Modeling time series of counts with COM-Poisson INGARCH models

Frequently count time series exhibit overdispersion, but the opposite phenomenon of underdispersion is well documented in some situations, thus may be encountered in real applications. The INGARCH model is a popular tool for modeling time series of counts. The Poisson and negative binomial models can only deal with overdispersion, and the double Poisson and generalized Poisson models can treat both of them, but the latter two models have some shortcomings or limitations. The revived COM-Poisson distribution is flexible in modeling a wide range of overdispersion and underdispersion with only two parameters, while possessing properties that make it methodologically appealing and useful in practice, thus we introduce a new INGARCH model based on this distribution. We give an approximate condition for stationarity and expressions for mean, variance and autocorrelation function and test the accuracy of these expressions via simulations and real examples. We discuss the maximum likelihood estimation procedure for the parameters of interest. In addition, we also try to find influences of different distributions on INGARCH models using three real data sets. The results show that the generalized Poisson and COM-Poisson models perform well in most cases and the COM-Poisson model is a powerful competitor compared with the generalized Poisson model.