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

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  • Borkovec, Milan

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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  • Borkovec, Milan, 2000. "Extremal behavior of the autoregressive process with ARCH(1) errors," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 189-207, February.
    • Handle: RePEc:eee:spapps:v:85:y:2000:i:2:p:189-207
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    1. Andrew A. Weiss, 1984. "Arma Models With Arch Errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 5(2), pages 129-143, March.
    2. 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.
    3. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    4. 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.
    5. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    1. Christopher Withers & Saralees Nadarajah, 2011. "The distribution of the maximum of a first order autoregressive process: the continuous case," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(2), pages 247-266, September.
    2. Djogbenou, Antoine & Inan, Emre & Jasiak, Joann, 2023. "Time-varying coefficient DAR model and stability measures for stablecoin prices: An application to Tether," Journal of International Money and Finance, Elsevier, vol. 139(C).
    3. Min Chen & Dong Li & Shiqing Ling, 2014. "Non-Stationarity And Quasi-Maximum Likelihood Estimation On A Double Autoregressive Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(3), pages 189-202, May.
    4. F. Laurini & J. A. Tawn, 2006. "The extremal index for GARCH(1,1) processes with t-distributed innovations," Economics Department Working Papers 2006-SE01, Department of Economics, Parma University (Italy).
    5. 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.
    6. Nielsen, Heino Bohn & Rahbek, Anders, 2014. "Unit root vector autoregression with volatility induced stationarity," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 144-167.
    7. 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.
    8. 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.

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