Negative Binomial Autoregressive Process with Stochastic Intensity

We introduce negative binomial-60 autoregressive (NBAR) processes with stochastic intensity for (univariate and bivariate) count processes. The univariate NBAR process is defined jointly with an underlying intensity process, which is autoregressive gamma. The resulting count process is Markov, with negative binomial conditional and marginal distributions. The process is then extended to the bivariate case with a Wishart autoregressive matrix intensity process. The NBAR processes are compound autoregressive, which allows for simple stationarity condition and quasi-closed form nonlinear forecasting formulae at any horizon, as well as a computationally tractable generalized method of moment estimator. The model is applied to a pairwise analysis of weekly occurrence counts of a contagious disease between the greater Paris region and other French regions.