Testing Serial Dependence in Time Series Models of Counts

In the course of the analysis of time series of counts the need to test for the presence of a dependence structure arises regularly. Suitable tests for this purpose are analysed in this paper. Their size and power properties are evaluated under various alternatives among which the INARMA-processes play a prominent role. The results can be summarized as follows. (1) All the tests considered but one are robust against extra binomial variation in the data. (2) Newly proposed tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer-valued first- order and second-order autoregressive as well as first-order moving average processes. (3) The tests considered are not powerful enough to distinguish between higher-order integer-valued autoregressive processes and the popular parameter-driven processes where a dynamic latent process introduces the serial dependence into the counts. The methods and findings of this study are applied to three data sets: the so called Furth-data already analysed in the branching process literature, data on worker absenteeism and to polio incidence data.