# Error-Function-Based Penalized Quantile Regression in the Linear Mixed Model
## 4 Financial Data Analysis  
The DEFF-QR method is developed in the financial statements and balance sheets of 12 companies from 2009 to 2023 [25]. Due to missing data, we select 10 companies from 2009 to 2022 for simulation. The revenue is regarded as the response variable. We aim to find the factors influencing the revenue. Consider 20 variables as explanatory variables as in Table 5.  
Figure 4: Boxplots of different methods in the dense model. Figure 3: Boxplots of different methods in the sparse model. In Table 5, \(X_{20}\), \(X_{21}\), \(X_{22}\), \(X_{23}\), \(X_{24}\), and \(X_{25}\) denote the numerical variables for the company's six categories. Specifically, each category is represented using binary indicators, where applicable categories are marked with a 1 and non-applicable ones with a 0. The other explanatory variables are continuous. Consider the following linear mixed-effects model:  
\[Y_{ij}=x_{ij}^{\intercal}\beta+z_{ij}^{\intercal}a_{i}+\epsilon_{ij},\quad i=1,2,\ldots,10,\quad j=1,2,\ldots,14, \tag{14}\]  
where \(Y_{ij}\) denotes the revenue of company \(i\) in year \(j\). For the random-effects coefficients \(\alpha_{i}\), we consider that random effects exist in all variables, i.e., \(z_{ij}^{\intercal}=x_{ij}^{\intercal}\). We empirically evaluated DERF-QR under \(\sigma=0.2\) and \(\sigma=2\), at quantile levels \(\tau=0.25\), 0.5, and 0.75. For comparison, we also included the DLQR method. The mean squared error (MSE) is defined as  
\[\text{MSE}=\frac{1}{nm}\sum_{i=1}^{n}\sum_{j=1}^{m}\left(Y_{ij}-\hat{Y}_{ij} \right)^{2}\]  
and the number of non-zero coefficients (model size, MS) is used to assess variable selection performance. Figure 5 presents the estimated coefficients across quantiles. Both DERF-QR and DLQR successfully eliminate most irrelevant variables. Notably, all methods consistently selected \(X_{6}\) and \(X_{9}\), while the other variables are very small or shrink to zero. This indicates that gross profit and EBITDA have a significant impact on revenue. We find that variable selection differs across quantile levels. At \(\tau=0.75\), all three methods tend to select \(X_{6}\), \(X_{9}\), and \(X_{14}\). Table 6 shows the MSE and MS for each method. When \(\sigma=0.2\), DERF-QR achieves the lowest MSE at \(\tau=0.25\) and \(\tau=0.75\). With \(\sigma=2\), DERF-QR attains the lowest MSE at \(\tau=0.5\) and also yields the smallest MS at both \(\tau=0.5\) and \(\tau=0.75\). DERF-QR consistently yields the lowest MSE across quantile levels. Overall, DERF-QR consistently achieves superior estimation accuracy and variable selection across quantile levels. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{\(\tau=0.25\)} & \multicolumn{2}{c}{\(\tau=0.5\)} & \multicolumn{2}{c}{\(\tau=0.75\)} \\ \cline{2-7}  & **MSE** & **MS** & **MSE** & **MS** & **MSE** & **MS** \\ \hline DERFQR (\(\sigma=0.2\)) & 0.230 & 11 & 0.048 & 11 & 0.060 & 13 \\ DERFQR (\(\sigma=2\)) & 0.256 & 11 & 0.032 & 11 & 0.061 & 11 \\ DLQR & 0.266 & 10 & 0.046 & 15 & 0.063 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: MSE and MS for different methods at various quantile levels.