Testing the dispersion structure of count time series using Pearson residuals

Pearson residuals are a widely used tool for model diagnostics of count time series. Despite their popularity, little is known about their distribution such that statistical inference is problematic. Squared Pearson residuals are considered for testing the conditional dispersion structure of the given count time series. For two popular types of Markov count processes, an asymptotic approximation for the distribution of the test statistics is derived. The performance of the novel tests is analyzed and compared to relevant competitors. Illustrative data examples are presented, and possible extensions of our approach are discussed.