A simultaneous test for conditional mean and conditional variance functions in time series models with martingale difference innovations

We consider here, in contrast to classical time series models where innovations are assumed to be independent and identically distributed (iid), a class of nonlinear semi-parametric models in which the innovations are stationary ergodic conditionally martingale differences. We establish the local asymptotic normality associated with these models. From this result, an efficient simultaneous locally asymptotic test is derived for testing the conditional mean and the conditional variance functions without a specified error law. The main result shows that the test statistic built by substituting consistent estimated residuals and parameters for theoretical ones is asymptotically normal. Its asymptotic power is also obtained under local alternatives. The performances of the proposed test are illustrated by means of some simulations.