📊 statistics

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

This paper introduces the marginalized state-space model (M-SSM) framework to provide a theoretically grounded representation of INGARCH models that naturally accommodates covariates and missing data, while demonstrating how this approach facilitates the derivation of observation-driven state-space representations for establishing weak stationarity in insurance applications.

Original authors: Jae Youn Ahn, Hong Beng Lim, Mario V. Wüthrich

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jae Youn Ahn, Hong Beng Lim, Mario V. Wüthrich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an insurance company trying to predict how many car accidents or house fires will happen next year for a specific group of people. You have a history of past claims (counts of accidents), but the situation is tricky:

The "Who" changes: Different people have different risk levels (heterogeneity).
The "When" changes: Risk isn't static; it evolves based on recent history (if you had an accident last month, you might be more likely to have one this month).
The Data is messy: Sometimes you don't have data for certain years because a policyholder didn't have coverage that year.

This paper introduces a new mathematical tool to solve these problems. It takes an existing, popular method called INGARCH (which is good at tracking how past counts influence future counts) and upgrades it by wrapping it inside a State-Space Model (a framework that separates "what we see" from "what is actually happening underneath").

Here is the breakdown using simple analogies:

1. The Problem: The "Two-Hat" Dilemma

In the old way of doing things (the standard INGARCH model), the "conditional mean" (the predicted average number of claims) had to wear two hats at once:

Hat A (The Driver): It had to drive the future, deciding how the risk level evolves over time based on the past.
Hat B (The Scale): It had to set the size of the current observation, accounting for things like how many cars a person owns or their age.

The Metaphor: Imagine trying to drive a car while simultaneously acting as the road map. If you need to change the road map (because the driver is now older or driving a different car), you have to stop driving to redraw the map. This makes it very hard to handle changing circumstances (covariates) or to ensure the driving remains stable over time.

2. The Solution: The "Shadow Driver" (State-Space)

The authors propose a new model that splits these two jobs. They introduce a latent state (a hidden "shadow driver") that does the driving, while a separate a priori rate (the road map) sets the scale.

The Shadow Driver ( $\Theta_t$ ): This is the hidden risk level. It evolves smoothly over time based on past accidents. The authors prove that this shadow driver can be "weakly stationary," meaning its behavior is stable and predictable, even if the real-world data looks chaotic.
The Road Map ( $\lambda_t$ ): This is the "a priori rate." It handles all the external factors: the driver's age, the type of car, the city they live in, or even if they had no car at all this year (zero exposure).

The Metaphor: Now, the Shadow Driver just focuses on driving the car (evolving the risk). The Road Map tells the driver, "Today, you are driving a truck in a snowstorm, so adjust your speed accordingly." If the driver has no car today (missing data), the Road Map just says "0 speed," and the Shadow Driver waits patiently without getting confused.

3. The "Negative Binomial" Twist

Insurance claims are "count data" (you can't have 2.5 accidents; it's 0, 1, 2, etc.). The paper specifically uses a Negative Binomial distribution.

Why? Real-world insurance data is "overdispersed." This means the variance (the spread of accidents) is much higher than the average. Some people have zero accidents for years; others have a cluster of them. The Negative Binomial is like a "stretchy" ruler that can handle this wild variability better than a standard Poisson model.

4. Handling Missing Data (The "Zero Exposure" Trick)

One of the biggest headaches in insurance is when a customer drops a policy for a year.

Old Way: You have to do complex math to "fill in the blanks" or integrate over missing possibilities.
New Way: Because the model separates the "scale" ( $\lambda_t$ ) from the "driver," if a customer has no policy, you simply set the scale to zero.
The Result: The math automatically knows that "Zero exposure = Zero claims." The model skips the complicated calculation for that year and just lets the "Shadow Driver" evolve naturally to the next year. It's like pausing a video game without losing your progress.

5. The Proof: Stability and Prediction

The authors didn't just build the model; they proved it works mathematically.

