A test for counting sequences of integer-valued autoregressive models

The integer autoregressive (INAR) model is one of the most commonly used models in nonnegative integer-valued time series analysis and is a counterpart to the traditional autoregressive model for continuous-valued time series. To guarantee the integer-valued nature, the binomial thinning operator or more generally the generalized Steutel and van Harn operator is used to define the INAR model. However, the distributions of the counting sequences used in the operators have been determined by the preference of analyst without statistical verification so far. In this paper, we propose a test based on the mean and variance relationships for distributions of counting sequences and a disturbance process to check if the operator is reasonable. We show that our proposed test has asymptotically correct size and is consistent. Numerical simulation is carried out to evaluate the finite sample performance of our test. As a real data application, we apply our test to the monthly number of anorexia cases in animals submitted to animal health laboratories in New Zealand and we conclude that binomial thinning operator is not appropriate.