Modeling Insurance Claim Count Dynamics using Regime-Switching INGARCH Model
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Zabibu Afazali
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Bård Støve
Abstract
This paper analyzes insurance claim count data using a Regime-Switching Integer-Valued Generalized Autoregressive Conditional Heteroscedasticity (RS-INGARCH) model. Weekly claim counts from a Ugandan insurance company, spanning 2020–2024 and covering Motor Private and Motor Commercial lines of business, are used. The study examines both the COVID-19 period (March 2020–January 2022) and the post-COVID-19 period, which saw major structural shifts in claims. The RS-INGARCH model outperforms benchmark models such as the Integer-Valued Autoregressive (INAR) and single-regime INGARCH models, based on Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Mean Squared Error (MSE), and out-of-sample forecasting accuracy. Performance gains were especially notable when the COVID-19 period was included, suggesting the RS-INGARCH(1,1) model better captures abrupt changes in claim dynamics. The Autocorrelation Function (ACF) plots and Ljung–Box test confirms reduced residual autocorrelation under RS-INGARCH(1,1), and the INGARCH(1,1) specification performed best among alternative lag choices. A two-regime specification was preferred over three regimes, as the latter introduced instability and overfitting due to limited data in infrequent regimes. The study recommends RS-INGARCH(1,1) models for forecasting and risk evaluation during periods of structural change, especially in emerging insurance markets.
Funding source: Direktoratet for Utviklingssamarbeid
Award Identifier / Grant number: NORAD, project no 68105
Acknowledgments
We thank our sponsors, NORHED II, and the principal investigators of the MATH4SDG project, Prof. John Mango Magero, Prof. Eunice Mureithi and Prof. Guttorm Alendal. We appreciate the insurance company for providing data and thank Geir D. Berentsen for his feedback that enhanced the quality of this work. We further extend our sincere appreciation to the two anonymous reviewers and the editor for their constructive comments and guidance throughout the review process.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: Zabibu Afazali conceptualized the study, conducted the primary analysis, and drafted the manuscript as part of her PhD work under the supervision and guidance of Bård Støve, Juma Kasozi, and Saint Kizito Omala, who provided critical feedback. Bård Støve provided methodological input and extensively engaged throughout the data analysis, interpretation, writing, and approval of the results for the study. All authors reviewed the results, approved the final manuscript, and consented to its submission.
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Use of Large Language Models, AI and Machine Learning Tools: During the preparation of this work, the authors used Grammarly for grammar and language editing. After applying this tool, the authors carefully reviewed, edited, and took full responsibility for the publication’s content.
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Conflict of interest: The authors declare no conflicts of interest.
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Research funding: This work has been supported by the Mathematics for Sustainable Development (MATH4SDG) project, which is a research and development project running in the period 2021–2026 at Makerere University-Uganda, the University of Dar es Salaam-Tanzania, and the University of Bergen-Norway, funded through the NORHED II program under the Norwegian Agency for Development Cooperation (NORAD, project no 68105).
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Data availability: Data are available upon request and subject to approval. The insurance claim data used in this study are confidential and cannot be publicly shared due to privacy agreements. The insurance company’s name is not disclosed per the provider’s request.
A Analysis Using Post-COVID-19 Data Only (January 2022–December 2024)
As shown in Figure 8, the two-regime RS-INGARCH model differentiates low and high claim periods, while the three-regime model adds an intermediate state for a more detailed view of claim dynamics. Both models track claim patterns effectively, with regime probabilities showing shifts over time. The three-regime model provides smoother transitions and better captures variability in claim intensity, especially for Motor Private claims from early 2023 onwards (Figure 8).
Two-regime three-regime RS-INGARCH model results for weekly motor commercial (ugMC) and motor private (ugMP) claim counts. Top panels show predictions (black), bottom panels display regime probabilities, highlighting claim shift dynamics. (a) 2-RS-INGARCH. (b) 3-RS-INGARCH.
The model comparison results based on AIC, BIC, and MSE indicate that the regime-switching INGARCH models outperform both the standard INGARCH(1,1) and INAR(1) models for both Motor Commercial (MC) and Motor Private (MP) claims. Specifically, the 2-RS-INGARCH(1,1) model achieves the lowest AIC and MSE values for MC claims, suggesting superior fit and predictive accuracy. For MP claims, the 2-RS-INGARCH(1,1) also attains the lowest AIC and MSE, although the 3-RS-INGARCH(1,1) exhibits comparable performance (Table 6). Overall, the findings highlight the importance of accounting for regime shifts in modeling claim count dynamics (Figure 8).
Model comparison based on AIC, BIC, and MSE for motor commercial (MC) and motor private (MP).
| Model | Motor commercial (MC) | Motor private (MP) | ||||
|---|---|---|---|---|---|---|
| AIC | BIC | MSE | AIC | BIC | MSE | |
| INAR(1) | 776.00 | 781.64 | 3.655 | 691.80 | 697.42 | 2.255 |
| INGARCH(1,1) | 730.46 | 738.93 | 3.665 | 604.92 | 613.36 | 1.719 |
| 2-RS-INGARCH(1,1) | 641.69 | 661.44 | 1.573 | 594.17 | 613.86 | 1.312 |
| 3-RS-INGARCH(1,1) | 648.80 | 688.29 | 1.510 | 606.92 | 646.29 | 1.349 |
Figure 9 shows the predictive performance of the INAR(1), INGARCH(1,1), 2-RS-INGARCH, and 3-RS-INGARCH models for Motor Private (MP) and Motor Commercial (MC) claim counts. While regime-switching models display reasonable regime identification in the in-sample period (as seen in Figure 8), they fail to capture volatility and regime changes in the out-of-sample period. Among the models, the 3-RS-INGARCH offers relatively better fit, but all models underestimate sharp fluctuations in MP and short-term bursts in MC, reflecting limited adaptability in the post-COVID-19 environment (Figure 9).
Observed weekly claim counts (colored lines) and model predictions (black lines) for motor private (MP) and motor commercial (MC) using INAR(1), INGARCH(1,1), two-regime RS-INGARCH, and three-regime RS-INGARCH models.
Figure 10 presents the MSFE and Logarithmic Score for Motor Private (MP) in panel (a) and Motor Commercial (MC) in panel (b). For MP, the INGARCH model achieves the lowest MSFE, while the 2-RS-INGARCH and 3-RS-INGARCH start with higher errors but gradually converges toward the INGARCH. In terms of Logarithmic Score, however, both RS models outperform INGARCH by assigning higher probabilities to the realized outcomes. For MC, the 3-RS-INGARCH performs poorly under MSFE, showing the highest errors across the horizon, but achieves the best Logarithmic Score, outperforming both INGARCH and 2-RS-INGARCH. These results highlight a trade-off: MSFE favors simpler models, while FS reveals that regime-switching models sometimes better capture the underlying predictive distribution. While RS models didn’t consistently outperform simpler ones, they may still capture relevant dynamics. Limited data complicates estimating switching points and transition probabilities, leading to unstable regime classifications. Although regime-switching models have theoretical advantages for structural changes, their empirical performance may be limited by data scarcity or stability environments.
Out-of-sample forecast accuracy with upper panels showing Mean Square Forecasting Error (MSFE) and lower panels showing Logarithmic Scores. (a) MP. (b) MC.
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Supplementary Material
This article contains supplementary material (https://doi.org/10.1515/apjri-2025-0011).
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