4.1. In-sample forecast evaluation

In this subsection, we evaluate the in-sample predictive power of the combined model forecast —the FAANN model—and other fitted models which include AR (benchmark model), DFM, and ANN and best combined forecasts of the DFM and ANN models. To achieve this, a full sample from January 1992 to December 2011, giving a total of 240 observations of the three datasets—deposit rate, gold mining share prices, and long-term interest rate—is used to estimate the forecasting models in order to check the robustness of in-sample results of competed models and compare it to the AR benchmark model. In-sample forecasting is most useful when it comes to investigate the true relationship between the independent variables and the forecast of dependent variable. Table 1 reports the root-mean-square error (RMSE)<sup>3</sup> of the in-sample forecasting results. The FAANN model outperformed all other models. The maximum reduction in RMSE over the AR benchmark model is around 24%, while the

<sup>2</sup> The data sources are the South Africa Reserve Bank, ABSA Bank, Statistics South Africa, National Association of Automobile Manufacturers of South Africa (NAAMSA), South African Revenue Service (SARS), Quantec, and World Bank.

<sup>3</sup> The RMSE statistic can be defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N <sup>P</sup> Ytþ<sup>n</sup> �tY<sup>b</sup> <sup>t</sup>þ<sup>n</sup> � �<sup>2</sup> <sup>r</sup> , where Yt þ n denotes the actual value of a specific variable in period t þ n andtYb <sup>t</sup>þ<sup>n</sup> is the forecast made in period t for t þ n.


Table 1. The RMSE of the in-sample forecasts.

minimum reduction is around 14% considering all variables. Regarding the in-sample forecasting, the FAANN model provides lower RMSE with a reduction of between 9 and 19% for all variables compared to the DFM. Despite that the same factors are augmented to AR and ANN to produce the DFM and the FAANN models, the in-sample results provide significant differences between estimation methods which favor the nonlinear method over the linear one. This is potentially due to the flexibility and property of the ANN models as universal approximators that can be used to different time series in order to obtain accurate forecasts. Comparing the forecasting performance of the FAANN and standard ANN model, the FAANN model produced lower RMSE of 6–19% for all variables. These results indicate the importance of the factors—which summarized 228 related series into five factors—that are used as input to the ANN to produce the FAANN model. Regarding the in-sample forecasting performance of the forecasts of combined models or information—the FAANN model—compared to the best forecast combination of the DFM and ANN models, the FAANN model outperforms the best forecast combination with reduction in the RMSE around 0.01–13% for all variables. These results confirm the superiority of the combination of information or models when a precise estimation method is used to estimate the combined information over the combined forecasts of individual models.

### 4.2. Out-of-sample forecast evaluation of individual models

In this subsection, we estimate the individual forecasts of the AR, DFM, and ANN and the best combined forecasts of the DFM and ANN models and the FAANN model that combine information of the factors and ANN for the three variables of interest, namely, deposit rate, gold mining share prices, and long-term interest rate, over the in-sample period January 1992 to December 2006 using monthly data, and then compute the out of sample for 3-, 6-, and 12-month-ahead forecasts for the period of January 2007 to December 2011. We employ iterative forecast technique to compute the RMSE for the three forecasting horizons used for the three variables across all of the different models in order to compare the forecast accuracy generated by the models. The starting date of the in-sample period depends on data availability of some important financial series. The out-of-sample period includes the occurrence of the financial crisis that affected economies and financial sectors in particular. Thus, we used this period as out of sample in order to show the suitability and efficiency of the combination of information—FAANN model—to produce accurate forecasts for such data that exhibits inherent nonlinearity or the data that faced fluctuations during the financial crisis. The result of each single variable can be summarized as follows:



Note: The last row reports the RMSE for the AR benchmark model; the remaining rows represent the ratio of the RMSE for the forecasting model to the RMSE for the AR. Bold entries indicate the forecasting model with the lowest RMSE.

