Some comments on specification tests in nonparametric absolutely regular processes

In this note several aspects of specification tests in nonparametric models driven by an absolutely regular process are discussed, which were recently proprosed in the literature. 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 of goodness-of-fit tests are incorrectly stated in the literature. Moreover, our result indicates that recent results on the asymptotic equivalence between nonparametric autoregression and nonparametric regression cannot be used for the asymptotic analysis of goodness-of-fit tests under fixed alternatives.