In this
paper, we address the problem of estimating the average treatment
effect (ATE) and the average treatment effect for the treated (ATT)
in observational studies when the number of potential confounders is
possibly much greater than the sample size. In particular, we
develop a robust method to estimate the propensity score via
covariate balancing in high-dimensional settings. Since it is
usually impossible to obtain the exact covariate balance in high
dimension, we propose to estimate the propensity score by balancing
a carefully selected subset of covariates that are predictive of the
outcome under the assumption that the outcome model is linear and
sparse. The estimated propensity score is, then, used for the
Horvitz-Thompson estimator to infer the ATE and ATT. We prove that
the proposed methodology has the desired properties such as sample
boundedness, root-$n$ consistency, asymptotic normality, and
semiparametric efficiency. We then extend these results to the case
where the outcome model is a sparse generalized linear model. In
addition, we show that the proposed estimator remains root-$n$
consistent and asymptotically normal even when the propensity score
model is misspecified. Finally, we conduct simulation studies to
evaluate the finite-sample performance of the proposed methodology,
and apply it to estimate the causal effects of college attendance on
adulthood political participation.
Open-source software is
available for implementing the proposed methodology. (Last Revised,
June 2017)