possibly much greater than the sample size. We first use a class of penalized
M-estimators for the propensity score and outcome models. We then calibrate the
initial estimate of the propensity score by balancing a carefully selected
subset of covariates that are predictive of the outcome. Finally, the estimated
propensity score is used to construct the inverse probability weighting
estimator. We prove that the proposed estimator, which has the sample
boundedness property, is root-n consistent, asymptotically normal, and
semiparametrically efficient when the propensity score model is correctly
specified and the outcome model is linear in covariates. More importantly, we
show that our estimator remains root-n consistent and asymptotically normal so
long as either the propensity score model or the outcome model is correctly
specified. We provide valid confidence intervals in both cases and further
extend these results to the case where the outcome model is a generalized
linear model. In simulation studies, we find that the proposed methodology
often estimates the average treatment effect more accurately than the existing
methods. We also present an empirical application, in which we estimate the
average causal effect of college attendance on adulthood political
participation. Open-source software is available for implementing the proposed
methodology.
