Based on the local asymptotic normality (LAN) of the log-likelihood ratio statistic, we proposed some distribution-free tests for examining simultaneously hypotheses about the conditional mean and the conditional variance functions in time series models. Our results are established under stationarity and ergodicity conditions with unspecified "innovation" densities under the null hypothesis as well as under the alternatives. First, we establish the contiguity of a class of nonlinear autoregressive process of order d with ARCH errors (AR(d)-ARCH(d)). Based on these results, an efficient test for the non-parametric form of the mean and the variance functions is then obtained and its asymptotic power is explicitly provided. Additionally, we discuss a Score type statistic for testing hypotheses about the parameters appearing in the mean and the variance functions when a part of them are nuisance parameters. Both the null and non-null limiting distributions of these tests are also derived. Results are illustrated by some simulations.
(joint with Professor Naamane LAÏB (Université Paris 6))