Integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) models are a popular framework for modeling serial dependence in count time-series. While convenient for modeling, prediction, and estimation, INGARCH models lack a clear theoretical justification for the evolution step. This limitation not only makes interpretation difficult and complicates the inclusion of covariates, but can also make the handling of missing data computationally burdensome. Consequently, applying such models in an insurance context, where covariates and missing observations are common, can be challenging. In this paper, we first introduce the marginalized state-space model (M-SSM), defined solely through the marginal distribution of the observations, and show that INGARCH models arise as special cases of this framework. The M-SSM formulation facilitates the natural incorporation of covariates and missing data mechanisms, and this representation in turn provides a coherent way to incorporate these elements within the INGARCH model as well. We then demonstrate that an M-SSM can admit an observation-driven state-space model (O-SSM) representation when suitable assumptions are imposed on the evolution of its conditional mean. This lifting from an M-SSM to an O-SSM provides a natural setting for establishing weak stationarity, even in the presence of heterogeneity and missing observations. The proposed ideas are illustrated through the Poisson and the Negative-Binomial INGARCH(1,1) models, highlighting their applicability in predictive analysis for insurance data.

翻译：整数值广义自回归条件异方差（INGARCH）模型是用于建模计数时间序列序列依赖性的流行框架。尽管该模型在建模、预测和估计方面具有便利性，但其演化步骤缺乏清晰的理论依据。这一局限性不仅导致解释困难且难以纳入协变量，还可能使缺失数据的处理在计算上变得复杂。因此，在协变量和缺失观测普遍存在的保险情境中应用此类模型颇具挑战。本文首先引入基于观测边际分布定义的边缘化状态空间模型（M-SSM），并证明INGARCH模型是该框架的特例。M-SSM公式自然促进了协变量和缺失数据机制的融入，这种表示方式进而为在INGARCH模型中统一整合这些要素提供了连贯路径。进一步研究表明，当对条件均值的演化施加适当假设时，M-SSM可转化为观测驱动状态空间模型（O-SSM）。从M-SSM到O-SSM的升维为建立弱平稳性提供了自然框架，即使在存在异质性和缺失观测的情况下亦如此。本文通过泊松和负二项INGARCH(1,1)模型演示所提出的思想，凸显其在保险数据预测分析中的适用性。