## 4 Empirical Study
### Model Implementation
(2021) [30], and it is given by:  
\[\sigma_{je}^{2}=\frac{V(\mathbf{x}^{S})}{\rho_{j(x^{S},x^{S*})}^{2}}-V( \mathbf{x}^{S}) \tag{36}\]  
Therefore, the selection of cluster-specific random correlations \(\rho_{1(x^{S},x^{S*})},\cdots\rho_{j(x^{S},x^{S*})}\) can lead to variations in the error severity in the NDB covariate \(\mathbf{x}^{S*}\). We generate three simulation datasets, based on the LGPIF dataset, corresponding to different error rate \(R_{e_{x}}\) scenarios in \(\mathbf{x}^{S*}\), with error rates set at 1%, 10%, and 40% (we raise the question of whether an NDB covariate with an excessive error rate (e.g., \(>\)50%) is worth including at all, suggesting that it might be better to exclude such a covariate). The definition of error rate that we use is:  
\[R_{e_{x}}=\frac{\Sigma_{h=1}^{H}|x_{h}^{S*}-x_{h}^{S}|}{\Sigma_{h=1}^{H}x_{h}^ {S}}: \begin{cases}0.01\ (1\%)\text{ error rate in }\mathbf{x}^{S*}\text{ for dataset A.}\\ 0.1\ (10\%)\text{ error rate in }\mathbf{x}^{S*}\text{ for dataset B.}\\ 0.4\ (40\%)\text{ error rate in }\mathbf{x}^{S*}\text{ for dataset C.}\end{cases}  \tag{37}\]  
**Candidate models:** As noted earlier, our goal is to determine the optimal scaling factor \(\zeta\), which varies depending on the error rate \(R_{e_{x}}\)--\(1\%,10\%,40\%\)--in the NDB covariate \(\mathbf{x}^{S*}\). The performance of the hierarchical GLM, with the corresponding optimal \(\zeta\) for each error rate, is then compared to that of the SIMEX method. Figure 4 presents the development process for four distinct risk premium models, facilitating a comprehensive comparison between the Gustafson correction method and SIMEX within the Bayesian hierarchical GLM and traditional GLM frameworks. Each model, labeled (A) through (D), is designed to systematically assess the impact of different error rates in the NDB covariate \(\mathbf{x}^{S*}\). Model (A), built using the true covariates within the hierarchical GLM framework, serves as the gold standard and provides a performance benchmark. This benchmark is critical for evaluating the effectiveness of the correction methods. In contrast, Model (B) highlights the detrimental effect (model risk) of using the NDB covariate \(\mathbf{x}^{S*}\) in the same hierarchical GLM framework. It is essential to compare this error-prone Model (B) with Model (C), which incorporates [PERSON]'s correction, and Model (D), which applies the SIMEX correction with conventional GLM. The purpose of this comparison is to assess how well these correction methods mitigate the negative effects of NDB errors in the covariate \(\mathbf{x}^{S*}\). **Choice of hyperparameters:** To run the Gibbs sampler for the hierarchical models--Model (A), -(B), and -(C)--in Figure 4, flat priors are selected on the hyperparameters \(\mathbf{\phi:}\)\(\{m_{0},\delta,q_{0},\Lambda,\rho_{u1},\rho_{u2},\rho_{v1},\rho_{v2},c_{0},d_{0},g_{0},h_{ 0}\}\) as outlined in Equation (21) through (24) and Equation (26):  
\[ \begin{split}\{\underline{m}_{0}=\mathbf{\beta}_{GLM}^{F},\ m_{0}= \mathbf{\beta}_{GLM}^{S},\ \delta=0.01,\ \delta=0.01\}&\text{for $\mathbf{\beta}_{0}^{F}$ and $\mathbf{\beta}_{0}^{S}$}\\ \{\underline{q}_{0}=p+2,\ q_{0}=p+2,\ \underline{\Lambda}=\Sigma_{ GLM}^{F},\ \Lambda=\Sigma_{GLM}^{S}\}&\text{for $\Sigma_{\beta_{0}}^{F}$ and $\Sigma_{\beta_{0}}^{S}$}\\ \{\underline{\rho}_{u1}=0.125,\ \underline{\rho}_{u1}=0.125,\ \underline{\rho}_{u2}=15,\ \underline{\rho}_{u2}=15\}&\text{for $\mathbf{u}_{0}^{F}$ and $\mathbf{u}_{0}^{S}$}\\ \{\underline{\rho}_{v1}=8,\ \rho_{v1}=8,\ \underline{\rho}_{v2}=1,\ \rho_{v2}=1\}&\text{for $\mathbf{v}_{0}^{F}$ and $\mathbf{v}_{0}^{S}$}\\ \{c_{0}^{F}=0.5,\ c_{0}^{S}=0.5,\ d_{0}^{F}=0.5,\ d_{0}^{S}=0.5\}& \text{for $\lambda^{2F}$ and $\lambda^{2S}$}\\ \{g_{0}^{F}=0.5,\ g_{0}^{S}=0.5,\ h_{0}^{F}=0.5,\ h_{0}^{S}=0.5\}& \text{for $\pi^{F}$ and $\pi^{S}$}\end{split}  \tag{38}\]  
where \(\mathbf{\beta}_{GLM}^{F},\mathbf{\beta}_{GLM}^{S}\) denote the naive GLM coefficient vectors, serving as initial values for the Gibbs sampler.