5. Conclusion

Table 6 presents the RMSE ratios of the FAANN model and the best forecast combination to the AR benchmark model over the out-of-sample period. Compared to the DFM, the results indicate that the FAANN model generates accurate forecasts for all variables and with all forecast horizons. The improvement of the FAANN model is compared to the DFM between 2 and 10% reduction in RMSE for all variables and horizons. Thus, these results indicate the superiority of augmentation of factors to nonlinear method (FAANN) over the linear one

Table 6. Forecast results of the best combination of DFM and ANN model and FAANN-RMSE for variables (January

Forecasting model h = 3 h = 6 h = 12

AR (benchmark model) 0.1862 0.1949 0.2314 FAANN 0.7465 0.6373 0.5359 Combined forecasts of DFM and ANN 0.907 0.882 0.835

AR (benchmark model) 1.7743 1.7924 1.8187 FAANN 0.9053 0.9121 0.8227 Best combined forecasts of DFM and ANN 0.921 0.929 0.911

AR (benchmark model) 0.2052 0.2140 0.2308 FAANN 0.7281 0.6051 0.5498 Best combined forecasts of DFM and ANN 0.827 0.815 0.804

Deposit rate

Gold mining share prices

Long-term interest rate

To confirm the RMSE results, the test of equal forecast accuracy of Diebold and Mariano [14] is used to evaluate forecasts. The test of equal forecast accuracy employed here is given by

forecasts of DFM and ANN. The S statistic follows a standard normal distribution asymptotically. Note, a significant negative value of S means that the FAANN model outperforms the other model in out-of-sample forecasting. Table 7 shows the result of the Diebold and Mariano test between the FAANN and the AR benchmark and between the FAANN and the best combined forecasts of DFM and ANN. The test results confirm that the FAANN models provide the lowest RMSEs. In summary the FAANN models provide significantly better forecasts at the 5% and 10% level compared to the AR and the best combined forecasts of DFM and

� � is the mean difference of the squared prediction error

2t denotes the forecast errors from the AR benchmark model or the best combined

1t denotes the forecast errors from the FAANN

(DFM) across the three different series and three different time horizons.

<sup>S</sup> <sup>¼</sup> <sup>d</sup>ffiffiffi b V p ð Þ<sup>d</sup>

and Vb d

model, and e2

Note: See note to Table 2.

90 Advanced Applications for Artificial Neural Networks

2007–December 2011).

ANN models.

, where <sup>d</sup> <sup>¼</sup> <sup>1</sup>

T P T t¼1 e2 1t � e2 2t

� � is the estimated variance. Here, e<sup>2</sup>

In this chapter we aim to evaluate the forecasting performance of the model combination and forecast combination for the ANN and DFM models. In the model combination, we merge the factors that were extracted from a large dataset—288 series in our case—with ANN which produces the FAANN model. For the forecast combination, we used different linear and nonlinear combination methods to combine the individual forecasts of the DFM and the ANN models. Using the period of January 1992 to December 2006 as in-sample period and January 2007 to December 2011 as out-of-sample period, we compare the forecast performance of the FAANN with DFM, ANN, and AR benchmark model for 3-, 6-, and 12-month-ahead forecast horizons for three variables, namely, for deposit rate, gold mining share prices, and long-term interest rate. The study has provided evidence using both the RMSE and Diebold and Mariano test as the comparison criteria that FAANN models best fit the three considered variables over the 3-, 6-, and 12-month-ahead forecast horizons.

Tables 2–4 show the ability of the model combination—FAANN model—to produce accurate forecast that outperforms DFM and ANN and their best forecast combination results. The results seem not contradicted with in-sample model forecast performance as in Table 1. The FAANN model outperformed the AR benchmark model with large reduction in RMSE of around 25–46% considering all variables and forecast horizons. Compared to the DFM and ANN models, the FAANN model produces more accurate forecasts that yielded a decrease in RMSE of around 6–43% and 5–40%, respectively. We attribute the superiority of the FAANN to the flexibility of ANN to account for potentially complex nonlinear relationships that are not easily captured by linear models and the contribution of the factors to the model. On the other hand, the ANN and the DFM outperformed the AR benchmark with a reduction in the RMSEs of around 1–17% and 2–10%, respectively, for all variables and across all forecast horizons. Table 6 shows comparison results of the forecasting performance of the combined models the FAANN—and the best forecast combination of the DFM and the ANN models. The results indicate that the combined models or information produced forecasts that are better than the best combined forecasts of the DFM and the ANN models. In other words, the nonlinear model that uses large dataset of economic and financial variables in addition to the lags of the interested variable improves the forecasting performance over models that are estimated separately—the DFM and the ANN. We also observed that the FAANN residual decreases as the forecast horizon increases.
