Validation of linear regression models

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 subsp ace U spanned by the regression functions to be verified. The asymptotic di stribution of the test statistic is shown to be normal with parameters depe nding 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 t est is proposed for the hypothesis that "g is not in a preassigned L-2-neig hborhood of U," which allows the "verification" of the model U at a control led type I error rate. The suggested procedure is very easy to apply becaus e of its asymptotic normal law and the simple form of the test statistic. I n particular, it does not require nonparametric estimators of the regressio n function and hence, the test does not depend on the subjective choice of smoothing parameters.