We propose a general class of INteger-valued Generalized AutoRegressive Conditionally Heteroskedastic (INGARCH) processes by allowing time-varying mean and dispersion parameters, which we call time-varying dispersion INGARCH (tv-DINGARCH) models. More specifically, we consider mixed Poisson INGARCH models and allow for a dynamic modeling of the dispersion parameter (as well as the mean), similarly to the spirit of the ordinary GARCH models. We derive conditions to obtain first and second order stationarity, and ergodicity as well. Estimation of the parameters is addressed and their associated asymptotic properties established as well. A restricted bootstrap procedure is proposed for testing constant dispersion against time-varying dispersion. Monte Carlo simulation studies are presented for checking point estimation, standard errors, and the performance of the restricted bootstrap approach. The inclusion of covariates is also addressed and applied to the daily number of deaths due to COVID-19 in Ireland. Insightful results were obtained in the data analysis, including a superior performance of the tv-DINGARCH processes over the ordinary INGARCH models.

中文翻译：

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时变离散整数值 GARCH 模型

我们通过允许时变均值和色散参数提出了一类通用整数值广义自回归条件异方差 (INGARCH) 过程，我们将其称为时变色散 INGARCH (tv-DINGARCH) 模型。更具体地说，我们考虑混合 Poisson INGARCH 模型，并允许对离散参数（以及平均值）进行动态建模，类似于普通 GARCH 模型的精神。我们推导出获得一阶和二阶平稳性以及遍历性的条件。解决了参数的估计问题，并建立了它们相关的渐近特性。提出了一种受限制的引导程序来测试恒定色散与时变色散的关系。蒙特卡罗模拟研究用于检查点估计、标准误差、以及受限引导方法的性能。协变量的包含也适用于爱尔兰因 COVID-19 导致的每日死亡人数。在数据分析中获得了深刻的结果，包括 tv-DINGARCH 过程优于普通 INGARCH 模型的性能。