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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