Copula-Based Dynamic Models for Multivariate Time Series

In this paper, we propose an intuitive way to couple several dynamic time series models by inducing dependence between the so-called generalized errors of each model. This extends previous work for modelling dependance between innovations of stochastic volatility models. We consider time-independent and time-dependent copula models and we study the asymptotic behavior of some empirical processes constructed from pseudo-observations, as well as the behavior of pseudo-maximum likelihood estimators of the associated copula parameters. The results show that even if the generalized errors depend on unknown parameters, the limiting behaviour of many processes of interest do not depend on the estimation errors. One can easily perform tests of change-point on the full distribution, and the margins or the copula, as if the generalized errors were observed. For some interesting parametric models of time-dependent copulas, the same behavior is observed: one can work with the pseudo-observations, as if we were observing the generalized errors. This interesting property makes it possible to construct consistent tests of specification for the dependence models, without having to consider the dynamic time series models. An example of application with financial data is given.