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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

    (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

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    2. 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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    Citations

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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. Douissi, Soukaina & Es-Sebaiy, Khalifa & Alshahrani, Fatimah & Viens, Frederi G., 2022. "AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 886-918.
    6. 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.
    7. 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.
    8. 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

    Keywords

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

    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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