After having fitted a model to a given count time series, one has to check the adequacy of this model fit. The (standardized) Pearson residuals, being easy to compute and interpret, are a popular diagnostic approach for this purpose. But which types of model inadequacy might be uncovered by which statistics based on the Pearson residuals? In view of being able to apply such statistics in practice, it is also crucial to ask for the properties of these statistics under model adequacy. We look for answers to these questions by means of a comprehensive simulation study, which considers diverse types of count time series models and inadequacy scenarios. We illustrate our findings with two real-data examples about strikes in the U.S., and about corporate insolvencies in the districts of Rhineland–Palatinate. We conclude with a theoretical discussion of Pearson residuals.

Model fitting for count time series is of great relevance for many economic applications. Here, we focus on the step of model selection, where information criteria like AIC and BIC are commonly used in practice. Previous studies about their model selection abilities concentrated on real-valued time series, but here, we comprehensively investigate AIC and BIC in a count time series context. In our simulations, we consider diverse scenarios of model selection, like the identification of serial (in)dependence, overdispersion, zero inflation or a trend, the order selection within a given model family as well as the model selection also across model families. We apply our findings to economic count time series about monthly numbers of strikes in the US, and about monthly numbers of corporate insolvencies in the districts of Rhineland-Palatinate.

While most of the literature about INARMA models (integer-valued autoregressive moving-average) concentrates on the purely autoregressive INAR models, we consider INARMA models that also include a moving-average part. We study moment properties and show how to efficiently implement maximum likelihood estimation. We analyze the estimation performance and consider the topic of model selection. We also analyze the consequences of choosing an inadequate model for the given count process. Two real-data examples are presented for illustration.