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Neural network linear forecasts for stock returns Angelos Kanas, страница 6

DJ index

FT index

Constant

0.008*

0.009*

(2.55)

(2.20)

1

−0.01

0.001

(−0.48)

(0.05)

2

−0.02

0.004

(−0.50)

(0.10)

DW

1.88

1.92

Q

26.92

35.70

(p-value)

(0.32)

(0.48)

LM

0.066

2.38

(p-value)

(0.99)

(0.99)]

RESET

5.77*

5.40*

(p-value)

(0.00)

(0.00)

R2

0.04

0.04

Q is the test for higher-order serial correlation in the residuals. t-statistics in the parentheses.

LM is the Lagrange multiplier test for heteroskedasticity.

RESET is the F-test version of the Ramsey RESET test for correct functional form. The null hypothesis is that the functional form (linearity) is correct. As shown in the table, the null hypothesis is rejected for both indices, as the corresponding p-values are lower than 0.05. * Statistically significant at the 5% level of significance.

linear model for the ‘training’ period (January 1980–December 1994), one-step ahead forecasts were generated from both models.

We compare the out-of-sample forecasts using two different testing approaches, namely directional accuracy (DA) and forecast encompassing. To examine the directional accuracy of the two competing models, we employ the Pesaran and Timmermann (1992) test. This test is based on the comparison of sign of the forecast observation, yˆn+i, with the true withheld observation, yn+i, for the out-of-sample observations i=1, 2, . . . , m. We define the success ratio (SR) as

m

                   SR=m−1  Ii[yn+iyˆn+i0]                                                                                                                       (3)

i=1

where Ii[ . ] is an indicator function that takes the value of 1 when its argument is true and 0 otherwise. We also define

m

P=m−1  Ii[yn+i0]

i=1

and

(4)

m

P =m−1  Ii[yˆn+i0]

i=1

The success rate in the case of independence (SRI) of yn+i and yˆn+i is given by

(5)

SRI=P*P +(1−P)(1−P )

with variance

(6)

var(SRI)=m−2[m(2P −1)2P(1−P)+m(2P−1)2P (1−P )+4PP (1−P)(1−P )]

The variance of SR is given by

(7)

var(SR)=m−1SRI(1−SRI)

On the basis of the above, the Pesaran and Timmermann (1992) DA test is given by

(8)

DA=[var(SR)−var(SRI)]−1/2(SR−SRI)

(9)

Pesaran and Timmermann (1992) show that under the null hypothesis that yn+i and yˆn+i are independently distributed, the DA test follows the standard normal distribution. The results from the DA tests are reported in Table 4. As shown in the table, the null hypothesis that the ANN model based forecasts and the actually observed returns are independent cannot be rejected for either the FT or the DJ index. Similarly, we cannot reject the null hypothesis that the linear model based forecasts and the observed returns are independent. Thus, we conclude that neither the ANN nor the linear model can predict the directional change of the actually observed out-of-sample stock returns for both the DJ and the FT indices.

Table 4. DA tests: period: January 1995–June 1999

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