Estimation and forecasting for binomial and negative binomial INAR(1) time series

To model count time series Al-Osh & Alzaid (1987) and McKenzie (1988) introduced the INteger-valued AutoRegressive process (INAR). Usually the innovation term is assumed to follow a Poisson distribution. However, other distributional assumptions may be used instead. In this work we discuss the issue of estimating and forecasting in case of INAR(1) time series with over and under-dispersion, resorting respectively to the Binomial and Negative Binomial distributions. We calculate the maximum likelihood functions for the considered cases and via a Monte Carlo experiment we show that the resulting estimators have a good performance. Moreover, we also concentrate on the problem of producing coherent predictions based on estimates of the p-step ahead predictive mass functions assuming Binomial and Negative Binomial distributions of the error term. Finally, we compare the forecast accuracy of Binomial and Negative Binomial INAR with that of Poisson INAR and ARMA models with a Monte Carlo experiment.