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Extremal behavior of the autoregressive process with ARCH(1) errors

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  • Borkovec, Milan
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    Abstract

    We investigate the extremal behavior of a special class of autoregressive processes with ARCH(1) errors given by the stochastic difference equationwhere are i.i.d. random variables. The extremes of such processes occur typically in clusters. We give an explicit formula for the extremal index and the probabilities for the length of a cluster.

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    File URL: http://www.sciencedirect.com/science/article/B6V1B-3Y2WWD8-2/2/1d84497692cfd41001cbdb3b1ebdd069
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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 85 (2000)
    Issue (Month): 2 (February)
    Pages: 189-207

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    Handle: RePEc:eee:spapps:v:85:y:2000:i:2:p:189-207

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    Related research

    Keywords: ARCH model Autoregressive process Compound Poisson process Coupling Extremal behavior Extremal index Frechet distribution Heavy tail Heteroscedastic homogeneous Markov process Recurrent Harris chain Separating sequence Strong mixing;

    References

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    1. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
    2. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
    3. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
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    Cited by:
    1. Klüppelberg, Claudia & Pergamenchtchikov, Serguei, 2007. "Extremal behaviour of models with multivariate random recurrence representation," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 432-456, April.
    2. Cline, Daren B.H., 2007. "Regular variation of order 1 nonlinear AR-ARCH models," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 840-861, July.
    3. Christopher Withers & Saralees Nadarajah, 2011. "The distribution of the maximum of a first order autoregressive process: the continuous case," Metrika, Springer, vol. 74(2), pages 247-266, September.
    4. Collamore, Jeffrey F. & Vidyashankar, Anand N., 2013. "Tail estimates for stochastic fixed point equations via nonlinear renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3378-3429.

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