Predictive Tests for Stability

The analysis-of-variance test that we have discussed is also commonly referred to as the Chow test, although it had been known much earlier. Chow suggests another test that can be used even when 2< (k + 1). This is the predictive test for stability. The idea behind is this: We use the first atj observations to estimate the regression equation and use it to get predictions for the next observations. Then we test the hypothesis that the prediction errors have mean zero. If «2 = 1. we just use the method described in Section 4.7. If «2 > F the F-test is given by

"G. C. Chow, "Tests of Equality Between Subsets of Coefficients in Two Linear Regression Models," Econometrica, 1960, pp. 591-605. The paper suggests two tests: the analysis-of-variance test and the predictive test. The former test, although referred to as the Chow test, was discussed earlier in C. R. Rao, Advanced Statistical Methods in Biometric Research (New York: Wiley, 1952), and S. KuUback and H. M. Rosenblatt, "On the Analysis of Multiple Regression in Ic Categories," Biometrika, 1957, pp. 67-83. Thus it is the second test-the predictive test-that should be called the Chow test.

Suppose that we estimate the production function (4.23) for the two periods I929-I948 and 1949-1967 separately. We get the following results:

1929-1948

log X = -4.058 + 1.617 log L, + 0.220 log (4.25)

(0 357) (0 209) (0 230)

R~ = 0.9759 RSS = 0.03555 d.f. = 17 (4.26)

1949-1967

logZ = -2.498 + 1.009 log £, + 0.579 log A:,

(0 531) (0 144) (0 055)

= 0.9958 RSS = 0.00336 d.f. = 16 Applying the test for stabihty (4.22), we get

URSS = 0.03555 + 0.00336 = 0.0389 with d.f. = 17 + 16 = 33

RRSS = 0.0434 [from (4.24) earlier] with d.f. = 36 Thus .

(0.0434 - 0.0389)/3 0.0389/33

From the F-tables with 3 and 33 d.f. we find that the 5% point is 2.9. Thus at the 5% significance level we do not reject the hypothesis of stability.

Again, looking at the individual coefficient estimates for the two periods, this result is perplexing. We will consider some other tests for stability and see whether these tests confirm this result.

(RSS - RSS.Vn,

RSS,/(,T, - A: - 1)

which has an F-distribution with d.f. «2 and «, - - I. Here

RSS = residual sum of squares from the regression based on /ij + «2

observations; this has («, + «2) - ( + 1) d.f. RSS, = residual sum of squares from the regression based on observations; this has - k ~ \ d.f.

The proof is given in the appendix to this chapter." In Chapter 8 we give a dummy variable method of applying the same test. We will first illustrate the use of this test and then discuss its advantages and limitations.

Illustrative Example

Consider the example of demand for food that we considered earlier. We have, for equation 1, from Table 4.4:

For 7927-/94/.• RSS, = 0.1151 For 1948-1962: RSS = 0.0544 Combined data: RSS = 0.2866

Considering the predictions for 1948-1962 using the estimated equation for 1927-1941, we have

(RSS - RSS,)/n2 (0.2866 - 0.1151)715 ~ RSS,/(«, - - l)~ 0.1151/12

From the F-tables with d.f 15 and 12 we find that the 5% point is 2.62. Thus at the 5% level of significance, we do not reject the hypothesis of stability. The analysis of variance test led to the opposite conclusion. For equation 2 we have (from Table 4.4):

RSS = 0.1151 RSS2 = 0.0535 RSS = 0.2412

The F-test is

(0.2412-0.1150/15 F 0.80

Thus at the 5% significance level we do not reject the hypothesis of stability. Thus the conclusions of the predictive test does not seem to be different from that of the analysis-of-variance test, for this equation.

However, for the predictive test, we can also reverse the roles of samples 1 and 2. That is, we can also ask the question of how well the equation fitted for the second period predicts for the first period. If the coefficients are stable, we should do well. The F-test for this is now (interchanging 1 and 2)

"The proof given in the appendix follows Maddala, Econometrics, pp. 459-460.

Comments

1. Wilson" argues that though the Chow test (the predictive test) has been suggested only for the case , < (A: + 1), that is, for the case when the analysis-of-variance test cannot be used, the test has desirable power properties when there are some unknown specification errors. Hence it should be used even when 2> (k + 1), that is, even in those cases where the analysis-of-variance test can be computed. We have illustrated how the predictive test can be used in two ways in this case.

2. Rea argues that in the case 2 < (k + 1) the Chow test cannot be considered a test for stability. All it tests is that the prediction error has mean zero, that is, the predictions are unbiased. If the coefficients are stable, the prediction error will have zero mean. But the converse need not be true in the case 2 < (k + 1). The prediction error can have a zero mean even if the coefficients are not stable, if the explanatory variables have moved in an offsetting manner. Reas conclusion is that "the Chow test is incapable of testing the hypothesis of equality against that of inequality. It can never be argued from the Chow test itself that the two sets [of regression coefficients] are equal, although at times it may be possible to conclude that they are unequal." This does not mean that the Chow test is not useful. Instead of calling it a test for stability we would call it a test for unbiasedness in prediction. Note that in the case both n, and

*A. L. Wilson, "When Is the Chow Test UMP?" The American Statistician, Vol. 32, No. 2, May 1978, pp. 66-68.

J. D. Rea, "Indeterminacy of the Chow Test When the Number of Observations Is Insufficient," Econometrica, Vol. 46, No. 1, January 1978, p. 229.

(RSS - RSS,)/n,

" RSSV(«3 -k-l)

which has an F-distribution with d.f. n, and 2 - - 1. For Equation 2 we have

(0.2412 - 0.0535)/15 0.0535/11

From the F-tables with d.f. 15 and 11 we see that the 5% point is 2.72 and the 1% point is 4.25. Thus, even at the 5% significance level, we cannot reject the hypothesis of stability of the coefficients.

Chow suggested the predictive test for the case where «2 is less than k + \. In this case the regression equation cannot be estimated with the second sample and thus the analysis-of-variance test cannot be applied. In this case only the predictive can be used. He also suggested that the predictive test can be used even when (k + 1) but that in this case the analysis-of-variance test should be preferred because it is more powerful.

In our example we have 2> (k + 1), but we have also used two predictive tests. In practice it is desirable to use both predictive tests as illustrated in our example.