NONASYMPTOTIC ANALYSIS OF SEMIPARAMETRIC REGRESSION MODELS WITH HIGH-DIMENSIONAL PARAMETRIC COEFFICIENTS

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 is an element of [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.