  Consider the problem of estimating average treatment effects when a large
number of covariates are used to adjust for possible confounding through
outcome regression and propensity score models. The conventional approach of
model building and fitting iteratively can be difficult to implement, depending
on ad hoc choices of what variables are included. In addition, uncertainty from
the iterative process of model selection is complicated and often ignored in
subsequent inference about treatment effects. We develop new methods and theory
to obtain not only doubly robust point estimators for average treatment
effects, which remain consistent if either the propensity score model or the
outcome regression model is correctly specified, but also model-assisted
confidence intervals, which are valid when the propensity score model is
correctly specified but the outcome regression model may be misspecified. With
a linear outcome model, the confidence intervals are doubly robust, that is,
being also valid when the outcome model is correctly specified but the
propensity score model may be misspecified. Our methods involve regularized
calibrated estimators with Lasso penalties, but carefully chosen loss
functions, for fitting propensity score and outcome regression models. We
provide high-dimensional analysis to establish the desired properties of our
methods under comparable conditions to previous results, which give valid
confidence intervals when both the propensity score and outcome regression are
correctly specified. We present a simulation study and an empirical application
which confirm the advantages of the proposed methods compared with related
methods based on regularized maximum likelihood estimation.
