A unifying class of compound Poisson integer‐valued ARMA and GARCH models

INAR (integer‐valued autoregressive) and INGARCH (integer‐valued GARCH) models are among the most commonly employed approaches for count time series modeling, 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 maximum likelihood, Gaussian quasi‐likelihood, and moment‐based approaches, along with likelihood ratio tests to distinguish between selected instances of our class. 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, including both established and newly introduced models, are applied to weekly case numbers of measles and mumps in Bavaria, Germany.