Joint extremal behavior of hidden and observable time series with applications to GARCH processes

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• 作者：Andree Ehlert ; Ulf-Rainer Fiebig ; Anja Jan?en ; Martin Schlather
• 关键词：ARCH processes ; (asymmetric) GARCH processes ; Extremal index ; Joint extremal behavior ; Multivariate regular variation ; Tail chain ; Time series ; Primary-0G70 ; Secondary-0J05 ; 60J22 ; 91G60
• 刊名：Extremes
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：18
• 期：1
• 页码：109-140
• 全文大小：557 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics
  Statistics
  Quality Control, Reliability, Safety and Risk
  Civil Engineering
  Hydrogeology
  Environmental Management
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer U.S.
• ISSN：1572-915X

文摘

For a class of generalized hidden Markov models \((X_{t},Y_{t})_{t \in \mathbb {Z}}\) we analyze the limiting behavior of the (suitably scaled) unobservable part \((Y_{t})_{t\in \mathbb Z}\) under an observable extreme event |X 0|>x, as \(x \to \infty \) . We discuss sufficient conditions for the existence of this limit and characterize its special structure. Our approach gives rise to an efficient and flexible algorithm for the Monte Carlo evaluation of extremal characteristics (such as the extremal index) of the observable process. Further, our setup allows to evaluate extremal measures which depend on the extremal behavior of X ?,X ?,- i.e. before X 0. An application to financial asset returns is given by the asymmetric GARCH(1,1) model whose extremal behavior has not been considered before. Our results complement the findings of Segers on the tail chains of single time series (Segers 2007).