The Half-sample Method for Testing Parametric Regressive and Autoregressive Models of Order 1

Abstract

The half-sample method has been introduced by Stephens (1978) for testing parametric models of distribution functions. In this paper, we present a similar method for testing the goodness-of-fit of linear or nonlinear regression or autoregression functions for parametric models of order $1$, under minimal stationarity and ergodicity assumptions. Our procedure is based on a measure of the cumulated deviation process $\hat{A}_{n}$ between a weighted marked process of residuals and a parametric estimator of the cumulated conditional mean function (i.e.\ cumulated regression or autoregression function), under the null hypothesis $H_{0}$. We establish a functional limit theorem under $H_{0}$ for a variant $\hat{A}^{(\kappa)}_{n}$, $0 < \kappa \le 1$, of the process $\hat{A}_{n}$. The half-sample method corresponds to $\kappa = 1/2$. We show that the limiting distribution of $\hat{A}^{(1/2)}_{n}$ under $H_{0}$ takes a very simple form. Several easily implemented goodness-of-fit tests can be based on this result. We provide simple conditions under which their power converges to 1 as the sample size goes to $\infty$. Finally, we investigate the asymptotic behavior of $\hat{A}^{(\kappa)}_{n}$ as $n \to \infty$ under sequences of $O(n^{-1/2})$ local alternatives. This allows us to compare the corresponding local powers of tests based on $\hat{A}^{(1/2)}_{n}$ and on $\hat{A}^(1)_{n}$