A moment-based test for conditional Poissonity in integer-valued conditional autoregressive models

The Poisson distribution, a fundamental tool in the analysis of discrete data, holds a position of importance comparable to that of the normal distribution in continuous data analysis. This distribution has been widely employed in many time series models developed for count data. However, the question of whether the Poisson assumption is valid in these models has received limited attention. This study specifically addresses testing for conditional Poissonity within integer-valued conditional autoregressive models. To this end, we propose a moment-based test statistic and derive its asymptotic null distribution under regularity conditions. We also extend the test to evaluate conditional negative binomiality. Simulation results confirm the validity and effectiveness of the test. Finally, we illustrate its application through a real data analysis.