## Example

STRUCTURAL DIFFERENCES IN THE U.S. SAVINGS-INCOME REGRESSION, THE DUMMY VARIABLE APPROACH

Before we proceed further, let us first present the regression results of model (9.5.1) applied to the U.S. savings-income data.

Y, = 1.0161 + 152.4786Dt + 0.0803Xt - 0.0655(DtXt) se = (20.1648) (33.0824) (0.0144) (0.0159) (9.5.4)

where * indicates p values less than 5 percent and ** indicates p values greater than 5 percent.

As these regression results show, both the differential intercept and slope coefficients are statistically significant, strongly suggesting that the savings-income regressions for the two time periods are different, as in Figure 9.3d.

From (9.5.4), we can derive equations (9.5.2) and (9.5.3), which are:

Savings-income regression, 1970-1981:

Savings-income regression, 1982-1995:

These are precisely the results we obtained in (8.8.1a) and (8.8.2a), which should not be surprising. These regressions are already shown in Figure 8.3.

The advantages of the dummy variable technique [i.e., estimating (9.5.1)] over the Chow test [i.e., estimating the three regressions (8.8.1), (8.8.2), and (8.8.3)] can now be seen readily:

1. We need to run only a single regression because the individual regressions can easily be derived from it in the manner indicated by equations (9.5.2) and (9.5.3).

2. The single regression (9.5.1) can be used to test a variety of hypotheses. Thus if the differential intercept coefficient a2 is statistically insignificant, we may accept the hypothesis that the two regressions have the same intercept, that is, the two regressions are concurrent (see Figure 9.3c). Similarly, if the differential slope coefficient fS2 is statistically insignificant but a2 is significant, we may not reject the hypothesis that the two regressions have the same slope, that is, the two regression lines are parallel (cf. Figure 9.3b). The test of the stability of the entire regression (i.e., a2 = = 0, simultaneously) can be made by the usual F test (recall the restricted least-squares F test). If this hypothesis is not rejected, the regression lines will be coincident, as shown in Figure 9.3a.

310 PART ONE: SINGLE-EQUATION REGRESSION MODELS

EXAMPLE 9.4 (Continued)

3. The Chow test does not explicitly tell us which coefficient, intercept, or slope is different, or whether (as in this example) both are different in the two periods. That is, one can obtain a significant Chow test because the slope only is different or the intercept only is different, or both are different. In other words, we cannot tell, via the Chow test, which one of the four possibilities depicted in Figure 9.2 exists in a given instance. In this respect, the dummy variable approach has a distinct advantage, for it not only tells if the two are different but also pinpoints the source(s) of the difference—whether it is due to the intercept or the slope or both. In practice, the knowledge that two regressions differ in this or that coefficient is as important as, if not more than, the plain knowledge that they are different.

4. Finally, since pooling (i.e., including all the observations in one regression) increases the degrees of freedom, it may improve the relative precision of the estimated parameters. Of course, keep in mind that every addition of a dummy variable will consume one degree of freedom. 