Local asymptotic normality of general conditionally heteroskedastic and score-driven time-series models

The paper establishes the Local Asymptotic Normality (LAN) property for general conditionally heteroskedastic time series models of multiplicative form, $\epsilon_t=\sigma_t(\btheta_0)\eta_t$, where the volatility $\sigma_t(\btheta_0)$ is a parametric function of $\{\epsilon_{s}, s< t\}$, and $(\eta_t)$ is a standardized i.i.d. noise endowed with a density $f_{\btheta_0}$. In contrast with earlier results, the finite dimensional parameter $\btheta_0$ enters in both the volatility and the density specifications. To deal with non-differentiable functions, we introduce a conditional notion of the familiar quadratic mean differentiability condition which takes into account parameter variation in both the volatility and the errors density. Our results are illustrated on two particular models: the APARCH with Asymmetric Student-$t$ distribution, and the Beta-$t$-GARCH model.