Neural network linear forecasts for stock returns Angelos Kanas, страница 7

The table reports the DA statistics (Pesaran and Timmermann, 1992). The DA test follows the standard normal distribution.

We next employ the forecast encompassing testing approach for our out-of-sample forecasts.⁸ Following Clements and Hendry (1998), the formal forecast encompassing test is based on a set of OLS regressions. To illustrate, let 1=ANN model, 2=linear model, E_i denote the forecast error for model i (i=1, 2), and D denote the difference between the forecasts from the two models. Given forecasts from the two models, we can test the null hypothesis that neither model encompasses the other by running two regressions: the first involves regressing the forecast error from the ANN model on the difference of forecasts, i.e.

E1,t=0+1Dt+wt                   (10) and obtain the estimated coefficient  ₁. The second involves regressing the forecast error from the linear model (i.e. model 2) on the difference of forecasts, i.e.

E2,t=d0+d1Dt+ut              (11) and obtain the estimated coefficient d ₁. If  ₁ is not statistically significant and d ₁ is statistically significant, then we reject the null hypothesis that neither model encompasses the other in favor of the alternative hypothesis that the ANN model encompasses the linear model. If  ₁ is significant and d ₁ is not significant, then the linear model encompasses the ANN. If both  ₁ and d ₁ are significant or if both are not significant, then we fail to reject the null hypothesis that neither model encompasses the other. Table 5 reports the results from the forecast encompassing tests. This Table reports the heteroskedasticity robust t-statistics of the estimated coefficients  ₁ and d ₁ from regressions (10) and (11) and the corresponding p-values. If the p-values of both estimated coefficients are lower than 0.05 then the null hypothesis should be accepted (namely, neither model encompasses the other). If the p-value of  ₁ is lower than 0.05 and the p-value of d ₁ is higher than 0.05, then the null should be rejected in favor of the alternative hypothesis that the ANN model encompasses the linear. In the opposite case where the p-value of  ₁ is higher than 0.05 and the p-value of d ₁ is lower than 0.05, the null is rejected in favor of the alternative that the linear model encompasses the ANN.

As shown in Table 5, the null hypothesis is rejected for both indices, in favor of the alternative that the ANN model encompasses the linear model. This implies that in those cases where the linear model fails to forecast the FT and the DJ index return correctly, this failure can be accounted for by the ANN model. Moreover, the fact that the ANN model is not encompassed by the linear model suggests that in the cases where the ANN model fails to correctly forecast the FT and DJ return, this failure cannot be accounted for by the linear model. This result is compatible with the RESET test (in Table 3) which suggests that the linear functional form for both indices can be rejected. The conclusion from the forecast encompassing tests in the ANN model explains the forecast error of the linear model in both cases, whereas the linear model cannot explain the forecast error the ANN in either case. This indicates the superiority of the nonlinear ANN-based stock index forecasts over the linear-based forecasts.⁹

Table 5. Forecast encompassing tests: period: January 1995–June 1999