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Source: Economic Report of the President, 1997, Table B-28, p. 332.

Source: Economic Report of the President, 1997, Table B-28, p. 332.

simple savings function that relates savings (Y) to disposable personal income DPI (X). Since we have the data, we can obtain an OLS regression of Y on X. But if we do that, we are maintaining that the relationship between savings and DPI has not changed much over the span of 26 years. That may be a tall assumption. For example, it is well known that in 1982 the United States suffered its worst peacetime recession. The civilian unemployment rate that year reached 9.7 percent, the highest since 1948. An event such as this might disturb the relationship between savings and DPI. To see if this happened, let us divide our sample data into two time periods: 1970-1981 and 1982-1995, the pre- and post-1982 recession periods.

Now we have three possible regressions:

Time period 1970-1981: Yt = + X2Xt + u1t n1 = 12 (8.8.1)

Time period 1982-1995: Yt = n + K2Xt + U2t n2 = 14 (8.8.2)

Time period 1970-1995: Yt = a1 + a2Xt + ut n = (n1 + n2) = 26 (8.8.3)

Regression (8.8.3) assumes that there is no difference between the two time periods and therefore estimates the relationship between savings and DPI for the entire time period consisting of 26 observations. In other words, this regression assumes that the intercept as well as the slope coefficient remains the same over the entire period; that is, there is no structural change. If this is in fact the situation, then a1 = = y1 and a2 = k2 = y2.

Regressions (8.8.1) and (8.8.2) assume that the regressions in the two time periods are different; that is, the intercept and the slope coefficients are different, as indicated by the subscripted parameters. In the preceding regressions, the us represent the error terms and the ns represent the number of observations.

CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 275

For the data given in Table 8.9, the empirical counterparts of the preceding three regressions are as follows:

Yt = 1.0161 + 0.0803 Xt t = (0.0873) (9.6015) (8.8.1a)

Yt = 153.4947 + 0.0148Xt t = (4.6922) (1.7707) (8.8.2a)

Yt = 62.4226 + 0.0376 Xt + ■■■ t = (4.8917) (8.8937) + ■■■ (8.8.3a)

In the preceding regressions, RSS denotes the residual sum of squares, and the figures in parentheses are the estimated t values.

A look at the estimated regressions suggests that the relationship between savings and DPI is not the same in the two subperiods. The slope in the preceding savings-income regressions represents the marginal propensity to save (MPS), that is, the (mean) change in savings as a result of a dollar's increase in disposable personal income. In the period 1970-1981 the MPS was about 0.08, whereas in the period 1982-1995 it was about 0.02. Whether this change was due to the economic policies pursued by President Reagan is hard to say. This further suggests that the pooled regression (8.8.3a)—that is, the one that pools all the 26 observations and runs a common regression, disregarding possible differences in the two subperiods may not be appropriate. Of course, the preceding statements need to be supported by appropriate statistical test(s). Incidentally, the scattergrams and the estimated regression lines are as shown in Figure 8.3.

Now the possible differences, that is, structural changes, may be caused by differences in the intercept or the slope coefficient or both. How do we find that out? A visual feeling about this can be obtained as shown in Figure 8.2. But it would be useful to have a formal test.

This is where the Chow test comes in handy.15 This test assumes that:

1. u\t ~ N(0, a2) and u2t ~ N(0, a2). That is, the error terms in the sub-

period regressions are normally distributed with the same (homoscedastic)

variance a2.

15Gregory C. Chow, "Tests of Equality Between Sets of Coefficients in Two Linear Regressions," Econometrica, vol. 28, no. 3, 1960, pp. 591-605.

276 PART ONE: SINGLE-EQUATION REGRESSION MODELS 