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. Our
method consists of the three steps. 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 we call the
high-dimensional covariate balancing propensity score, 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.
An
open-source software
package is available for implementing the proposed
methodology.