  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.
