A negative binomial autoregression with a linear conditional variance-To-mean function

Abstract

A general integer-valued time-series model with a conditional variance proportional to the conditional mean is proposed. Specifically, the conditional distribution is a Poisson mixture with a dependent mixing sequence, which results in a negative binomial distribution with a linear conditional variance-To-mean relationship. In addition, the conditional mean is specified as a general parametric function of past observations. We first propose stationarity, ergodicity, and finite moment conditions for the model. Furthermore, the parameters are estimated using the Poisson quasi-maximum likelihood estimate, whose asymptotic properties are studied under weak conditions. Illustrations of the proposed methodology on simulated and actual time series of counts are given.