Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach

• 成果类型: Article
• 署名作者: Straumann, Daniel; Mikosch, Thomas
• 署名单位: University of Copenhagen
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000803
• 发表日期: 2006
• 页码: 2449-2495
• 关键词: garch processes; ASYMPTOTIC THEORY; models; Consistency; heteroskedasticity; coefficients; igarch(1; 1); normality; ARCH

摘要

This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form X-t = sigma(t)Z(t), where the unobservable volatility sigma(t) is a parametric function of (Xt-1,..., Xt-p, sigma(t-1),..., sigma(t-q)) for some p, q >= 0, 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.