  It is important to make robust inference of the conditional average treatment
effect from observational data, but this becomes challenging when the
confounder is multivariate or high-dimensional. In this article, we propose a
double dimension reduction method, which reduces the curse of dimensionality as
much as possible while keeping the nonparametric merit. We identify the central
mean subspace of the conditional average treatment effect using dimension
reduction. A nonparametric regression with prior dimension reduction is also
used to impute counterfactual outcomes. This step helps improve the stability
of the imputation and leads to a better estimator than existing methods. We
then propose an effective bootstrapping procedure without bootstrapping the
estimated central mean subspace to make valid inference.
