Extended Poisson INAR(1) processes with equidispersion, underdispersion and overdispersion

TL;DR

This paper introduces two novel extensions of the Poisson INAR(1) process to effectively model count time series exhibiting equidispersion, underdispersion, and overdispersion, with applications demonstrated on real data.

Contribution

The paper develops new INAR(1) models based on binomial thinning that handle various dispersion levels and derives their properties and estimation methods.

Findings

Models successfully fit overdispersed and underdispersed data

Estimation methods are asymptotically consistent

A dispersion test is proposed and validated

Abstract

Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule-Walker and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of number of weekly sales and of cases of family violence illustrating its capabilities in challenging cases of overdispersed and underdispersed count data.