Cross-Validation for High-Dimensional and Nonparametric Models in Econometrics

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

It is now well-known in empirical economics that nonparametric/high-dimensional methods are important because these methods substantially reduce misspecification of the economic models, thereby minimizing the possibility of inconsistent estimation of model parameters. The implementation of nonparametric/high-dimensional methods, however, typically requires the practitioners to select a smoothing parameter. Nevertheless, adequate data-driven procedures for selecting smoothing parameters are not readily available, which might explain the slow adoption of nonparametric methods in applied work. In addition, estimators based on an inappropriate choice of the smoothing parameter may have large estimation or approximation error, and hence lead to detrimental mistakes in policy recommendations. This project therefore focuses on how to select a smoothing parameter for Lasso estimators and sieve estimators, because Lasso estimators are commonly used to estimate high-dimensional models using "big data" and machine learning techniques, and sieve estimators are one of important approaches to estimate non-parametric models. The investigators will derive theoretical results for these estimators and provide theoretically justified procedures to select the smoothing parameter. This project investigates two lines of research: first, convergence rates of cross-validated Lasso estimator, and second, convergence rates of cross-validated sieve estimator. In the first line of research, the investigators will provide novel and practical cross-validation-based procedure to choose the regularization parameter for the Lasso estimator and show that the cross-validated Lasso estimator achieves the fastest possible rate of convergence up-to the logarithmic factor under certain conditions. In the second line of research, the investigators demonstrate that cross-validation produces sieve estimators with optimal rates of convergence for a large class of sieve estimators. This project will thus derive theoretical results for a large class of non-linear nonparametric/high-dimensional estimators when the smoothing/regularization parameter for the estimators is selected using cross-validation. Therefore, this project will provide a data-driven procedure for selecting the smoothing parameter, which should be of interest to researchers in diverse academic fields as well as in industry.