Inference in Additively Separable Models With a High-Dimensional Set of Conditioning Variables

Abstract

This paper studies nonparametric series estimation and inference for the effect of a single variable of interest x on an outcome y in the presence of potentially high-dimensional conditioning variables z. The context is an additively separable model E[y|x, z] = g0(x) + h0(z). The model is high-dimensional in the sense that the series of approximating functions for h0(z) can have more terms than the sample size, thereby allowing z to have potentially very many measured characteristics. The model is required to be approximately sparse: h0(z) can be approximated using only a small subset of series terms whose identities are unknown. This paper proposes an estimation and inference method for g0(x) called Post-Nonparametric Double Selection which is a generalization of Post-Double Selection. Standard rates of convergence and asymptotic normality for the estimator are shown to hold uniformly over a large class of sparse data generating processes. A simulation study illustrates finite sample estimation properties of the proposed estimator and coverage properties of the corresponding confidence intervals. Finally, an empirical application to college admissions policy demonstrates the practical implementation of the proposed method.