Stationarity: They showed that even though the real-world data changes (people get older, cars change), the hidden "Shadow Driver" remains stable. This is crucial for insurance because you need to know that your risk model won't blow up over time.
Real-World Test: They tested this on real data from the Wisconsin Local Government Property Insurance Fund (1,234 local governments over 5 years).
- The Result: Their new model performed just as well as (and sometimes better than) existing models in predicting future claims.
- The Bonus: It was faster and easier to compute because it didn't require complex numerical integration to handle the hidden states.

Summary

This paper is about untying a knot. It takes a complex insurance prediction model, untangles the "evolving risk" from the "changing circumstances," and wraps it in a mathematically stable package.

For the Actuary: It means you can easily add new variables (like a new law or a change in car type) without breaking the model's stability.
For the Data: It handles missing years (no policy) effortlessly.
For the Math: It proves that the hidden "risk engine" is stable, giving insurers confidence in their long-term pricing.

The paper concludes that this "Heterogeneous NB-INGARCH" model is a practical, robust tool for setting fair insurance prices in a changing world.

Technical Summary: State-Space Representations of Negative-Binomial INGARCH Models for Insurance Ratemaking

1. Problem Statement

The paper addresses specific limitations of Integer-valued Generalized Autoregressive Conditional Heteroskedastic (INGARCH) models when applied to insurance ratemaking, particularly for property and casualty (P&C) insurance. While INGARCH models are popular for modeling serial dependence in count time series due to their tractable likelihood and affine recursion, they face two primary challenges in insurance contexts:

Incorporation of Covariates: Standard INGARCH formulations define the conditional mean as a linear combination of past observations and past conditional means. Introducing exogenous covariates (to capture policyholder heterogeneity) is difficult because the conditional mean must remain strictly positive. Adding linear covariate terms requires restrictive non-negativity constraints on coefficients and regressors, while log-link alternatives disrupt the affine structure and the associated credibility interpretation.
Stationarity vs. Heterogeneity: Insurance panels typically feature short time series (e.g., $\le 10$ periods) and time-varying policyholder characteristics. Standard stationarity assumptions for the observed count process are inappropriate when the scale of observations changes systematically over time due to covariates. However, a suitable notion of stationarity is essential for ratemaking. In classical INGARCH, the conditional mean serves a dual role: governing temporal evolution and determining the observation scale, making it difficult to separate these dynamics to define stationarity at a latent level.

2. Methodology

The authors propose a Heterogeneous Negative-Binomial INGARCH(1,1) model for insurance ratemaking. The core methodological innovation is the decomposition of the predictive mean into a time-varying a priori rate ( $\lambda_t$ ) and a baseline conditional mean ( $M_{t|t-1}$ ).

Key Methodological Steps:

Model Decomposition: The expected claim count is defined as $E[Z_t | Z_{\le t-1}] = \lambda_t M_{t|t-1}$ $E [Z_{t} ∣ Z_{\leq t - 1}] = λ_{t} M_{t ∣ t - 1}$ .
- $\lambda_t$ : Captures time-varying heterogeneity via exogenous covariates (e.g., $\lambda_t = \exp(x_t^\top w)$ ).
- $M_{t|t-1}$ : Follows an affine INGARCH(1,1) recursion ( $M_{t+1|t} = \beta^{[0]}_t + \beta^{[1]}_t Z_t + \beta^{[2]}_t M_{t|t-1}$ ), preserving the dynamics of the underlying risk state.
State-Space Equivalence (M-SSM): The authors demonstrate that the proposed Heterogeneous NB-INGARCH model is equivalent to a Marginalized State-Space Model (M-SSM) with:
- Observation Equation: $Z_t | \Theta_t \sim \text{Pois}(\lambda_t \Theta_t)$ .
- Marginal Evolution: $\Theta_{t+1} | Z_{\le t} \sim \text{Gamma}(a_{t+1|t}, b_{t+1|t})$ .
- Here, $\Theta_t$ represents the latent risk state. The equivalence allows the model to handle time-varying heterogeneity naturally through the observation layer while modeling serial dependence through the latent state dynamics.
Weak Stationarity via Lifting: A critical theoretical contribution is addressing the fact that M-SSMs specify only conditional marginal distributions, leaving the covariance structure of the latent process undefined.
- The authors show that under specific parameter restrictions (specifically relating the evolution parameters $\Delta_t$ and $q_t$ ), the M-SSM can be "lifted" to a full Linear State-Space Model (LSSM).
- In this LSSM lift, the latent state $\Theta_t$ satisfies a linear conditional mean evolution: $E[\Theta_{t+1} | \Theta_t, Z_{\le t}] = \Delta_t \Theta_t + (1-\Delta_t)$ .
- Crucially, the authors prove that while the full joint evolution (the "lift") is not unique, the covariance structure of the latent process is uniquely determined by the M-SSM specification. Consequently, if $\Delta_t \equiv \Delta \in (0, 1]$ , the latent state process is weakly stationary, even in the presence of time-varying heterogeneity in the observations.

