# Recurrent Neural Networks and ARIMA Models for Euro/Dollar Exchange Rate Forecasting
## 3 Results and Discussion
### Accuracy Analysis  
By using the first expression for U1 given in [33], we obtained [PERSON]'s U coefficients from the forecasting set to evaluate the accuracy of the models. [PERSON]'s coefficients are bounded by 0 and 1, where the lower boundary indicates a perfect forecast, and the upper boundary indicates unreliable forecasting. Coefficients close to 1 represent fully impractical situations for exchange forecasting. The use of these values would involve repeatedly performing the forecasting process to find negative forecasts or negative exchanges. [PERSON]'s U coefficients for the three forecasting methods are listed in Table 2.  
The three models give values close to 0. Therefore, the three techniques provide reliable predictions where the coefficient for the LSTM network is lower than that of the ELMAN and ARIMA models.  
two-sided  
\(H_{0}\): The two techniques offer the same prediction precision.  
\(H_{1}\): The two techniques do not offer the same prediction precision.  
_greater_  
\(H_{0}\): The two techniques offer the same prediction precision.  
\(H_{1}\): Technique 2 is more accurate than technique 1.  
As shown in Table 3, by comparing the \(p\)-values obtained with a significance level of \(\alpha=0.05\), we can deduce the following results:  
ARIMA vs. Elman: For two-sided, the value of \(p=1.694\times 10^{-4}<\alpha\); therefore, \(H_{0}\) is rejected, and it can be concluded that the two techniques did not have the same accuracy. For _greater_, \(p=8.468\times 10^{-5}<\alpha\), and \(H_{0}\) is rejected; thus, it can be concluded that the predictions performed with Elman were more precise than those made with ARIMA.  
ARIMA vs. LSTM: For two-sided, \(p=4.287\times 10^{-5}<\alpha\), and \(H_{0}\) is rejected; thus, it can be concluded that the two techniques did not have the same accuracy. For _greater_, \(p=2.144\times 10^{-5}<\alpha\); thus, \(H_{0}\) is rejected, and it can be concluded that the predictions performed with LSTM were more accurate than those made with ARIMA.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Scale-Dependent Errors**} & \multicolumn{6}{c}{**Percentage Errors**} \\ \hline
**Model** & **MSE** & **RMSE** & **MAE** & **MAE** & **MAPE** & **sMAPE** & **RMSPE** & **RMdSPE** & **MdAPE** & **SMdAPE** \\ \hline
**ARIMA** & 0.0001102 & 0.010496 & 0.009673 & 0.0084 & 0.869\% & 0.2183\% & 0.9414\% & 0.7572\% & 0.7567\% & 0.1898\% \\
**ELMAN** & 3.391 \(\times 10^{-5}\) & 0.00582 & 0.00494 & 0.0053 & 0.444\% & 0.111\% & 0.5226\% & 0.483\% & 0.483\% & 0.121\% \\
**LSTM** & 1.615 \(\times 10^{-5}\) & 0.004 & 0.0031 & 0.0020 & 0.282\% & 0.0706\% & 0.3607\% & 0.181\% & 0.181\% & 0.0453\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance metrics for the three prediction models.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**ARIMA** & **ELMAN** & **LSTM** \\ \hline
0.004743 & 0.002625 & 0.001808 \\ \hline \hline \end{tabular}
\end{table}
Table 2: [PERSON]’s U coefficient.  
Elman vs. LSTM: For two-sided, the value \(p=0.0154<\alpha\), and \(H_{0}\) is rejected; thus, the two techniques did not have the same accuracy. For _greater_, \(p=0.0077<\alpha\), and \(H_{0}\) is rejected; thus, it can be concluded that the forecasting performed with LSTM was more accurate than that performed with Elman.  
Based on these results, the predictions made with the LSTM network were more accurate than the forecasts performed with the ARIMA (0, 1, 0) model and the Elman network for short-term forecasting.