In this paper, we propose an intuitive way to couple several dynamic time series models even when there are no innovations. This extends previous work for modeling dependence 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 univariate dynamic models depend on unknown parameters, the limiting behavior of many processes of interest do not depend on the estimation errors. One can easily perform tests of change point on the full distribution, the margins or the copula, as if the parameters of the dynamic models were known. This is also true for some interesting parametric models of time-dependent copulas. 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. Monte Carlo simulations are used to demonstrate the power of the proposed goodness-of-fit test for finite samples. An example of application with financial data is given.
Published December 2018 , 21 pages