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426 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

EXAMPLE 11.10 (Continued)

is about 0.074. If this p value is deemed sufficiently low, the White test also suggests that there is heteroscedasticity.

In sum, then, on the basis of the residual graphs and the Park, Glejser, and White tests, it seems that our R&D regression (11.7.3) suffers from heteroscedasticity. Since the true error variance is unknown, we cannot use the method of weighted least squares to obtain heteroscedasticity-corrected standard errors and t values. Therefore, we will have to make some educated guesses about the nature of the error variance.

Looking at the residual graphs given in Figure 11.13, it seems that the error variance is proportional to sales as in Eq. (11.6.7), that is, the square root transformation. Effecting this transformation, we obtain the following results.

Sales Salesi se = (381.1285) (0.0071) R2 = 0.3648 (11.7.7)

If you want, you can multiply the preceding equation by VSales, to get back to the original model. Comparing (11.7.7) with (11.7.3), you can see that the slope coefficients in the two equations are about the same, but their standard errors are different. In (11.7.3) it was 0.0083, whereas in (11.7.7) it is only 0.0071, a decrease of about 14 percent.

To conclude our example, we present below White's heteroscedasticity-consistent standard errors, as discussed in Section 11.6.

R&Di = 192.9931 + 0.0319 Sales, se = (533.9931) (0.0101) r2 = 0.4783 (11.7.8)

Comparing with the original (i.e., without correction for heteroscedasticity) regression (11.7.3), we see that although the parameter estimates have not changed (as we would expect), the standard error of the intercept coefficient has decreased and that of the slope coefficient has slightly increased. But remember that the White procedure is strictly a large-sample procedure, whereas we only have 18 observations.

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