Validation of linear regression models

Abstract

A new test is proposed in order to verify that a regression function, say $g$, has a prescribed (linear) parametric form. This procedure is based on the large sample behavior of an empirical $L^2$-distance between $g$ and the subspace $U$ spanned by the regression functions to be verified. The asymptotic distribution of the test statistic is shown to be normal with parameters depending only on the variance of the observations and the $L^2$-distance between the regression function $g$ and the model space $U$. Based on this result, a test is proposed for the hypothesis that "$g$ is not in a preassigned $L^2$-neighborhood of $U$," whichallows the "verification" of the model $U$ at a controlled type I error rate. The suggested procedure is very easy to apply because of its asymptotic normal law and the simple form of the test statistic. In particular, it does not require nonparametric estimators of the regression function and hence, the test does not depend on the subjective choice of smoothing parameters.