Specification tests indexed by bandwidths

In this note we consider several goodness-of-fit tests for model specification in non- parametric regression models which are based on kernel methods. In order to circumvent the problem of choosing a bandwidth for the corresponding test statistic we propose to consider the statistics as stochastic processes indexed with bandwidths proportional to the asymptotically optimal bandwidth for the estimation of the regression function. We prove weak convergence of these processes to centered Gaussian processes and suggest to use functionals of these processes as test statistics for the problem of model specification. A bootstrap test is proposed to obtain a good approximation of the nominal level. The results are illustrated by means of a simulation study and the new test is compared with some of the currently available procedures.