Before revision:  We prove a consistency rate result of our estimatorin the partially linear regression model, and en route we provide a consistency analysis for a general framework of performing generalized method of moments (GMM) estimation .
After revision:  We prove a consistency rate and establish asymptotic normality for our estimator. We show that under mild assumption on the consistency rate of the nuisance estimator, we can achieve the same error rate as an oracle with a priori knowledge of these nuisance parameters. We show that when the nuisance functions have a locally sparse parametrization, then a local l1-penalized regression achieves the required rate. We apply our method to estimate heterogeneous treatment effects from observational data with discrete treatments or continuous treatments, and we show that, unlike prior work, our method provably allows to control for a high-dimensional set of variables under standard sparsity conditions .