Inference on autoregressive moving average models for count data

Abstract

In the analysis of count time series at equally spaced intervals with covariate information, Poisson Autoregressive (AR) or Integer-Valued Autoregressive (INAR) models have been widely discussed in the literature, with their fundamental properties and estimation methods thoroughly explored. However, when time series data exhibits both long-term dependencies (autocorrelation) and moving average effects, capturing both of these elements is essential for more effective modeling and forecasting. To address this, we introduce autoregressive moving average (ARMA) models of order (1,1) for count time series. We first consider the case where the offspring random variable follows a Bernoulli distribution, meaning that each individual in the population at time t - 1 can produce only one or zero offspring at time t. Additionally, we extend this model to incorporate the possibility of any individual producing multiple offspring at a given time point, resulting in a binomial offspring random variable. We derive the key properties of these models, present methods for parameter estimation and forecasting function. The performance of the proposed methods are assessed through simulation studies.