High-Dimensional Sparse Econometric Models, an Introduction

In this chapter we discuss conceptually high dimensional sparse econometric models as well as estimation of these models using l1-penalization and postl1-penalization methods. Focusing on linear and nonparametric regression frameworks, we discuss various econometric examples, present basic theoretical results, and illustrate the concepts and methods with Monte Carlo simulations and an empirical application. In the application, we examine and confirm the empirical validity of the Solow-Swan model for international economic growth. 1 The High Dimensional Sparse Econometric Model We consider linear, high dimensional sparse (HDS) regression models in econometrics. The HDS regression model has a large number of regressors p, possibly much larger than the sample size n, but only a relatively small number s < n of these regressors are important for capturing accurately the main features of the regression function. The latter assumption makes it possible to estimate these models effectively by searching for approximately the right set of the regressors, using l1-based penalization methods. In this chapter we will review the basic theoretical properties of these procedures, established in the works of [8, 10, 18, 17, 7, 15, 13, 27, 26], among others (see [20, 7] for a detailed literature review). In this section, we review the modeling foundations as well as motivating examples for these procedures, with emphasis on applications in econometrics. Let us first consider an exact or parametric HDS regression model, namely, Alexandre Belloni Duke University, Fuqua School of Business, 100 Fuqua Drive, Durham, NC, e-mail: [email protected] Victor Chernozhukov Massachusetts Institute of Technology, Department of Economics, 50 Memorial Drive, Cambridge, MA e-mail: [email protected]