Program Evaluation with High-Dimensional Data

Abstract

We consider estimation of general modern moment-condition problems in econometrics in a data-rich environment where there may be many more control variables available than there are observations. The framework allows for a continuum of target parameters and for Lasso-type or Post-Lasso type methods to be used as estimators of a continuum of high-dimensional nuisance functions. As an important leading example of this environment, we first provide results on estimation and inference for relevant treatment effects, such as local average and quantile treatment effects. The setting is designed expressly to handle many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and function-valued outcomes. An approximate sparsity condition permits estimation and inference to proceed after data-driven selection of control variables. We provide conditions under which post-selection inference is uniformly valid across a wide-range of models 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 apply the methods to estimate the effect of 401(k) participation on accumulated assets. We also generalize the results from the treatment effects setting to accommodate more general moment condition models. 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. We also establish rate results for continua of Lasso type estimators for continua of regression functions.