Table 2. Out-of-sample (January 2007–December 2011) RMSE for deposit rate.

minimum reduction is around 14% considering all variables. Regarding the in-sample forecasting, the FAANN model provides lower RMSE with a reduction of between 9 and 19% for all variables compared to the DFM. Despite that the same factors are augmented to AR and ANN to produce the DFM and the FAANN models, the in-sample results provide significant differences between estimation methods which favor the nonlinear method over the linear one. This is potentially due to the flexibility and property of the ANN models as universal approximators that can be used to different time series in order to obtain accurate forecasts. Comparing the forecasting performance of the FAANN and standard ANN model, the FAANN model produced lower RMSE of 6–19% for all variables. These results indicate the importance of the factors—which summarized 228 related series into five factors—that are used as input to the ANN to produce the FAANN model. Regarding the in-sample forecasting performance of the forecasts of combined models or information—the FAANN model—compared to the best forecast combination of the DFM and ANN models, the FAANN model outperforms the best forecast combination with reduction in the RMSE around 0.01–13% for all variables. These results confirm the superiority of the combination of information or models when a precise estimation method is used to estimate the combined information over the

FAANN DFM ANN AR Best combined forecasts of DFM and ANN

In this subsection, we estimate the individual forecasts of the AR, DFM, and ANN and the best combined forecasts of the DFM and ANN models and the FAANN model that combine information of the factors and ANN for the three variables of interest, namely, deposit rate, gold mining share prices, and long-term interest rate, over the in-sample period January 1992 to December 2006 using monthly data, and then compute the out of sample for 3-, 6-, and 12-month-ahead forecasts for the period of January 2007 to December 2011. We employ iterative forecast technique to compute the RMSE for the three forecasting horizons used for the three variables across all of the different models in order to compare the forecast accuracy generated by the models. The starting date of the in-sample period depends on data availability of some important financial series. The out-of-sample period includes the occurrence of the financial crisis that affected economies and financial sectors in particular. Thus, we used this period as out of sample in order to show the suitability and efficiency of the combination of information—FAANN model—to produce accurate forecasts for such data that exhibits inherent nonlinearity or the data that faced fluctuations during the financial crisis. The result of each

combined forecasts of individual models.

Variable Model

86 Advanced Applications for Artificial Neural Networks

Table 1. The RMSE of the in-sample forecasts.

single variable can be summarized as follows:

4.2. Out-of-sample forecast evaluation of individual models

Deposit rate 0.1687 0.1849 0.1793 0.1954 0.1694 Share prices for gold mining 1.5922 1.7782 1.7787 1.8187 1.6215 Long-term interest rate 0.1253 0.1537 0.1546 0.1640 0.1438


Table 3. Out-of-sample (January 2007–December 2011) RMSE for gold mining share prices.


Table 4. Out-of-sample (January 2007–December 2011) RMSE for long-term interest rate.

RMSE relative to the AR benchmark model of 10–18%. The average of the RMSE reduction over the forecast horizons is 12%. On average the FAANN outperforms the ANN and DFM models with reduction in RMSE of 6 and 8%, respectively.

• Long-term interest rate: for estimation purpose the same package and algorism that are used with previous variables are implemented. Thus, the optimal network in abbreviated form is Nð Þ <sup>8</sup>�3�<sup>1</sup> . Table 4 results show the performance of the FAANN model where the model produces more accurate forecasts compared to all competing model on both the single-level forecast horizons and the average of these horizons. Compared to the AR benchmark, the FAANN provides a reduction in the RMSE range from 45–27%, while the average RMSE reduction is around 38%. The performance of the FAANN model stands out followed by the ANN and the DFM with average reduction in RMSE of 9 and 5%, respectively, relative to the AR benchmark model. Comparing the FAANN performance to the ANN and the DFM, the FAANN model RMSE reduction is around 28 and 32%, respectively.
