A note on the Bickel-Rosenblatt test in autoregressive time series

In a recent paper Lee and Na [2002. Statist. Probab. Lett. 56(1), 23-25] introduced a test for the parametric form of the distribution of the innovations in autoregressive models, which is based on the integrated squared error of the nonparametric density estimate from the residuals and a smoothed version of the parametric fit of the density. They derived the asymptotic distribution under the null-hypothesis, which is the same as for the classical Bickel-Rosenblatt [1973. Ann. Statist. 1, 1071-1095] test for the distribution of i.i.d. observations. In this note we first extend the results of Bickel and Rosenblatt to the case of fixed alternatives, for which asymptotic normality is still true but with a different rate of convergence. As a by-product we also provide an alternative proof of the Bickel and Rosenblatt result under substantially weaker assumptions on the kernel density estimate. As a further application we derive the asymptotic behaviour of Lee and Na's statistic in autoregressive models under fixed alternatives. The results can be used for the calculation of the probability of the type II error if the Bickel-Rosenblatt test is used to check the parametric form of the error distribution or to test interval hypotheses in this context.