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Bayesian Estimation and Particle Filter for Max-Stable Processes

Author

Listed:
  • Tsuyoshi Kunihama

    (Graduate School of Economics, University of Tokyo)

  • Yasuhiro Omori

    (Faculty of Economics, University of Tokyo)

  • Zhengjun Zhang

    (Department of Statistics, University of Wisconsin Madison)

Abstract

Extreme values are often correlated over time, for example, in a financial time series, and these values carry various risks. Max-stable processes such as maxima of moving maxima (M3) processes have been recently considered in the literature to describe timedependent dynamics, which have been difficult to estimate. This paper first proposes a feasible and efficient Bayesian estimation method for nonlinear and non-Gaussian state space models based on these processes and describes a Markov chain Monte Carlo algorithm where the sampling efficiency is improved by the normal mixture sampler. Furthermore, a unique particle filter that adapts to extreme observations is proposed and shown to be highly accurate in comparison with other well-known filters. Our proposed algorithms were applied to daily minima of high-frequency stock return data, and a model comparison was conducted using marginal likelihoods to investigate the time-dependent dynamics in extreme stock returns for financial risk management.

Suggested Citation

  • Tsuyoshi Kunihama & Yasuhiro Omori & Zhengjun Zhang, 2010. "Bayesian Estimation and Particle Filter for Max-Stable Processes," CIRJE F-Series CIRJE-F-757, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2010cf757
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    References listed on IDEAS

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    1. Chib, Siddhartha, 2001. "Markov chain Monte Carlo methods: computation and inference," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 57, pages 3569-3649, Elsevier.
    2. Hall, Peter & Peng, Liang & Yao, Qiwei, 2002. "Moving-maximum models for extrema of time series," LSE Research Online Documents on Economics 6084, London School of Economics and Political Science, LSE Library.
    3. Sylvia FrüHwirth-Schnatter & Helga Wagner, 2006. "Auxiliary mixture sampling for parameter-driven models of time series of counts with applications to state space modelling," Biometrika, Biometrika Trust, vol. 93(4), pages 827-841, December.
    4. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 361-393.
    5. Chib, Siddhartha & Greenberg, Edward, 1996. "Markov Chain Monte Carlo Simulation Methods in Econometrics," Econometric Theory, Cambridge University Press, vol. 12(3), pages 409-431, August.
    6. A. Martins & H. Ferreira, 2005. "The multivariate extremal index and the dependence structure of a multivariate extreme value distribution," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(2), pages 433-448, December.
    7. Nakajima, Jouchi & Kunihama, Tsuyoshi & Omori, Yasuhiro & Frühwirth-Schnatter, Sylvia, 2012. "Generalized extreme value distribution with time-dependence using the AR and MA models in state space form," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3241-3259.
    8. Zhengjun Zhang, 2009. "On approximating max-stable processes and constructing extremal copula functions," Statistical Inference for Stochastic Processes, Springer, vol. 12(1), pages 89-114, February.
    9. Omori, Yasuhiro & Chib, Siddhartha & Shephard, Neil & Nakajima, Jouchi, 2007. "Stochastic volatility with leverage: Fast and efficient likelihood inference," Journal of Econometrics, Elsevier, vol. 140(2), pages 425-449, October.
    10. Chib S. & Jeliazkov I., 2001. "Marginal Likelihood From the Metropolis-Hastings Output," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 270-281, March.
    11. Deheuvels, Paul, 1983. "Point processes and multivariate extreme values," Journal of Multivariate Analysis, Elsevier, vol. 13(2), pages 257-272, June.
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    Cited by:

    1. Nakajima, Jouchi & Kunihama, Tsuyoshi & Omori, Yasuhiro & Frühwirth-Schnatter, Sylvia, 2012. "Generalized extreme value distribution with time-dependence using the AR and MA models in state space form," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3241-3259.

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