The Negative Binomial Process: A Tractable Model with Composite Likelihood-Based Inference

We propose a log-linear Poisson regression model driven by a stationary latent gamma autoregression. This process has negative binomial (NB) marginals to analyze overdispersed count time series data. Estimation and statistical inference are performed using a composite (CL) likelihood function. We establish theoretical properties of the proposed count model, in particular, the strong consistency and asymptotic normality of the maximum CL estimator. A procedure for calculating the standard error of the parameter estimator and confidence intervals is derived based on the parametric bootstrap. Monte Carlo experiments were conducted to study and compare the finite-sample properties of the proposed estimators. The simulations demonstrate that, compared to the approach that combines generalized linear models with the ordinary least squares method, the proposed composite likelihood approach provides satisfactory results for estimating the parameters related to the correlation structure of the process, even under model misspecification. An empirical illustration of the NB process is presented for the monthly number of viral hepatitis cases in Goiânia (capital and largest city of the Brazilian state of Goiás) from January 2001 to December 2018.