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

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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.

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Bibliographic Info

Paper provided by Institute for Monetary and Economic Studies, Bank of Japan in its series IMES Discussion Paper Series with number 09-E-32.

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Date of creation: Nov 2009
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Handle: RePEc:ime:imedps:09-e-32

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

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  1. Sangjoon Kim & Neil Shephard, 1994. "Stochastic volatility: likelihood inference and comparison with ARCH models," Economics Papers 3., Economics Group, Nuffield College, University of Oxford.
  2. Fruhwirth-Schnatter, Sylvia & Fruhwirth, Rudolf, 2007. "Auxiliary mixture sampling with applications to logistic models," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 51(7), pages 3509-3528, April.
  3. Chib S. & Jeliazkov I., 2001. "Marginal Likelihood From the Metropolis-Hastings Output," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 270-281, March.
  4. Sylvia Fr�Hwirth-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, Biometrika Trust, vol. 93(4), pages 827-841, December.
  5. Yasuhiro Omori & Toshiaki Watanabe, 2007. "Block Sampler and Posterior Mode Estimation for Asymmetric Stochastic Volatility Models," CIRJE F-Series, CIRJE, Faculty of Economics, University of Tokyo CIRJE-F-507, CIRJE, Faculty of Economics, University of Tokyo.
  6. Deheuvels, Paul, 1983. "Point processes and multivariate extreme values," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 13(2), pages 257-272, June.
  7. Toshiaki Watanabe, 2004. "A multi-move sampler for estimating non-Gaussian time series models: Comments on Shephard & Pitt (1997)," Biometrika, Biometrika Trust, Biometrika Trust, vol. 91(1), pages 246-248, March.
  8. J. Durbin, 2002. "A simple and efficient simulation smoother for state space time series analysis," Biometrika, Biometrika Trust, Biometrika Trust, vol. 89(3), pages 603-616, August.
  9. Chib, Siddhartha, 2001. "Markov chain Monte Carlo methods: computation and inference," Handbook of Econometrics, Elsevier, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 5, chapter 57, pages 3569-3649 Elsevier.
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Cited by:
  1. Stéphane Auray & Aurélien Eyquem & Fréderic Jouneau-Sion, 2012. "Modelling Tails of Aggregated Economic Processes in a Stochastic Growth Model," Working Papers, Centre de Recherche en Economie et Statistique 2012-29, Centre de Recherche en Economie et Statistique.

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