Program evaluation with high-dimensional data

In this paper, 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 we consider 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 highdimensional nuisance functions. As an important leading example of this environment, we first provide detailed results on estimation and inference for relevant treatment effects, such as local average and quantile treatment effects. The setting we work in is designed expressly to handle many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and possibly function-valued outcomes. To make informative inference possible, we assume that key reduced form predictive relationships are approximately sparse. That is, 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 to the researcher. This condition permits estimation and inference to proceed after datadriven 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 illustrate the use of the proposed methods with an application to estimating the effect of 401(k) participation on accumulated assets. We generalize the results from the treatment effects setting to accommodate more general moment condition models in a second part of the paper. In particular, we establish a functional central limit theorem for robust estimators of a continuum of target parameters that holds uniformly in a wide range of data generating processes (dgp's), i.e. over dgp's P € P where P include dgp's where perfect model selection is theoretically impossible. We prove that the use of orthogonal moment conditions is key to achieving uniform validity. We also establish a functional central limit theorem for the multiplier bootstrap that resamples first-order approximations to the proposed estimators that holds uniformly over P € P. We propose a notion of differentiability, together with a functional delta method, that allows us to derive approximate distributions for smooth functionals of a continuum of target parameters that hold uniformly over P € P and to establish the validity of the multiplier bootstrap for approximating these distributions uniformly over P € P. Finally, we establish rate and consistency results for continua of Lasso or Post-Lasso type estimators for continua of (nuisance) regression functions and provide practical, theoretically justified choices for the penalty parameters used in these methods. Each of these results is new and of independent interest.