Joint extremal behavior of hidden and observable time series with applications to GARCH processes
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In: Extremes, Vol. 18, No. 1, 03.2015, p. 109-140.
Research output: Journal contributions › Journal articles › Research › peer-review
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TY - JOUR
T1 - Joint extremal behavior of hidden and observable time series with applications to GARCH processes
AU - Ehlert, Andree
AU - Fiebig, Ulf Rainer
AU - Janßen, Anja
AU - Schlather, Martin
PY - 2015/3
Y1 - 2015/3
N2 - For a class of generalized hidden Markov models (Xt,Yt)t∈ℤ we analyze the limiting behavior of the (suitably scaled) unobservable part (Yt)t∈ℤ under an observable extreme event |X0|>x, as x→∞. 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−1,X−2,…, i.e. before X0. 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).
AB - For a class of generalized hidden Markov models (Xt,Yt)t∈ℤ we analyze the limiting behavior of the (suitably scaled) unobservable part (Yt)t∈ℤ under an observable extreme event |X0|>x, as x→∞. 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−1,X−2,…, i.e. before X0. 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).
KW - (asymmetric) GARCH processes
KW - ARCH processes
KW - Extremal index
KW - Joint extremal behavior
KW - Multivariate regular variation
KW - Tail chain
KW - Time series
KW - Economics
UR - http://www.scopus.com/inward/record.url?scp=84925485593&partnerID=8YFLogxK
U2 - 10.1007/s10687-014-0206-9
DO - 10.1007/s10687-014-0206-9
M3 - Journal articles
AN - SCOPUS:84925485593
VL - 18
SP - 109
EP - 140
JO - Extremes
JF - Extremes
SN - 1386-1999
IS - 1
ER -