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Extremes of Markov chains with tail switching potential

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  • Paola Bortot
  • Stuart Coles

Abstract

Summary. A recent advance in the utility of extreme value techniques has been the characteri‐ zation of the extremal behaviour of Markov chains. This has enabled the application of extreme value models to series whose temporal dependence is Markovian, subject to a limitation that prevents switching between extremely high and extremely low levels. For many applications this is sufficient, but for others, most notably in the field of finance, it is common to find series in which successive values switch between high and low levels. We term such series Markov chains with tail switching potential, and the scope of this paper is to generalize the previous theory to enable the characterization of the extremal properties of series displaying this type of behaviour. In addition to theoretical developments, a modelling procedure is proposed. A simulation study is made to assess the utility of the model in inferring the extremal dependence structure of autoregressive conditional heteroscedastic processes, which fall within the tail switching Markov family, and generalized autoregressive conditional heteroscedastic processes which do not, being non‐Markov in general. Finally, the procedure is applied to model extremal aspects of a financial index extracted from the New York Stock Exchange compendium.

Suggested Citation

  • Paola Bortot & Stuart Coles, 2003. "Extremes of Markov chains with tail switching potential," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 851-867, November.
    • Handle: RePEc:bla:jorssb:v:65:y:2003:i:4:p:851-867
    DOI: 10.1046/j.1369-7412.2003.00419.x
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    Cited by:

    1. Phong Nguyen & Wei-han Liu, 2017. "Time-Varying Linkage of Possible Safe Haven Assets: A Cross-Market and Cross-asset Analysis," International Review of Finance, International Review of Finance Ltd., vol. 17(1), pages 43-76, March.
    2. Stan Tendijck & Philip Jonathan & David Randell & Jonathan Tawn, 2024. "Temporal evolution of the extreme excursions of multivariate k$$ k $$th order Markov processes with application to oceanographic data," Environmetrics, John Wiley & Sons, Ltd., vol. 35(3), May.
    3. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    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. Paola Bortot & Carlo Gaetan, 2014. "A Latent Process Model for Temporal Extremes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(3), pages 606-621, September.
    6. janssen, Anja & Segers, Johan, 2013. "Markov Tail Chains," LIDAM Discussion Papers ISBA 2013017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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