## 7 Directions for Future Research
Prediction errors (measured by RTE) for classical (BE) and penalized (NNG, ALASSO, RLASSO, and Lasso) methods using beta-type D across different sample sizes (n = 100 and 400) in RLASSO, and Lasso) methods using beta-type D across different sample sizes (n = 100 and 400) in low correlation (C1 and C2) and high correlation (C3 and C4) settings. Figure S5. Comparison of prediction errors quantified using RTE for classical and penalized methods, using CV, AIC, and BIC criteria across small (n = 100, upper panel) and large (n = 400, lower panel) sample sizes with low correlation (C1) using beta-type B. Figure S6. Comparison of prediction errors quantified using RTE for classical and penalized methods, using CV, AIC, and BIC criteria across small (n = 100, upper panel) and large (n = 400, lower panel) sample sizes with low correlation (C4) using beta-type B.  
Conceptualization, E.K. and W.S. ; software, E.K. ; data analysis, E.K. ; writing--original draft preparation, E.K. and W.S. ; writing--review and editing, E.K. and W.S. All authors have read and agreed to the published version of the manuscript. This work was supported by the German Research Foundation (DFG), grant number SA580/10-3 to W.S. ([[https://www.dfg.de/en/](https://www.dfg.de/en/)]([https://www.dfg.de/en/](https://www.dfg.de/en/)) (accessed on 26 June 2025)). The codes used to generate the datasets analyzed in the simulation study will be made available in a repository at [[https://github.com/[PERSON]/varselectionpredict](https://github.com/[PERSON]/varselectionpredict)]([https://github.com/[PERSON]/varselectionpredict](https://github.com/[PERSON]/varselectionpredict)). This repository will also contain the Ozone data analyzed in our study. The authors are grateful to [PERSON] (Medical Center-University of Freiburg, Germany) for her invaluable administrative support. We also thank the four anonymous reviewers for their insightful and constructive comments, which helped us improve the manuscript. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} BSS & best subset selection \\ BE & backward elimination \\ FS & forward selection \\ NNG & nonnegative garrote \\ ALASSO & adaptive lasso \\ \end{tabular}   
Figure 17: Comparison of prediction errors measured using RTE for NNG (R, CV) and ALASSO (R, CV) against Ridge (CV). The **upper** and **lower** panels shows results for low (C2) and high (C3) correlation, respectively. The **left** and **right** panels show results for small (\(n=100\)) and large (\(n=400\)) sample sizes, respectively. Figure 14: The average number of variables selected by NNG (L, CV) and ALASSO (L, CV) compared to Lasso (CV) for beta-type A. The **upper panel** represents low correlation (C2) and the lower **panel** represents high correlation (C3). The **left panel** shows results for small sample sizes (\(n=100\)), and the **right panel** shows results for large sample sizes (\(n=400\)). The horizontal dashed line indicates the number of true signal variables (7). Figure 16: Average number of variables selected by different approaches using beta-type A. The **upper panel** represents low correlation (C2) and the **lower panel** represents high correlation (C3). The **left panel** shows results for small sample sizes (\(n=100\)), and the **right panel** shows results for large sample sizes (\(n=400\)). The horizontal dashed line indicates the number of true signal variables (7).