Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients

Abstract

We consider a two-step projection based Lasso procedure for estimating a partially linear regression model where the number of coefficients in the linear component can exceed the sample size and these coefficients belong to the $l_{q}$-“balls” for $q\in[0,1]$. Our theoretical results regarding the properties of the estimators are nonasymptotic. In particular, we establish a new nonasymptotic “oracle” result: Although the error of the nonparametric projection per se (with respect to the prediction norm) has the scaling $t_{n}$ in the first step, it only contributes a scaling $t_{n}^{2}$ in the $l_{2}$-error of the second-step estimator for the linear coefficients. This new “oracle” result holds for a large family of nonparametric least squares procedures and regularized nonparametric least squares procedures for the first-step estimation and the driver behind it lies in the projection strategy. We specialize our analysis to the estimation of a semiparametric sample selection model and provide a simple method with theoretical guarantees for choosing the regularization parameter in practice.