(RRSS - URSS)/(A: + 1)

URSS/(,T, + n, - 2A; - 2)

which has an F-distribution with degrees of freedom (k + \) and (at, + 2 - 2k - 2). This test is derived in the appendix to this chapter.

Example I: Stability of the Demand for Food Function

Consider the data in Table 4.3, where we fitted separate demand functions for 1927-1941 and 1948-1962 and for the entire period. Suppose that we want to test the stability of the parameters in the demand function between the two periods. The required numbers are given in Table 4.4. For equation 1 we have

URSS = sum of RSSs from the two separate regressions for 1927-1941 and 1948-1962 = 0.1151 + 0.0544 = 0.1695 with d.f. = 12 -b 12 = 24 RRSS = RSS from a regression for the pooled data = 0.2866 with d.f. = 27

This regression from the pooled data imposes the restriction that the parameters are the same in the two periods. Hence

(0.2866 - 0.1695)/3 0.1695/24

From the F-tables with d.f. 3 and 24 we see that the 5% point is about 3.01 and the 1% point is about 4.72. Thus, even at the 1% level of significance, we reject the hypothesis of stability. Thus there is no case for pooling.

Note that if we look at the individual coefficients, p, is almost the same for the two regressions. Thus it appears that the price elasticity has been constant but it is the income elasticity that has changed in the two periods. In Chapter 8 we discuss procedures of testing the stability of individual coefficients using the dummy variable method.

Consider now equation 2. We now have

URSS = 0.1151 + 0.0535 = 0.1686 with d.f. = 11 + 11 = 22 RRSS = 0.2412 with d.f. = 26 Hence

(0.2412 - 0.1686)/4 0.1686/22

From the F-tables with d.f. 4 and 22 we see that the 5% point is about 2.82. So, at the 5% significance level, we do not reject the hypothesis of stability.

One can ask how this result came about. If we look at the individual coefficients for Equation 2 for the two periods 1927-1941 and 1948-1962 separately.

Example 2: Stability of Production Functions

Consider the data in Table 3.11 on output and labor and capital inputs in the United States for 1929-1967." The variables are:

X = index of gross domestic product (constant dollars)

Li = labor input index (number of persons adjusted for hours of work and educational level)

L2 = persons engaged

Ki = capital input index (capital stock adjusted for rates of utilization)

K2 = capital stock in constant dollars

We will estimate regression equations of the form

logZ = a + p, logL + 2iog + (4.23)

First, considering the two measures of labor and capital inputs, we obtain the following results:

logX = -3.938 + 1.451 log Li + 0.384 log / , (4.24)

(0 237) (0 083) (0 048)

R2 = 0.9946 = 0.9943 RSS = 0.0434 d = 0.001205

logX = -6.388 -t- 2.082 log Fj -t- 0.571 log K2

(0 294) (0 100) (0 067)

R = 0.9831 RSS = 0.1363

Figures in parentheses are standard errors. Since the dependent variable is the same and the number of independent variables is the same, the Rs are comparable. A comparison of the Rs indicates that F, and Ki are better ex-

"The data are from L. R. Christensen and D. W. Jorgenson, "U.S. Real Product and Real Factor Input 1929-67," Review of Income and Wealth, March 1970.

we notice that the f-ratios are very small, that is, the standard errors are very high relative to the magnitudes of the coefficient estimates. Thus the observed differences in the coefficient estimates between the two periods would not be statistically significant. When we look at the regression for the pooled data we notice that the coefficient of the interaction term is significant, but the estimates for the two periods separately, as well as the rejection of the hypothesis of stability for equation 1, casts doubt on the desirability of including the interaction term.

planatory variables than and K- Hence we will conduct further analysis with L and only.

One other thing we notice is that all variables increase with time. Regressing each of the variables on time we find:

log X = 4.897 + 0.0395/ = 0.9549

(0 032) (0 0014)

log L, = 4.989 + 0.0185/ = 0.9238

(0.020) (0 0009)

logX, = 4.171 + 0.0324/ R2 = 0.9408

(0 031) (0 0013)

We can eliminate these time trends from these variables and then estimate the production function with the trend-adjusted data. But this is the same as including / as an extra explanatory variable (see the discussion in Section 4.4)." Thus we get the result

logX = -3.015 4- 1.341 logL, + 0.292 log X, + 0.0052/

(0 091) (0 060) (0 0022)

/?2 = 0.9954 = 0.9949 RSS = 0.0375 = 0.001072

Comparing this result with (4.24), we notice that p, + 02 has gone down from 1.451 + 0.384 = 1.835 to 1.341 4- 0.292 = 1.633. , + 02 measures returns to scale. Also, the has increased, or equivalently, has decreased. This is to be expected because the /-value for the last variable is greater than 1.

Although we do not need to test the hypothesis of constant returns to scale, that is p, + P2 = 1, we will illustrate it. The estimated variances and covariances (from the SAS regression package used) were

estimate of ( ,) = 0.008353 estimate of V) = 0.003581

estimate of cov(pI, P2) = -0.001552

Thus

estimate of V(0, -t- P2 - 1) = 0.008353 + 0.003581 - 2(0.001552)

= 0.008830

Hence

SE(p, + p2 - 1) = V0.008830 = 0.094 The /-statistic to test p, + p. - 1 = 0 is

, = (Pi + Pa - 1) - (0) 0633 SE(p, + p2 - 1) 0.094

From the /-tables with 34 d.f. we find that the 5% significance point is 2.03 and the 1% significance point is 2.73. Thus we reject the hypothesis of constant returns to scale even at the 1% level of significance.

"This result is commonly known as the Frisch-Waugh theorem.