Goodness-of-fit Test Using Residuals In Infinite-order Nonlinear Autoregressive Models

It is of importance to consider the goodness-of-fit tests of the error distribution in time series models. In autoregressive time series models, the goodness-of-fit tests based on the residual empirical process has been extensively studied by many authors including Boldin(1982) and Koul(2002).Recently, Lee and Na(2002) considered the Bickel-Rosenblatt test based on the integrated squared error of the true error density function and kernel density esti-mate from the residuals in linear autoregressive models. They derived the asymptotic distribution under the null-hypothesis, which is the same as for the classical Bickel-Rosenblatt test for the distribution of i.i.d observations. Bachmann and Dette(2005) extended 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 further application,they derived the asymptotic behavior of the test statistic in autoregressive models under fixed alternatives.In this paper, we extend the results, from the set up of linear autoregressive models to the case of infinite-order nonlinear autoregressive time series models. We study the asymptotic behavior of the Bickel-Rosenblatt test statistic. It is proved that without knowing the nonlinear autoregressive function, and under some conditions, the test statistic behaves asymptotically the same as the one based on the true errors.