The paper establishes the local asymptotic normality property for general conditionally heteroskedastic time series models of multiplicative form, $\epsilon _t=\sigma _t(\boldsymbol {\theta }_0)\eta _t$, where the volatility $\sigma _t(\boldsymbol {\theta }_0)$ is a parametric function of $\{\epsilon _{s}, s< t\}$, and $(\eta _t)$ is a sequence of i.i.d. random variables with common density $f_{\boldsymbol {\theta }_0}$. In contrast with earlier results, the finite dimensional parameter $\boldsymbol {\theta }_0$ enters in both the volatility and the density specifications. To deal with nondifferentiable 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, and are extended to handle a conditional mean.

本文建立了乘法形式的一般条件异方差时间序列模型的局部渐近正态性，$\epsilon _t=\sigma _t(\boldsymbol {\theta }_0)\eta _t$，其中波动率$\sigma _t(\ boldsymbol {\theta }_0)$是$\{\epsilon _{s}, s< t\}$的参数函数，$(\eta _t)$是具有共同密度$f_的 iid 随机变量序列{\boldsymbol {\theta}_0}$。与之前的结果相比，有限维参数$\boldsymbol {\theta }_0$输入挥发性和密度规格。为了处理不可微函数，我们引入了熟悉的二次均值可微条件的条件概念，该条件考虑了波动性和误差密度的参数变化。我们的结果在两个特定模型上进行了说明：具有不对称 Student- t分布的 APARCH和 Beta- t -GARCH 模型，并扩展到处理条件均值。