条件异方差时间序列的拟极大似然估计：一种随机递归方程方法
Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach
摘要：
本文研究了一般乘型$X_t=σ_tZ_t$的条件异方差时间序列模型的拟极大似然估计，其中不可观测波动率$σ_t$是$(X_{t-1},...,X_{t-p},σ_{t-1},... ,σ_{t-q})$的参数函数，对于某些$p,q\ge0$，且$(Z_t)$是标准化的i.i.d.噪声我们假设这些模型是随机递归方程的解，满足收缩（随机Lipschitz系数）属性。这些假设对于流行的Gestival过程、非对称Gestival过程和指数Gestival过程都是成立的。利用压缩性质，给出了随机递归方程严格平稳解$(X_t)$存在唯一的条件，并建立了QMLE的相合性和渐近正态性.我们还讨论了这种时间序列模型的可逆性问题。
英文摘要：
This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form $X_t=σ_tZ_t$, where the unobservable volatility $σ_t$ is a parametric function of $(X_{t-1},...,X_{t-p},σ_{t-1},... ,σ_{t-q})$ for some $p,q\ge0$, and $(Z_t)$ is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution $(X_t)$ to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.
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