  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.
