Nonparametric tests for model selection with time series data

We consider a test for the selection of variables/covariates in a time series regression model based on a L-2 - measure of global deviation between the nonparametric estimates of the regression model obtained under the null and alternative hypotheses. Thus, the test can be viewed in the context of dimension reduction. Moreover, the test only requires, unlike others proposed for the same hypothesis testing problem, the choice of one bandwidth parameter. We show that our test has power against contiguous alternatives that converge to the null at a rate T-alpha, in contrast to alternative tests whose rates are T-alpha 1, where 1/4 < alpha 1 < alpha < 1/2. Thus the asymptotic relative efficiency of their test compared to ours is zero. Finally, the test is extended to the situation when the null follows a parametric model up to a finite set of parameters.