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

Author

Listed:
  • Jouchi Nakajima

    (Department of Statistical Science, Duke University)

  • Tsuyoshi Kunihama

    (Department of Statistical Science, Duke University)

  • Yasuhiro Omori

    (Faculty of Economics, University of Tokyo)

  • Sylvia Fruhwirth-Schnatter

    (Department of Applied Statistics, Johannes Kepler University Linz)

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 fol- low 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 an accu- rate 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 tted to a monthly series of minimum returns and the empirical results support strong evidence of time-dependence among the observed minimum returns.

Suggested Citation

  • Jouchi Nakajima & Tsuyoshi Kunihama & Yasuhiro Omori & Sylvia Fruhwirth-Schnatter, 2011. "Generalized Extreme Value Distribution with Time-Dependence Using the AR and MA Models in State Space Form," CIRJE F-Series CIRJE-F-782, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2010cf782
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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. 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.
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    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. 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.
    6. 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.
    7. 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.
    8. Deheuvels, Paul, 1983. "Point processes and multivariate extreme values," Journal of Multivariate Analysis, Elsevier, vol. 13(2), pages 257-272, June.
    9. 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.
    10. 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.
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    Cited by:

    1. Jouchi Nakajima & Tsuyoshi Kunihama & Yasuhiro Omori, 2017. "Bayesian modeling of dynamic extreme values: extension of generalized extreme value distributions with latent stochastic processes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(7), pages 1248-1268, May.
    2. 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.
    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.
    4. Tsuyoshi Kunihama & Yasuhiro Omori & Zhengjun Zhang, 2012. "Efficient estimation and particle filter for max‐stable processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(1), pages 61-80, January.
    5. Chen, Lei & Kou, Yingxin & Li, Zhanwu & Xu, An & Wu, Cheng, 2018. "Empirical research on complex networks modeling of combat SoS based on data from real war-game, Part I: Statistical characteristics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 754-773.
    6. Chao Huang & Jin-Guan Lin, 2014. "Modified maximum spacings method for generalized extreme value distribution and applications in real data analysis," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(7), pages 867-894, October.
    7. 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.

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    More about this item

    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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