Parametric inference and forecasting for continuously invertible volatility models

We introduce the notion of continuous invertibility on a compact set for volatility models driven by a Stochastic Recurrence Equation (SRE). We prove in this context the strong consistency and the asymptotic normality of the M-estimator associated with the Quasi-Likelihood criteria. We recover known results on univariate and multivariate GARCH type models where the estimator coincides with the classical QMLE. In EGARCH type models. our approach gives a strongly consistence and asymptotically normal estimator when the limiting covariance matrix exists. We provide a necessary and sufficient condition for the existence of this limiting covariance matrix in the EGARCH(1,1) model introduced in Nelson (1991). We exhibit for the first time sufficient conditions for the asymptotic normality of the estimation procedure used in practice since Nelson (1991). AMS 2000 subject classifications: Primary 62F12; secondary 60H25, 62F10, 62M20, 62M10, 91B84.