3. Key Contributions

Heterogeneous NB-INGARCH Formulation: The paper introduces a model that successfully integrates time-varying covariates (heterogeneity) into the INGARCH framework without violating the affine structure required for credibility interpretations.
M-SSM Representation: It establishes an equivalence between the Heterogeneous NB-INGARCH model and a Poisson-Gamma M-SSM, providing a unified framework where covariates affect the observation scale and dynamics affect the latent state.
Resolution of Stationarity Issues: The paper resolves the conflict between time-varying heterogeneity and stationarity by formulating weak stationarity at the level of the latent state process. It proves that the covariance structure of this latent process is well-defined and unique across all compatible LSSM lifts, justifying the use of weak stationarity assumptions in short-panel insurance data.
Handling Missing Data: The state-space formulation allows for a principled treatment of missing data (e.g., zero exposure periods) by setting the a priori rate $\lambda_t = 0$ . This renders the observation degenerate at zero, allowing the model to skip the filtering step and propagate uncertainty solely through the evolution equation, avoiding the need for numerical integration in the likelihood.

4. Results

The paper validates the model using real P&C insurance data from the Wisconsin Local Government Property Insurance Fund (LGPIF), covering 1,016 entities from 2006–2010, with a hold-out test set from 2011.

Comparative Performance: The proposed Heterogeneous NB-INGARCH model was compared against a stationary Poisson-AR(1) state-space model and a Poisson-Gamma random-effects model.
- Log-Likelihood: The proposed model achieved the highest out-of-sample log-likelihood (-176.94), outperforming the Poisson-AR(1) (-177.90) and the random-effects model (-179.02).
- MSE: The Poisson-AR(1) model had the lowest Mean Squared Error (0.184 vs. 0.193 for the proposed model), though the differences were marginal.
Parameter Estimates: The estimated persistence parameter $\hat{\Delta} \approx 0.925$ (less than 1) indicates that the data supports dynamic updating beyond a pure random-effects structure ( $\Delta=1$ ), confirming the necessity of the INGARCH dynamics.
Computational Efficiency: Unlike the Poisson-AR(1) model, which requires numerical integration or filtering over the latent state for likelihood evaluation, the proposed NB-INGARCH model yields a closed-form negative-binomial predictive distribution. This results in a simpler and faster estimation procedure.

5. Significance and Claims

The paper claims that its primary significance lies in bridging the gap between INGARCH models and State-Space Models (SSMs) for insurance applications.

Interpretability: By decoupling the observation scale (heterogeneity) from the latent dynamics, the model offers a clearer interpretation of risk evolution, aligning with actuarial credibility theory.
Practical Applicability: The framework is specifically designed for the constraints of insurance data: short time series, the necessity of covariates, and the need for a stationarity concept that accommodates time-varying policyholder characteristics.
Theoretical Justification: The paper provides the necessary theoretical underpinning (via the unique covariance structure of the LSSM lift) to justify using weak stationarity assumptions for latent states in heterogeneous settings, a feature previously difficult to reconcile in standard INGARCH formulations.

The authors conclude that while future work could extend these results to broader INGARCH classes, the proposed framework offers a robust, computationally efficient, and theoretically sound tool for predictive analysis in P&C ratemaking.