Alternative distributions for observation driven count series models

Observation-driven models provide a flexible framework for modelling time series of counts. They are able to capture a wide range of dependence structures. Many applications in this field of research are concerned with count series whose conditional distribution given past observations and explanatory variables is assumed to follow a Poisson distribution. This assumption is very convenient since the Poisson distribution is simple and leads to models which are easy to implement. On the other hand this assumption is often too restrictive since it implies equidispersion, the fact that the conditional mean equals the conditional variance. This assumption is often violated in empirical applications. Therefore more flexible distributions which allow for overdispersion or underdispersion should be used. This paper is concerned with the use of alternative distributions in the framework of observationdriven count series models. In this paper different count distributions and their properties are reviewed and used for modelling. The models under consideration are applied to a time series of daily counts of asthma presentations at a Sydney hospital. This data set has already been analyzed by Davis et al. (1999, 2000). The Poisson-GLARMA model proposed by these authors is used as a benchmark. This paper extends the work of Davis et al. (1999) to distributions which are nested in either the generalized negative binomial or the generalized Poisson distribution. Additionally the maximum likelihood estimation for observation-driven models with generalized distributions is presented in this paper.