State-Space Representation of INGARCH Models and Their Application in Insurance

2025-11-17

AI Summary

INGARCH models are widely used for count time series but suffer from weak theoretical foundations, limited interpretability, difficulties in incorporating covariates, and inadequate handling of missing data—especially hindering their application in insurance. Method: We propose a marginal distribution–defined degenerate state-space model (M-SSM), establishing for the first time a rigorous theoretical link between INGARCH and state-space representations. By transforming M-SSM into an observation-driven state-space model (O-SSM), we naturally integrate exogenous covariates and missing-data mechanisms while enabling weak stationarity analysis. The framework accommodates both Poisson and negative binomial marginal assumptions, ensuring interpretability and computational robustness. Contribution/Results: Empirical evaluation demonstrates that the proposed framework substantially enhances flexibility and predictive accuracy for insurance count sequences, empirically validating the practical efficacy of INGARCH(1,1) in real-world settings.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• INGARCH models lack theoretical justification for evolution step
• Covariate inclusion and missing data handling are computationally challenging
• Insurance applications face difficulties due to model limitations

Innovation

• Proposed marginalized state-space model for count time-series
• Enabled natural covariate integration and missing data handling
• Established weak stationarity under heterogeneity via state-space lifting