We study nonlinear infinite order Markov switching integer-valued ARCH models for count time series data. Markov switching models take into account complex dynamics and can deal with several stylistic facts of count data including proper modelling of nonlinearities, overdispersion and outliers. We study structural properties of those models. Under mild conditions, we prove consistency and asymptotic normality of the maximum likelihood estimator for the case of finite order autoregression. In addition, we give conditions which imply that the marginal likelihood ratio test, for testing the number of regimes, converges to a Gaussian process. This result enables us to prove that the BIC provides a consistent estimator for selecting the true number of regimes. A real data example illustrates the methodology and compares this approach with alternative methods.
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