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This paper considers 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 functionals are root- n consistent, asymptotically normal and eﬀicient; and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. We present Monte Carlo studies to compare the ﬁnite sample performance of the sieve MLE, the two-step estimator of Chen and Fan (2006), the correctly speciﬁed parametric MLE and the incorrectly speciﬁed parametric MLE. The simulation results indicate that our sieve MLEs perform very well; having much smaller biases and smaller variances than the two-step estimator for Markov models generated via Clayton, Gumbel and other tail dependent copulas.
Chen, Xiaohong; Wu, Wei Biao; and Yi, Yanping, "Efficient Estimation of Copula-based Semiparametric Markov Models" (2009). Cowles Foundation Discussion Papers. 2008.