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.

中文翻译：

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非线性泊松自回归的混合

我们研究了用于计数时间序列数据的非线性无限阶马尔可夫切换整数值 ARCH 模型。马尔可夫切换模型考虑了复杂的动力学，可以处理计数数据的几个风格事实，包括非线性、过度分散和异常值的正确建模。我们研究这些模型的结构特性。在温和条件下，我们证明了有限阶自回归情况下最大似然估计量的一致性和渐近正态性。此外，我们给出了条件，这些条件意味着边际似然比测试，用于测试制度的数量，收敛到一个高斯过程。这一结果使我们能够证明 BIC 为选择真实的制度数量提供了一致的估计量。