3 - Kernel estimators of regression functions

Summary

Abstract This chapter reviews the asymptotic properties of the Nadaraya-Watson kernel estimator of an unknown (multivariate) regression function. Conditions are set forth for pointwise asymptotic normality and uniform weak consistency. These conditions cover the standard i.i.d. case with continuously distributed regressors, as well as the cases that the distribution of all, or some, regressors is discrete and/or the data are generated by a class of strictly stationary time series processes. Moreover, attention is paid to the problem of how the kernel and the window width should be specified. Furthermore, the estimation procedure under review is illustrated by a numerical example.

Introduction

A large extent of applied econometric research involves the specification and estimation of regression models, where in most of the cases the linear regression model is used. The most crucial assumption underlying these models is that they represent the mathematical expectation of the dependent variable conditional on the regressors, which implies that the expectation of the error term conditional on the regressors equals zero with probability 1. If the dependent variable has finite absolute first moment this conditional expectation always exists. Compare Chung (1974, Theorem 9.1.1). Therefore, regression models are either true or false in the sense that they represent conditional expectations given the regressors, or not.