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Generalized Extreme Value Distribution with Time-Dependence Using the AR and MA Models in State Space Form

Author

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  • Jouchi Nakajima

    (Institute for Monetary and Economic Studies, Bank of Japan. Currently in the Personnel and Corporate Affairs Department ( studying at Duke University, E-mail: jouchi.nakajimaa@sstat.duke.edu))

  • Tsuyoshi Kunihama

    (Graduate student, Graduate School of Economics, University of Tokyo. (E-mail: ee097005@mail.ecc.u-tokyo.ac.jp))

  • Yasuhiro Omori

    (Professor, Faculty of Economics, University of Tokyo. (E-mail: omori@e.u-tokyo.ac.jp))

  • Sylvia Fruwirth-Scnatter

    (Professor, Department of Applied Statistics, Johannes Kepler University in Lintz. (E-mail: Sylvia.Fruehwirth-Schnatter@jku.at))

Abstract

A new state space approach is proposed to model the time- dependence in an extreme value process. The generalized extreme value distribution is extended to incorporate the time-dependence using a state space representation where the state variables either follow an autoregressive (AR) process or a moving average (MA) process with innovations arising from a Gumbel distribution. Using a Bayesian approach, an efficient algorithm is proposed to implement Markov chain Monte Carlo method where we exploit a very accurate approximation of the Gumbel distribution by a ten-component mixture of normal distributions. The methodology is illustrated using extreme returns of daily stock data. The model is fitted to a monthly series of minimum returns and the empirical results support strong evidence for time-dependence among the observed minimum returns.

Suggested Citation

  • Jouchi Nakajima & Tsuyoshi Kunihama & Yasuhiro Omori & Sylvia Fruwirth-Scnatter, 2009. "Generalized Extreme Value Distribution with Time-Dependence Using the AR and MA Models in State Space Form," IMES Discussion Paper Series 09-E-32, Institute for Monetary and Economic Studies, Bank of Japan.
  • Handle: RePEc:ime:imedps:09-e-32
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    References listed on IDEAS

    as
    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. Fruhwirth-Schnatter, Sylvia & Fruhwirth, Rudolf, 2007. "Auxiliary mixture sampling with applications to logistic models," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3509-3528, April.
    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. Omori, Yasuhiro & Watanabe, Toshiaki, 2008. "Block sampler and posterior mode estimation for asymmetric stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 2892-2910, February.
    5. 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.
    6. Deheuvels, Paul, 1983. "Point processes and multivariate extreme values," Journal of Multivariate Analysis, Elsevier, vol. 13(2), pages 257-272, June.
    7. J. Durbin, 2002. "A simple and efficient simulation smoother for state space time series analysis," Biometrika, Biometrika Trust, vol. 89(3), pages 603-616, August.
    8. Toshiaki Watanabe, 2004. "A multi-move sampler for estimating non-Gaussian time series models: Comments on Shephard & Pitt (1997)," Biometrika, Biometrika Trust, vol. 91(1), pages 246-248, March.
    9. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies, Oxford University Press, vol. 65(3), pages 361-393.
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    Cited by:

    1. Auray, Stéphane & Eyquem, Aurélien & Jouneau-Sion, Frédéric, 2014. "Modeling tails of aggregate economic processes in a stochastic growth model," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 76-94.
    2. repec:eee:phsmap:v:490:y:2018:i:c:p:754-773 is not listed on IDEAS
    3. Wang, Yixin & So, Mike K.P., 2016. "A Bayesian hierarchical model for spatial extremes with multiple durations," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 39-56.

    More about this item

    Keywords

    Extreme values; Generalized extreme value distribution; Markov chain Monte Carlo; Mixture sampler; State space model; Stock returns;

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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