## Figure

Public school teacher's salary (Y) in relation to per pupil expenditure on education (X).

9.5 THE DUMMY VARIABLE ALTERNATIVE TO THE CHOW TEST9

In Section 8.8 we discussed the Chow test to examine the structural stability of a regression model. The example we discussed there related to the relationship between savings and income in the United States over the period 1970-1995. We divided the sample period into two, 1970-1981 and 1982-1995, and showed on the basis of the Chow test that there was a difference in the regression of savings on income between the two periods.

However, we could not tell whether the difference in the two regressions was because of differences in the intercept terms or the slope coefficients or both. Very often this knowledge itself is very useful.

Referring to Eqs. (8.8.1) and (8.8.2), we see that there are four possibilities, which we illustrate in Figure 9.3.

1. Both the intercept and the slope coefficients are the same in the two regressions. This, the case of coincident regressions, is shown in Figure 9.3a.

2. Only the intercepts in the two regressions are different but the slopes are the same. This is the case of parallel regressions, which is shown in Figure 9.3b.

9The material in this section draws on the author's articles, "Use of Dummy Variables in Testing for Equality between Sets of Coefficients in Two Linear Regressions: A Note," and "Use of Dummy Variables . . . A Generalization," both published in the American Statistician, vol. 24, nos. 1 and 5, 1970, pp. 50-52 and 18-21.

CHAPTER NINE: DUMMY VARIABLE REGRESSION MODELS 307

Savings

(a) Coincident regressions

(a) Coincident regressions

Income

Savings

Savings

(b) Parallel regressions

Income

(b) Parallel regressions

Savings

Savings

(c) Concurrent regressions FIGURE 9.3 Plausible savings-income regressions.

Income

(c) Concurrent regressions FIGURE 9.3 Plausible savings-income regressions.

Savings

(d) Dissimilar regressions

Income

(d) Dissimilar regressions

3. The intercepts in the two regressions are the same, but the slopes are different. This is the situation of concurrent regressions (Figure 9.3c).

4. Both the intercepts and slopes in the two regressions are different. This is the case of dissimilar regressions, which is shown in Figure 9.3d.

The multistep Chow test procedure discussed in Section 8.8, as noted earlier, tells us only if two (or more) regressions are different without telling us what is the source of the difference. The source of difference, if any, can be pinned down by pooling all the observations (26 in all) and running just one multiple regression as shown below10:

Yt = ai + a2 Dt + Pi Xt + M DtXt) + ut where Y = savings X = income t = time

D = 1 for observations in 1982-1995

= 0, otherwise (i.e., for observations in 1970-1981)

10As in the Chow test, the pooling technique assumes homoscedasticity, that is, o 2 = a2 = a 2 •

308 PART ONE: SINGLE-EQUATION REGRESSION MODELS

 Observation Savings Income Dum