Abstract
In this article, we propose a general class of INteger-valued Generalized AutoRegressive Conditional Heteroskedastic (INGARCH) models based on a flexible family of mixed Poisson (MP) distributions. Our proposed class of count time series models contains the negative binomial (NB) INGARCH process as particular case and open the possibility to introduce new models such as the Poisson-inverse Gaussian (PIG) and Poisson generalized hyperbolic secant processes. In particular, the PIG INGARCH model is an interesting and robust alternative to the NB model. We explore first-order and second-order stationary properties of our MPINGARCH models and provide expressions for the autocorrelation function and mean and variance marginals. Conditions to ensure strict stationarity and ergodicity properties for our class of INGARCH models are established. We propose an Expectation-Maximization algorithm to estimate the parameters and obtain the associated information matrix. Further, we discuss two additional estimation methods. Monte Carlo simulation studies are considered to evaluate the finite-sample performance of the proposed estimators. We illustrate the flexibility and robustness of the MPINGARCH models through two real-data applications about number of cases of Escherichia coli and Campylobacter infections. This article contains a Supporting Information.