Program Evaluation with High-Dimensional Data

We consider estimation and inference on policy relevant treatment effects, such as local average and quantile treatment effects, in a data-rich environment where there may be many more control variables available than there are observations. The setting is designed to handle endogenous receipt of treatment, heterogeneous treatment effects, and possibly function-valued outcomes. We require that the relationship between the control variables and the outcome, treatment status, and instrument status can be captured up to a small approximation error by a small number of the control variables whose identities are unknown. We provide conditions under which post-selection inference is uniformly valid across a wide-range of data generating processes (dgps) and show that a key condition underlying the uniform validity of post-selection inference allowing for imperfect model selection is the use of orthogonal moment conditions. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) participation on accumulated assets. We also present a generalization of the treatment effect framework to a much richer setting, where possibly a continuum of target parameters is of interest and Lasso or Post-Lasso type methods are used to estimate a continuum of high-dimensional nuisance functions. We establish functional central limit theorems for the continuum of target parameters and for the multiplier bootstrap that holds uniformly over dgps. We propose a notion of the functional delta method that allows us to derive approximate distributions for smooth functionals of a continuum of target parameters and to establish the validity of the multiplier bootstrap for approximating these distributions uniformly over dgps. Finally, we establish rate results for continua of Lasso or Post-Lasso type estimators for continua of (nuisance) regression functions.