  We introduce a general approach for modeling the dynamic of multivariate time
series when the data are of mixed type (binary/count/continuous). Our method is
quite flexible and conditionally on past values, each coordinate at time $t$
can have a distribution compatible with a standard univariate time series model
such as GARCH, ARMA, INGARCH or logistic models whereas past values of the
other coordinates play the role of exogenous covariates in the dynamic. The
simultaneous dependence in the multivariate time series can be modeled with a
copula. Additional exogenous covariates are also allowed in the dynamic. We
first study usual stability properties of these models and then show that
autoregressive parameters can be consistently estimated equation-by-equation
using a pseudo-maximum likelihood method, leading to a fast implementation even
when the number of time series is large. Moreover, we prove consistency results
when a parametric copula model is fitted to the time series and in the case of
Gaussian copulas, we show that the likelihood estimator of the correlation
matrix is strongly consistent. We carefully check all our assumptions for two
prototypical examples: a GARCH/INGARCH model and logistic/log-linear INGARCH
model. Our results are illustrated with numerical experiments as well as two
real data sets.
