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 Conway-Maxwell-Poisson (CMP) distribution with time-varying conditional mean depending on covariates and lagged observations. This new class of Conway-Maxwell-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.

Type I multivariate zero‐inflated COM–Poisson regression model

“…(2021) develop a COM–Poisson model for inflated frequencies on two integers. For the time‐series data, Sellers and Young (2019) discuss zero‐inflated sum of COM–Poissons (ZISCMP) regression, MacDonald and Bhamani (2020) develop a hidden Markov model with COM–Poisson state‐dependent distributions, and Melo and Alencar (2020) present a generalized COM–Poisson autoregressive and moving averages time‐series model for equidispersed, underdispersed, and overdispersed count data. Recently, Sellers et al.…”

Modelagem de séries temporais de contagem usando a distribuição Conway-Maxwell-Poisson

“…Thus, to circumvent this problem, we chose the logarithmic link function, and to avoid calculating the logarithm of observations equal to zero, we replace y t−j in Equation (3.1) for y * t−j = max(y t−j , c), thus replacing y t−j = 0 by an arbitrary small value c such that 0 < c < 1. The structure of the model is the same as in Bayer et al (2018) The CMP-SARMA model extends the approach proposed by Melo and Alencar (2020) by incorporating a seasonal autoregressive moving average (SARMA) structure.…”

“…. , r + p + P + q + Q} (γ i = ν,), is given by and Alencar, 2020) the expected value of the derivative above is given by…”

“…In this section, we shall present the CMP-ARMA model proposed by Melo and Alencar (2020) for modeling underdispersed, equidispersed, and overdispersed count time series data. This class of models is very exible and contains the GARMA models with the Poisson, Geometric, and Bernoulli distributions as particular cases.…”

“…Conclusions and future4.3.1 Randomized quantile residualAs already mentioned, the distribution of the deviance and Pearson residuals are usually nonnormally distributed for discrete data with low tted means. For this reason,Melo and Alencar (2020) recommend the randomized quantile residual for CMP-ARMA models Feng et al (2017). showed theoretically that the randomized quantile residual follows a standard normal distribution under the true model.…”