Some comments on specification tests in nonparametric absolutely regular processes

In this note, several aspects of a recently proposed specification test in nonparametric models driven by an absolutely regular process are discussed. In particular, we give a more detailed asymptotic analysis of tests based on kernel methods under fixed alternatives using a central limit theorem for U-statistics with n-dependent nondegenerate kernel. As a by-product, it is demonstrated that several results regarding the asymptotic distribution or goodness-of-fit tests are incorrectly stated in the literature. Our result also indicates that results on the asymptotic equivalence of nonparametric autoregression and nonparametric regression cannot be used for the asymptotic analysis of goodness-of-fit tests under fixed alternatives. Copyright 2004 Blackwell Publishing Ltd.