A class of count time series models uniting compound Poisson INAR and INGARCH models

INAR (integer-valued autoregressive) and INGARCH (integer-valued GARCH) models are among the most commonly employed approaches for count time series modelling, but have been studied in largely distinct strands of literature. In this paper, a new class of generalized integer-valued ARMA (GINARMA) models is introduced which unifies a large number of compound Poisson INAR and INGARCH processes. Its stochastic properties, including stationarity and geometric ergodicity, are studied. Particular attention is given to a generalization of the INAR($p$) model which parallels the extension of the INARCH(p) to the INGARCH(p, q) model. For inference, we consider moment-based estimation and a maximum likelihood inference scheme inspired by the forward algorithm. Models from the proposed class have a natural interpretation as stochastic epidemic processes, which throughout the article is used to illustrate our arguments. In a case study, different instances of the class, including both established and newly introduced models, are applied to weekly case numbers of measles and mumps in Bavaria, Germany.