Autoregressive conditional negative binomial model applied to over-dispersed time series of counts

Integer-valued time series analysis offers various applications in biomedical, financial, and environmental research. However, existing works usually assume no or constant over-dispersion. In this paper, we propose a new model for time series of counts, the autoregressive conditional negative binomial model that has a time-varying conditional autoregressive mean function and heteroskedasticity. The location and scale parameters of the negative binomial distribution are flexible in the proposed set-up, inducing dynamic over-dispersion. We adopt Bayesian methods with a Markov chain Monte Carlo sampling scheme to estimate model parameters and utilize deviance information criterion for model comparison. We conduct simulations to investigate the estimation performance of this sampling scheme for the proposed negative binomial model. To demonstrate the proposed approach in modelling time-varying over-dispersion, we consider two types of criminal incidents recorded by New South Wales (NSW) Police Force in Australia. We also fit the autoregressive conditional Poisson model to these two datasets. Our results demonstrate that the proposed negative binomial model is preferable to the Poisson model.