Conway–Maxwell–Poisson Autoregressive Moving Average Model for Equidispersed, Underdispersed, and Overdispersed Count Data

Abstract

In this work, we propose a dynamic regression model based on the ConwayŮMaxwell–Poisson (CMP) distribution with time‐varying conditional mean depending on covariates and lagged observations. This new class of ConwayŮMaxwell–Poisson autoregressive moving average (CMP‐ARMA) models is suitable for the analysis of time series of counts. The CMP distribution is a two‐parameter generalization of the Poisson distribution that allows the modeling of underdispersed, equidispersed, and overdispersed data. Our main contribution is to combine this dispersion flexibility with the inclusion of lagged terms to model the conditional mean response, inducing an autocorrelation structure, usually relevant in time series. We present the conditional maximum likelihood estimation, hypothesis testing inference, diagnostic analysis, and forecasting along with their asymptotic properties. In particular, we provide closed‐form expressions for the conditional score vector and conditional Fisher information matrix. We conduct a Monte Carlo experiment to evaluate the performance of the estimators in finite sample sizes. Finally, we illustrate the usefulness of the proposed model by exploring two empirical applications.