Abstract: We investigate a test for the conditional variance of stationary and ergodic integer-valued time series. This hypothesis testing problem is motivated by the fact that the form of the conditional variance of the process is determined by the conditional distribution and the conditional mean. First, we estimate the unknown parameters of the intensity function using an M-estimator and prove strong consistency and asymptotic normality. Second, we show that the proposed test has asymptotic size α and is consistent. Finally, we discuss the nontrivial power of the proposed test for the local alternative. The proposed test statistic can be applied to various problems, such as specification tests for intensity functions, tests for overdispersion and underdispersion, and goodness-of-fit tests for ergodic and stationary integer-valued time series. A simulation study illustrates the finite-sample performance of the proposed test. Lastly, in a real-data application, we analyze the number of patients with Escherichia coli in Germany.

Key words and phrases: Conditional variance, integer-valued time series, intensity.