2. The two error terms u1t and u2t are independently distributed. The mechanics of the Chow test are as follows:

1. Estimate regression (8.8.3), which is appropriate if there is no parameter instability, and obtain RSS3 with df = (n1 + n2 — k), where k is the number of parameters estimated, 2 in the present case. For our example RSS3 = 23,248.30. We call RSS3 the restricted residual sum of squares (RSSr) because it is obtained by imposing the restrictions that À1 = y1 and

= Y2, that is, the subperiod regressions are not different.

2. Estimate (8.8.1) and obtain its residual sum of squares, RSS1, with df = (n1 — k). In our example, RSS1 = 1785.032 and df = 10.

3. Estimate (8.8.2) and obtain its residual sum of squares, RSS2, with df = (n2 — k). In our example, RSS2 = 10,005.22 with df = 12.

4. Since the two sets of samples are deemed independent, we can add RSS1 and RSS2 to obtain what may be called the unrestricted residual sum of squares (RSSUR), that is, obtain:

In the present case,

5. Now the idea behind the Chow test is that if in fact there is no structural change [i.e., regressions (8.8.1) and (8.8.2) are essentially the same], then the RSSR and RSSUR should not be statistically different. Therefore, if we form the following ratio:

then Chow has shown that under the null hypothesis the regressions (8.8.1) and (8.8.2) are (statistically) the same (i.e., no structural change or break) and the F ratio given above follows the F distribution with k and (ni + n2 — 2k) df in the numerator and denominator, respectively.

6. Therefore, we do not reject the null hypothesis of parameter stability (i.e., no structural change) if the computed F value in an application does not exceed the critical F value obtained from the F table at the chosen level of significance (or the p value). In this case we may be justified in using the pooled (restricted?) regression (8.8.3). Contrarily, if the computed F value exceeds the critical F value, we reject the hypothesis of parameter stability and conclude that the regressions (8.8.1) and (8.8.2) are different, in which case the pooled regression (8.8.3) is of dubious value, to say the least.

Returning to our example, we find that

From the F tables, we find that for 2 and 22 df the 1 percent critical F value is 5.72. Therefore, the probability of obtaining an F value of as much as or greater than 10.69 is much smaller than 1 percent; actually the p value is only 0.00057.

The Chow test therefore seems to support our earlier hunch that the savings-income relation has undergone a structural change in the United States over the period 1970-1995, assuming that the assumptions underlying the test are fulfilled. We will have more to say about this shortly.

Incidentally, note that the Chow test can be easily generalized to handle cases of more than one structural break. For example, if we believe that the savings-income relation changed after President Clinton took office in January 1992, we could divide our sample into three periods: 1970-1981, 1982-1991, 1992-1995, and carry out the Chow test. Of course, we will have four RSS terms, one for each subperiod and one for the pooled data. But the logic of the test remains the same. Data through 2001 are now available to extend the last period to 2001.

There are some caveats about the Chow test that must be kept in mind:

1. The assumptions underlying the test must be fulfilled. For example, one should find out if the error variances in the regressions (8.8.1) and (8.8.2) are the same. We will discuss this point shortly.

2. The Chow test will tell us only if the two regressions (8.8.1) and (8.8.2) are different, without telling us whether the difference is on account of the intercepts, or the slopes, or both. But in Chapter 9, on dummy variables, we will see how we can answer this question.

3. The Chow test assumes that we know the point(s) of structural break. In our example, we assumed it to be in 1982. However, if it is not possible to

_ (23,248.30 - 11,790.252)/2 = (11,790.252)/22 = 10.69

278 PART ONE: SINGLE-EQUATION REGRESSION MODELS

determine when the structural change actually took place, we may have to use other methods.16

Before we leave the Chow test and our savings-income regression, let us examine one of the assumptions underlying the Chow test, namely, that the error variances in the two periods are the same. Since we cannot observe the true error variances, we can obtain their estimates from the RSS given in the regressions (8.8.1a) and (8.8.2a), namely,

. RSS1 1785.032 , o __ = nT—Î = 10 = 1785032 (8-8'6)

Notice that, since there are two parameters estimated in each equation, we deduct 2 from the number of observations to obtain the df. Given the assumptions underlying the Chow test, of and 022 are unbiased estimators of the true variances in the two subperiods. As a result, it can be shown that if of = a22—that is, the variances in the two subpopulations are the same (as assumed by the Chow test)—then it can be shown that

(a12/a12) _ F (8 8 8) ~ F(n1— k),(n2—k) (8.8.8)

follows the F distribution with (n1 — k) and (n2 — k) df in the numerator and the denominator, respectively, in our example k = 2, since there are only two parameters in each subregression.

Of course, of = af, the preceding F test reduces to computing a 2

or 22

Note: By convention we put the larger of the two estimated variances in the numerator. (See Appendix A for the details of the F and other probability distributions.)

Computing this F in an application and comparing it with the critical F value with the appropriate df, one can decide to reject or not reject the null hypothesis that the variances in the two subpopulations are the same. If the null hypothesis is not rejected, then one can use the Chow test.

Returning to our savings-income regression, we obtain the following result:

16For a detailed discussion, see William H. Greene, Econometric Analysis, 4th ed., Prentice Hall, Englewood Cliffs, N.J., 2000, pp. 293-297.

Under the null hypothesis of equality of variances in the two subpopulations, this F value follows the F distribution with 12 and 10 df, in the numerator and denominator, respectively. (Note: We have put the larger of the two estimated variances in the numerator.) From the F tables in Appendix D, we see that the 5 and 1 percent critical F values for 12 and 10 df are 2.91 and 4.71, respectively. The computed F value is significant at the 5 percent level and is almost significant at the 1 percent level. Thus, our conclusion would be that the two subpopulation variances are not the same and, therefore, strictly speaking we should not use the Chow test.

Our purpose here has been to demonstrate the mechanics of the Chow test, which is used popularly in applied work. If the error variances in the two subpopulations are heteroscedastic, the Chow test can be modified. But the procedure is beyond the scope of this book.17

Another point we made earlier was that the Chow test is sensitive to the choice of the time at which the regression parameters might have changed. In our example, we assumed that the change probably took place in the recession year of 1982. If we had assumed it to be 1981, when Ronald Reagan began his presidency, we might find that the computed F value is different. As a matter of fact, in exercise 8.34 the reader is asked to check this out.

If we do not want to choose the point at which the break in the underlying relationship might have occurred, we could choose alternative methods, such as the recursive residual test. We will take this topic up in Chapter 13, the chapter on model specification analysis.

In Section 5.10 we showed how the estimated two-variable regression model can be used for (1) mean prediction, that is, predicting the point on the population regression function (PRF), as well as for (2) individual prediction, that is, predicting an individual value of Y given the value of the re-gressor X = X0, where X0 is the specified numerical value of X.

The estimated multiple regression too can be used for similar purposes, and the procedure for doing that is a straightforward extension of the two-variable case, except the formulas for estimating the variances and standard errors of the forecast value [comparable to (5.10.2) and (5.10.6) of the two-variable model] are rather involved and are better handled by the matrix methods discussed in Appendix C. Of course, most standard regression packages can do this routinely, so there is no need to look up the matrix formulation. It is given in Appendix C for the benefit of the mathematically inclined students. This appendix also gives a fully worked out example.

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