Integer-valued time series models have received growing attention in the past three decades, and the integer-valued GARCH models are successful and popular but most existing models assume a linear intensity process. There are many valuable nonlinear phenomena in real life which deserve to be studied, so we propose two classes of negative binomial integer-valued GARCH models with a nonlinear intensity process, which are named as threshold and hysteretic models, respectively. The hysteretic model is a generalization of the threshold one, which enjoys a more flexible regime-switching mechanism. Stability properties of these models are established. The estimation procedure is discussed in detail for the hysteretic model. As an application, we bring attention to some features of the daily number of trades of a stock which have been overlooked in previous studies.

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本报告也是国家天元数学东北中心统计学主题的系列报告之一。