Estimation of Copula Models for Time Series of Possibly Different Lengths By

The theory of conditional copulas provides a means of constructing °exible multivariate density models, allowing for time-varying conditional densities of each individual variable, and for timevarying conditional dependence between the variables. Further, the use of copulas in constructing these models often allows for the partitioning of the parameter vector into elements relating only to a marginal distribution, and elements relating to the copula. This paper presents a two-stage (or multi-stage) maximum likelihood estimator for the case that such a partition is possible. We extend the existing statistics literature on the estimation of copula models to consider data that exhibit temporal dependence and heterogeneity. The estimator is °exible enough that the case that unequal amounts of data are available on each variable is easily handled. We investigate the small sample properties of the estimator in a Monte Carlo study, and ̄nd that it performs well in comparisons with the standard (one-stage) maximum likelihood estimator. Finally, we present an application of the estimator to a model of the joint distribution of daily Japanese yen U.S. dollar and euro U.S. dollar exchange rates. We ̄nd some evidence that a copula that captures asymmetric dependence performs better than those that assume symmetric dependence.