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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

    (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.

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Bibliographic Info

Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-782.

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Length: 41pages
Date of creation: Jan 2011
Date of revision:
Handle: RePEc:tky:fseres:2010cf782

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  1. Sangjoon Kim, Neil Shephard & Siddhartha Chib, . "Stochastic volatility: likelihood inference and comparison with ARCH models," Economics Papers W26, revised version of W, Economics Group, Nuffield College, University of Oxford.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. Deheuvels, Paul, 1983. "Point processes and multivariate extreme values," Journal of Multivariate Analysis, 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, vol. 91(1), pages 246-248, March.
  8. 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, vol. 93(4), pages 827-841, December.
  9. 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.
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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 2012-29, Centre de Recherche en Economie et Statistique.

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