Estimation of multivariate models for time series of possibly different lengths
提出一种多阶段极大似然估计法，用于各变量数据量不等时的参数多元密度模型估计，相比仅用重叠数据的一阶段法更有效，并以日元美元和欧元美元汇率数据验证了条件Copula的时变性和极端事件下的更强依赖性。
Abstract 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.