ANDREW J. PATTON*

SUMMARY
We consider the problem of estimating parametric multivariate density models when unequal amounts of data
are available on each variable. We focus in particular on the case that the unknown parameter vector may
be partitioned into elements relating only to a marginal distribution and elements relating to the copula. In
such a case we propose using a multi-stage maximum likelihood estimator (MSMLE) based on all available
data rather than the usual one-stage maximum likelihood estimator (1SMLE) based only on the overlapping
data. We provide conditions under which the MSMLE is not less asymptotically efficient than the 1SMLE,
and we examine the small sample efficiency of the estimators via simulations. The analysis in this paper is
motivated by a model of the joint distribution of daily Japanese yen–US dollar and euro–US dollar exchange
rates. We find significant evidence of time variation in the conditional copula of these exchange rates, and
evidence of greater dependence during extreme events than under the normal distribution. Copyright  2006
John Wiley & Sons, Ltd.