## 5 Empirical Results  
In this section, we denote the three cases as follows: in sample (IS), stationary out-of-sample (OS), and nonstationary out-of-sample (NS). For each of these cases we report the FPAA and the OA. The results are reported in Table 1.  
We first examine the four experiments we conducted in the IS case. The OA declines as we predict larger subsets of the data for testing, albeit in-period. However, the chance that we do better than a coin toss over forecast blocks (FPAA) keeps increasing. As the test dataset increases, since the algorithm already shows greater than 50% accuracy, the average likelihood that any prediction sample will be predicted more accurately than 50% naturally increases, as is expected. (When you have a biased coin with more heads than tails, the number of experiments in which heads will exceed tails will increase in the length of the number of coins tossed in each experiment.) Ultimately, if the test dataset is the same as the training data, the small level of predictability is amplified to deliver very high levels of average accuracy. The histograms of all experiments are shown in Figure 9. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Experimental Case** & \(\mathbf{F=10}\) & \(\mathbf{F=30}\) & \(\mathbf{F=1000}\) & \(\mathbf{F=5000}\) \\ \hline \(A_{1}\). In-sample (IS) & & & & \\ (OA, FPAA) & (0.580, 0.579) & (0.556, 0.638) & (0.525, 0.882) & (0.522, 0.927) \\ No of \(\sigma\)s away from 0.527 & 0.31 & 0.30 & 0.12 & 0.31 \\ (Precision, Recall) & (0.570, 0.983) & (0.545, 0.986) & (0.525,0.991) & (0.526,0.999) \\ (AUC, F1) & (0.528, 0.702) & (0.517, 0.696) & (0.506,0.688) & (0.505,0.689) \\ \hline \(A_{2}\). Stationary (OS) & & & & \\ (OA, FPAA) & (0.592, 0.622) & (0.551, 0.619) & (0.524, 0.910) & - \\ No of \(\sigma\)s away from 0.527 & 0.38 & 0.25 & 0.17 & - \\ (Precision, Recall) & (0.580,0.978) & (0.541,0.984) & (0.525,0.999) & - \\ (AUC, F1) & (0.540,0.707) & (0.513,0.692) & (0.501,0.688) & - \\ \hline \(A_{3}\). Nonstationary (NS) & & & & \\ (OA, FPAA) & (0.582, 0.589) & (0.552, 0.667) & (0.534, 0.887) & - \\ No of \(\sigma\)s away from 0.527 & 0.36 & 0.28 & 0.26 & - \\ (Precision, Recall) & (0.573,0.981) & (0.546,0.992) & (0.535,0.999) & - \\ (AUC, F1) & (0.526,0.705) & (0.508,0.699) & (0.502,0.696) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Accuracy levels for three cases of experiments on stock market predictability. We report the overall accuracy (OA) and forecast period average accuracy (FPAA). The first number in each cell is the OA and the second number is the FPAA, computed at a threshold of 50%, i.e., \(FPAA=1\) if the percentage of correct forecasts in the prediction period (\(F\)) is greater than 0.5, else \(FPAA=0\). We may also compute FPAA for a threshold of 0.527, i.e., the baseline percentage of times the stock market rises, but we get identical results for \(F=10,30\) and slightly lower values for \(F=1000\). We also report how far from the threshold of 0.527 (in standard deviations) the OA is. In no case, is the predictive power significant. In addition, standard metrics such as precision, recall, AUC, and F1 scores are provided. Next, we look at the case where we use OS values from within the same period, i.e., the stationary (OS) case. The histograms of all experiments for the OS case are shown in Figure 10. The difference in accuracy between the IS and OS cases may be attributed to randomness because in both cases the training and the testing sample are drawn from within the same continuous period of time. In the third set of results for the nonstationary (NS) case, we see almost similar results to the stationary case. The histograms of all experiments for the NS case are shown in Figure 11.  
The patterns in Table 1 show that OA declines as the forecast period lengthens for all three cases of forecast experiments. The baseline is the percentage of days for which the market moved up, which for the current data sample is 52.7%.