Goodness-of-fit test using residuals in infinite-order autoregressive models

In this paper we study the asymptotic behavior of the Bickel-Rosenblatt test in infinite-order autoregressive models. Under some mild conditions, the test statistic based on residuals is shown to have the same limiting distribution as that based on true iid errors. It is also proved that this result remains the same when the nuisance parameters in the model are to be estimated, which means that the Bickel-Rosenblatt test is easily applicable in composite hypothesis goodness-of-fit testing unlike the Kolmogorov-Smirnov type tests. The result of a simulation study is supplemented to verify the result of this paper.