Neural network linear forecasts for stock returns Angelos Kanas, страница 5

a_j are the weights for the direct signals from each of the two input variables to the output variable, b_i is the weight for the signal from each of the six hidden units to the output variable, and c_j,i are the weights for the signals from each of the two input variables, j, to the hidden units, i. i=1, 2, . . . , 6, and j=1, 2.

where y_t is the stock returns series, Z_1,t−1 is the lagged percentage change in trading volume, Z_2,t−1 is the lagged percentage change in dividends, and w_t is the error term. The explanatory variables in the linear model are the same as those in the nonlinear ANN model in order for the nonlinear model to nest the linear model.⁷ Different explanatory variables in the linear and nonlinear models would entail non-nesting of the models.

The model in Equation (2) is estimated using ordinary least squares (OLS) for the same period as the ‘training’ period of the ANN. Table 3 reports the results. The lagged trading volume and lagged dividends are not statistically significant for both indices. Furthermore, according to the Ramsey’s RESET test for correct functional form, the null hypothesis that the examined (linear) functional form is correct is rejected for both indices. A comparison of the results between the linear model and the linear component of the ANN (parameters a₁ and a₂) reveals that the value of the estimated coefficients differs between the two models. The coefficients for the dividend variable are 0.008 (DJ) and 0.009 (FT) for the linear model whereas they are −8.41 (DJ) and −0.17 (FT) for the ANN. Similar comments apply to the trading volume variable. On the basis of these estimated coefficients, out-of-sample forecasts (i.e. linear forecasts) were generated and compared to the nonlinear forecasts for the subsequent period. The use of this traditional linear forecasting procedure and its comparison to nonlinear procedures is discussed in Clemen (1989) and Granger (1989).

4. OUT-OF-SAMPLE FORECASTING

This section focuses on the out-of-sample forecasting ability of the ANN and the linear models. It is important to note that we examine exclusively the out-of-sample forecasting ability of the two competing models and, therefore, we do not automatically favor the model with the higher flexibility of fit the data in-sample, i.e. the ANN model. The out-of-sample forecasts refer to the period January 1995–June 1999, which is the ‘testing’ period for the ANN. On the basis of the estimated coefficients of the ANN and the

Table 3. Linear model estimation: period: January 1980–December 1994