## 5 Results and Discussions  
In Table 2, we report out-of-sample portfolio performances of eleven portfolios on three evaluation metrics. We also report hypothesis testing results of each pair of those portfolios in the Table 2, including: Table A1 is \(p\)-values of the annual volatility, Table A2 is \(p\)-values of the Sharpe ratio and Table A3 is \(p\)-values of the portfolio turnover. Similarly in Table 3, we report out-of-sample portfolio performances of eleven portfolios but only on one hundred largest assets by market capitalization (\(N=100\)). Tables A4-A6 are \(p\)-values of the annual volatility, the Sharpe ratio and the portfolio turnover respectively for these portfolios.  
A first noteworthy observation is that the most basic method in our experiment, the sample portfolio, is significantly worse than all other portfolios on all of three metrics, this shows the instability of the sample covariance matrix. Three more complicated portfolios, using the linear shrinkage estimators, have no difference both on the volatility and the Sharpe ratio. However, on the portfolio turnover, the SSIM portfolio is significantly higher than the STIM and SCCM portfolios and these two portfolios have the same turnover.  
The results from our experiment show that the undersampling and ensemble learning applied to the sample covariance matrix estimation could improve significantly the portfolio performances on the annual volatility and the Sharpe ratio. These improvements are approximately the same as three portfolios of optimal shrinkage covariance estimations. Although on the portfolio turnover, it is better than the sample portfolio but still higher than the linear shrinkage portfolios.  
These shrinkage portfolios have similar performances, except that the portfolio turnover of the SSIM portfolio is worse than the portfolio turnover of the STIM and the SCCM portfolios. Applying our approach on the shrinkage to single-index model, the \(k\)-SSIM portfolio, improves significantly the portfolio turnover of the SSIM portfolio to the same level of the STIM and the SCCM portfolios. This suggests that the estimated covariance matrix from the SSIM estimation is less stable than the STIM and SCCM estimations.  
Furthermore, combinations of more than two different matrices have the lowest portfolio turnover in our experiment and in many cases, they are better than the others. This suggests that adding more matrices that are well-structured and domain-based could stabilize the portfolio. The exceptional estimator in this study, the sTS portfolio, is the combination of the sample matrix, the diagonal matrix and the single-index model and it has the best results in most cases. The diagonal matrix in this context could be seen as a combination of the sample portfolio with the equally-weighted portfolio and that helps to increase the portfolio diversification.  
Comparing the three portfolios of optimal shrinkage estimations with the three portfolios of estimators shows that they have comparable results. This suggests that the strength of the shrinkage technique mostly come from the combination with other matrices and the \"optimal\" shrinkage intensity calculation is an additional step to make sure its results are the best. Moreover, we conclude that the combinations do not have to use different matrices, the combination of multiple sample covariance matrices by using ensemble learning and undersampling technique produces the similar performances to the advanced and optimal shrinkage estimations.