This chapter reviews the asymptotic properties of the Nadaraya-Watson type kernel estimator of an unknown (multivariate) regression function. Conditions are set forth for pointwise weak and strong consistency, asymptotic normality, and uniform consistency. These conditions cover the standard i.i.d. case with continuously distributed regressors, as well as the cases where the distribution of all, or some, regressors is discrete. Moreover, attention is paid to the problem of how the kernel and the window width should be specified. This chapter is a modified and extended version of Bierens (1987b). For further reading and references, see the monographs by Eubank (1988), Hardle (1990), and Rosenblatt (1991), and for an empirical application, see Bierens and Pott-Buter (1990).

Introduction

The usual practice in constructing regression models is to specify a parametric family for the response function. Obviously the most popular parametric family is the linear model. However, one could consider this as choosing a parametric functional form from a continuum of possible functional forms, analogously to sampling from a continuous distribution, for often the set of theoretically admissible functional forms is uncountably large. Therefore the probability that we pick the true functional form in this way is zero, or at least very close to zero.

The only way to avoid model misspecification is to specify no functional form at all. But then the problem arises how information about the functional form of the model can be derived from the data. A possible solution to this problem is to use so-called kernel estimators of regression functions.