## 4 Empirical Study  
### Data: Local Government Property Insurance Fund  
We evaluate our hierarchical GLM using an insurance dataset from the Wisconsin Local Government Property Insurance Fund (LGPIF) (The Local Government Property Insurance Fund was established to provide insurance coverage to government properties not owned by the State of Wisconsin. The Fund made insurance available for local government property such as municipal buildings, schools, libraries and vehicles. For further details, refer to [[https://sites.google.com/a/wisc.edu/local-government-property-insurance-fund](https://sites.google.com/a/wisc.edu/local-government-property-insurance-fund)]([https://sites.google.com/a/wisc.edu/local-government-property-insurance-fund](https://sites.google.com/a/wisc.edu/local-government-property-insurance-fund)) accessed on 10 October 2024). Compiled by the actuarial research team at the University of Wisconsin, this dataset encompasses information on insurance coverage for \(H=1679\) policies pertaining to various government building units across Wisconsin. This dataset presents unique challenges, particularly regarding unobservable heterogeneity (RQ1) in the log-normal outcome variable influenced by the NDB covariate (RQ2). This paper employs a frequency-severity approach to risk premium modeling, incorporating four covariates: two for the claim count model (a binary covariate \(\mathbf{z}^{F}\) (_AC15_: 1 or 0) and a continuous covariate \(\mathbf{x}^{F}\) (_LnCoverage_)), and two for the claim amount model (also a binary covariate \(\mathbf{z}^{S}\) (_Fire5_: 1 or 0) and a continuous covariate \(\mathbf{x}^{S}\) (_lnDeduct_)). The outcome variables--claim count and claim amount--are represented as \(N_{h}\) and \(\widehat{Y}_{h}\), respectively. Therefore, the structure of this dataset is outlined as follows:  
\[\begin{array}{llll}Year_{1},&\ldots,&Year_{y}\\ \text{Policy }(h=a)\text{:}&\{(N_{a},\mathbf{X}_{a}^{F},Y_{a(1)},\ldots Y_{a(N_{ a})},\mathbf{X}_{a}^{S}),\ \ldots,(N_{a},\mathbf{X}_{a}^{F},Y_{a(1)},\ldots Y_{a(N_{a})},\mathbf{X}_{a}^{S} )\}\\ \text{Policy }(h=b)\text{:}&\{(N_{b},\mathbf{X}_{b}^{F},Y_{b(1)},\ldots Y_{b(N_{ b})},\mathbf{X}_{b}^{S}),\ \ldots,(N_{b},\mathbf{X}_{b}^{F},Y_{b(1)},\ldots Y_{b(N_{b})},\mathbf{X}_{b}^{S} )\}\\ &\hskip 28.452756 pt\vdots\\ \text{Policy }(h=H)\text{:}&\{(N_{H},\mathbf{X}_{H}^{F},Y_{H(1)},\ldots Y_{H(N_{ H})},\mathbf{X}_{H}^{S}),\ \ldots,(N_{H},\mathbf{X}_{H}^{F},Y_{H(1)},\ldots Y_{H(N_{H})},\mathbf{X}_{H}^{S })\}\end{array}\]  
The experiment concerns predicting the aggregate claim amount \(E[S_{h}|\mathbf{X}]\) using the frequency-severity principle to establish risk premiums for specific policies. Predictions are categorized into six distinct entity types: _city, county, school, town, village_, and _miscellaneous_.  
These classifications represent the source of the insured property, effectively organizing the policies into six fixed risk clusters (i.e., \(j=1,\ldots,6\)), representing six distinct entity types. This approach allows for the incorporation of varying risk characteristics associated with each entity type in the predictions. As discussed in the main text, the continuous covariate \(\mathbf{x}^{5}\) in the claim amount model is assumed to be influenced by NDB error.