COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1691 Efficient Estimation of Copula-based Semiparametric Markov Models Xiaohong Chen, Wei Biao Wu, and Yanping Yi February 2009 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. |
