## 4 Experimental Results
### Multi-Time-Step Prediction Comparisons between Proposed and Benchmark Models  
In this subsection, we evaluated the performance of the proposed models with various look-back periods and provided the experimental results to predict multi-time-step ahead for the three stock market indices. The look-back periods of 5, 21, and 42 days and the look-ahead period of five days were adopted for each period. The proposed and benchmark models were implemented as described in previous sections. The Adam optimizer and OHLV features were used for all methods in Table 6.  
Table 6 compares the proposed and benchmark models in terms of different look-back periods for five-time-step prediction, where the best performance results are marked in bold for each stock market index, period, and metric. From the table, the proposed models outperformed the benchmarks in 66.7% and 66.7% of cases for the period from 1 January 2000 through 31 December 2019; in 22.2% and 11.1% of cases for the period from 1 January 2017 through 31 December 2019, before the COVID-19 pandemic; and in 55.6% and 55.6% of cases for the period from 1 January 2019 through 31 December 2021, after the COVID-19 pandemic in terms of MSE and MAE, respectively. For long-term predictions, the MSE and MAE were not as good as for short-term predictions. Specifically, the results showed that the errors grew with the increase in prediction steps, highlighting that long-term predictions are more challenging than short-term ones. Nonetheless, the ensemble model still outperformed conventional benchmark models in long-term predictions. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Look-Back**} & \multirow{2}{*}{**Metric**} & \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**2000-2019 1**} & \multicolumn{3}{c}{**2017-2019**} & \multicolumn{3}{c}{**2019-2021**} \\ \cline{4-11}  & & & **DAX** & **DOW** & **S\&F500** & **DAX** & **DOW** & **S\&F500** & **DAX** & **DOW** & **S\&F500** \\ \hline  
We evaluated the performance of the proposed models with four different features and two different optimizers. The average MSE and MAE for the use of different features and optimizers of the proposed models over the three periods are shown in Tables 7 and 8, where the best performance results are marked in bold for each stock market index, a look-back period, and optimizer. For multi-time-step prediction, in terms of the average MSE over the three periods, Table 7 confirms that the CNN-LSTM, GRU-CNN, and ensemble models with the introduced medium feature outperformed the other models in 66.7%, 83.3%, and 83.3% of cases with the DAX dataset; 66.7%, 66.7%, and 50% of cases with the DOW dataset; and 66.7%, 50%, and 83.3% of cases with the S&P500 dataset. Further, for multi-time-step prediction, in terms of the average MAE over the three periods, Table 8 shows that the CNN-LSTM, GRU-CNN, and ensemble models with the novel medium feature outperformed the other models in 83.3%, 100%, and 83.3% of cases with the DAX dataset; 66.7%, 100%, and 66.7% of cases with the DOW dataset; and 66.7%, 50%, and 83.3% of cases with the S&P500 dataset. In addition, the proposed models were trained for 1500 epochs with the Adam optimizer and OHLV features to achieve higher performance than that of the model as described in Section 3.2.4. Figures 9-11 compare the actual and predicted close prices of the DAX, DOW, and S&P500 indices, respectively, with respect to the different look-back periods. In Figures 9-11, the left, middle, and right plots correspond to the look-back periods of 5, 21, and 42 days, respectively. The look-back period and stock market index also evidently affect the model performance.