Mathematics – Statistics Theory
Scientific paper
2009-01-07
Annals of Statistics 2009, Vol. 37, No. 6B, 4214-4253
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/09-AOS719 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/09-AOS719
This paper considers the efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are all geometrically ergodic. We then propose a sieve maximum likelihood estimation (MLE) for the copula parameter, the invariant distribution and the conditional quantiles. We show that the sieve MLEs of any smooth functional is root-$n$ consistent, asymptotically normal and efficient and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. Monte Carlo studies indicate that, even for Markov models generated via tail dependent copulas and fat-tailed marginals, our sieve MLEs perform very well.
Chen Xiaohong
Wu Wei Biao
Yi Yanping
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