As could be seen in the previous section, the dummy variable could work as an intercept shifter. Sometimes it is reasonable to believe that the shift should take place in the slope coefficient instead of the intercept. If we go back to the human capital model it is possible to argue that the difference in wage rate between men and women could be due to differences in their return to education. This would mean that men and women have slope coefficients that are different in size.

A model that control for differences in the slope coefficient for different categories of the qualitative variable could be expressed in the following way:

In this case the slope coefficient for x equals b1 when D=0 and B1+B2 when D=1. Hence, a way to test if the return to education differs between men and women would be to test if B2 is different from zero, which should be tested before going on to test if B1+B2 is different from zero. Observe that the coefficient for the cross product is interpreted differently if both variables had been continuous. Since d is binary, dx is only active when D=1, and the corresponding effect is therefore related to the category specified

by d=1.

Example 8.3

Use the same data set as in Example 8.2 and estimate the coefficients in (8.10). The results are presented below with standard errors within parenthesis:

In 7 = 4.11 + 0.024X + 0.014DX

< ( w ^ (8.11)

(0.031) (0.003) (o.ooi)

Use the regression results to investigate if there is a difference in the return to education between men and women. To answer that question we simply test the estimated coefficient for the cross product. Doing that, we receive a f-value of 10.3 which is above any critical values of conventional significance level. Hence, if this specification is correct, we can conclude that the returns to education differ between men and women.

8.2.1. A model will intercept and slope dummy variable

Whenever working with cross products it is very important to always include the involved variables separately to separate that kind of effect from the cross product. If it is the case that d in itself has a positive effect on the dependent variable, that unique effect will be part of the cross effect otherwise. Hence whenever including a cross product the model should be specified in the following way:

In Y = B0 + BlX + B2D + B3(DX) + U (8j2)

When we include the two variables, x and d separately and together with their product we allow for changes in both the intercept and the slope. If it turns out that the coefficient of B2 is not significant one can go on and reduce the specification to (8.10), but not otherwise.

Example 8.4

Extend the specification of (8.10) by including d separately. That is, estimate the parameters of the model given by (8.12) and interpret the results. Doing that, we received the following results, with standard errors within parenthesis.

In Y = 4.006 + 0.033X + 0.210Z) - 0.002DX

(8.12)

(0.045) (0.004) (0.062) (0.005)

By investigating the f-values we see that b and b2 are statistically significant from zero. But the f-value from the cross product is not significant any more. Since D alone has a significant effect on the dependent variable, there is little effect left from the cross product, and hence we conclude that there is no difference in the return to education between men and women.

Example 8.3 and 8.4 should convince you that it is very important to include the variables that appear in a cross product separately, since they might stand for the main effect. In Example 8.3 we did not include D even though it was relevant. In chapter 7 we learned that omitting relevant variables has consequences and bias the remaining coefficients. In this case it made us to draw the wrong conclusion about the return to education for men and women.

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