Test for conditional Poissonity in integer-valued conditional autoregressive models

The Poisson distribution is a representative distribution for discrete data, much like the normal distribution is for continuous data. Many time series models for count data have been developed based on the Poisson distribution. However, studies examining the validity of the Poisson assumption in these time series models have been relatively scarce. This study addresses the problem of testing for conditional Poissonity in integer-valued conditional autoregressive models. For this purpose, we introduce a test statistic based on the information matrix of the likelihood function. Under regularity conditions, it is shown that the proposed test has an asymptotic null distribution following a chi-square distribution. Simulation results demonstrate the validity of the proposed test. Additionally, a real data analysis is provided for illustration.