  This paper is concerned with modeling the dependence structure of two (or
more) time-series in the presence of a (possible multivariate) covariate which
may include past values of the time series. We assume that the covariate
influences only the conditional mean and the conditional variance of each of
the time series but the distribution of the standardized innovations is not
influenced by the covariate and is stable in time. The joint distribution of
the time series is then determined by the conditional means, the conditional
variances and the marginal distributions of the innovations, which we estimate
nonparametrically, and the copula of the innovations, which represents the
dependency structure. We consider a nonparametric as well as a semiparametric
estimator based on the estimated residuals. We show that under suitable
assumptions these copula estimators are asymptotically equivalent to estimators
that would be based on the unobserved innovations. The theoretical results are
illustrated by simulations and a real data example.
