An alternative hyper-Poisson integer-valued GARCH model with application to polio, internet protocol and COVID-19 data

Abstract

Time series of counts are observed widely in actuarial science, finance, epidemiology and biology. These time series may exhibit over-, equi- and under-dispersion. The Poisson distribution is commonly used in count time series models, but it is restricted by the equality of mean and variance. Other distributions such as the generalized Poisson, double Poisson, hyper-Poisson, and COM-Poisson distributions have been proposed to replace the Poisson distribution to model the different levels of dispersion in time series of counts. These models have certain limitations such as complex expressions for the mean and variance which complicate the formulation as GARCH models. In this study, we propose an alternative hyper-Poisson (AHP) distribution, with simple forms of conditional mean and variance, for an integer-valued GARCH (INGARCH) model for time series of counts that also exhibit the different levels of dispersion. We demonstrate that the AHP-INGARCH model is comparable to some existing INGARCH models. Additionally, the model can cover a wider range of dispersion. The maximum likelihood estimation can be used to estimate the parameters of the proposed model. Applications to three real-life data sets related to polio, internet protocol and daily COVID-19 new deaths underscore the usefulness of the proposed model in studying both over-dispersed and under-dispersed time series of counts.