# Hybrid Forecasting Models Based on the Neural Networks for the Volatility of Bitcoin
## 3 Hybrid Models and Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Activation Function** & **Nodes** & **MAE** & **RMSE** & **MAPE** \\ \hline GT-VIX-H & Relu & 10 & 0.016745304 & 0.022489019 & 48.02497968 \\  & & 20 & 0.01674541 & 0.02248914 & 48.0246867 \\  & & 30 & 0.016746443 & 0.022489882 & 48.02759455 \\  & & 40 & 0.016745041 & 0.022488647 & 48.02404413 \\  & & 50 & 0.0 Table 9 shows the results of three performance measures obtained by the GT-VIX-GARCH-H model. We can see the improvement in forecasting accuracy in Table 9. The results in Table 9 show that the hybrid models with selected GARCH models based on the HONN model for volatility forecasting of Bitcoin reduce the performance measures (MAE, RMSE, MAPE). That is, in all cases, the measures decrease compared to the measures of the other models. More specifically, compared to the GJR-GARCH(1,1) forecast, MAE is reduced by 11 %, MAPE is reduced by 30 %. Furthermore, we analyze the robustness of our results to determine whether the proposed models are statistically significant. For the analysis, we apply the MCS test [50] to GT-VIX-GARCH-H models. The detailed results of the MCS test, which can be interpreted as a level of confidence for the forecasts, are presented in Table 11. According to the results in Table 11, we can find that the GT-VIX-GARCH-H model with the Relu function and 30 nodes, which has the lowest MAE, is the best model for forecasting of Bitcoin volatility. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Loss Function**} & \multicolumn{4}{c}{**Input Variables**} \\
**Ranking** & **Model** & **Activation Function** & **Nodes** & **MAE** & **MCS** \\ \hline
1 & GT-VIX-GARCH-H & Relu & 30 & 0.016095555 & 1.000 \\
2 & GT-VIX-GARCH-H & Tanh & 20 & 0.01609808 & 0.991 \\
3 & GT-VIX-GARCH-H & Tanh & 30 & 0.016094577 & 0.991 \\
4 & GT-VIX-GARCH-H & Relu & 50 & 0.016097934 & 0.991 \\
5 & GT-VIX-GARCH-H & Tanh & 40 & 0.016098404 & 0.991 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Model confidence set. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Activation Function** & **Nodes** & **MAE** & **RMSE** & **MAPE** \\ \hline GT-VIX-GARCH-H & Relu & 10 & 0.016099377 & 0.022304876 & 41.97126277 \\  & & 20 & 0.016101827 & 0.02228648 & 42.07658833 \\  & & 30 & 0.016095555 & 0.022284853 & 42.03342476 \\  & & 40 & 0.016109686 & 0.022277813 & 42.13001676 \\  & & 50 & 0.016097934 & 0.02228606 & 42.05669279 \\ \hline \hline \multirow{10}{*}{\begin{tabular}{c} Tanh \\ \end{tabular} } & Tanh & 10 & 0.016098348 & 0.022302633 & 41.964812188 \\  & & 20 & 0.01609808 & 0.022231812 & 42.38508374 \\  & & 30 & 0.016094577 & 0.022298402 & 42.01919185 \\  & & 40 & 0.016098404 & 0.022302775 & 42.04282541 \\  & & 50 & 0.016096921 & 0.022285756 & 42.04785915 \\ \hline \hline \multirow{10}{*}{\
\begin{tabular}{c} Elu \\ \end{tabular} } & 10 & 0.016108832 & 0.02236164 & 41.71558473 \\  & & 20 & 0.016100976 & 0.022290669 & 42.06674165 \\  & & 30 & 0.016098687 & 0.022285274 & 42.06192355 \\  & & 40 & 0.016101002 & 0.022291082 & 42.0672782 \\  & & 50 & 0.016105162 & 0.022295976 & 42.088699 \\ \hline \hline \multirow{10}{*}{
\begin{tabular}{c} Sigmoid \\ \end{tabular} } & 10 & 0.016099661 & 0.022292957 & 42.06001306 \\  & & 20 & 0.016099905 & 0.022281962 & 42.06856524 \\ \cline{1-1}  & & 30 & 0.016100028 & 0.022278299 & 42.07671847 \\ \cline{1-1}  & & 40 & 0.016105448 & 0.022285267 & 42.1902812 \\ \cline{1-1}  & & 50 & 0.016096398 & 0.02229977 & 42.03760228 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: GT-VIX-GARCH-H model performance.