  We are studying the problems of modeling and inference for multivariate count
time series data with Poisson marginals. The focus is on linear and log-linear
models. For studying the properties of such processes we develop a novel
conceptual framework which is based on copulas. However, our approach does not
impose the copula on a vector of counts; instead the joint distribution is
determined by imposing a copula function on a vector of associated continuous
random variables. This specific construction avoids conceptual difficulties
resulting from the joint distribution of discrete random variables yet it keeps
the properties of the Poisson process marginally. We employ Markov chain theory
and the notion of weak dependence to study ergodicity and stationarity of the
models we consider. We obtain easily verifiable conditions for both linear and
log-linear models under both theoretical frameworks. Suitable estimating
equations are suggested for estimating unknown model parameters. The large
sample properties of the resulting estimators are studied in detail. The work
concludes with some simulations and a real data example.
