Stable limits of martingale transforms with application to the estimation of Garch parameters

• 成果类型: Article
• 署名作者: Mikosch, T; Straumann, D
• 署名单位: University of Copenhagen; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053605000000840
• 发表日期: 2006
• 页码: 493-522
• 关键词: AUTOREGRESSIVE PROCESSES; Regular Variation; stationarity; CONVERGENCE; ARCH

摘要

In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavy-tailed innovations. This means that the innovations are regularly varying with index alpha is an element of (2, 4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and root n-rates breaks down. This was recently observed by Hall and Yao [Econometrica 71 (2003) 285-317]. It is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavy-tailed case nevertheless very much parallels the normal asymptotic theory. In the light-tailed case, the limit theory is based on the CLT for stationary ergodic finite variance martingale difference sequences. In the heavy-tailed case such a general result does not exist, but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for GARCH processes. It is the aim of the paper to give a general structural result for infinite variance limits which can also be applied in situations more general than GARCH.