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

TL;DR

This paper introduces a new class of generalized integer-valued ARMA models that unifies compound Poisson INAR and INGARCH processes, studying their properties and applying them to epidemiological data.

Contribution

It unifies INAR and INGARCH models into a single GINARMA framework, extending their applicability and theoretical understanding.

Findings

Models exhibit stationarity and geometric ergodicity.

Application to measles and mumps data demonstrates practical utility.

New models provide better fit for epidemic count data.

Abstract

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.