CIRJE-F-782 "Generalized Extreme Value Distribution with Time-Dependence Using the AR and MA Models in State Space Form"
Author Name Nakajima, Jouchi, Tsuyoshi Kunihama, Yasuhiro Omori and Sylvia Frühwirth-Schnatter
Date January 2011
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Remarks  Revised version of CIRJE-F-689 (2009); subsequently published in Computational Statistics and Data Analysis, 56-11, 3241-3259. November 2012.

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.