The motivation behind the tests suggested by Davidson and MacKinnon is the following: The models given by and , can be combined into a single model

= (1 - a)fix + ( ) + (12.29)

and testing a = 0 versus a = 1. However, there is no way of estimating (3, y, and a from this model. All we get are estimates of (1 - a)p and ay. Davidson

"Two surveys of this extensive literature are: J. G. MacKinnon, "Model Specification Tests Against Non-nested Alternatives" (with discussion). Econometric Reviews, Vol. 2, 1983. pp. 85-158, and M. McAleer and M. H. Pesaran, "Statistical Inference in Non-nested Econometric Models," Applied Mathematics and Computation, 1986.

"R. Davidson and J. G. MacKinnon, "Several Tests for Model Specification in the Presence of Alternative Hypotheses," Econometrica. Vol. 49, 1981, pp. 781-793. The survey paper by MacKinnon contains additional references to the papers by Davidson and MacKinnon.

The Davidson and MacKinnon Test

There have been several tests that have been suggested for testing such nonnested hypotheses." The first such test is the Cox test. However, we will outline some simple alternative and asymptotically equivalent tests suggested by Davidson and MacKinnon" that are easy to apply. Their test procedure for testing against , ( is the maintained hypothesis) is as follows:

1. Estimate (12.26) by OLS. Let y, = be the predicted value of y.

2. Estimate the regression equation

= fix + ay, + (12.27)

and test the hypothesis a = 0.

If the hypothesis is not rejected, then is not rejected by H,. If the hypothesis is rejected, then is rejected by ,. A test of , against would be based on analogous steps.

1. Estimate (12.25) by OLS. Let = be the predicted value of y.

2. Estimate the regression equation

= + + V (12.28)

and test the hypothesis 8 = 0.

If the hypothesis is not rejected, then H, is not rejected by . If the hypothesis is rejected, then H, is rejected by . The outcome of these two tests can be summarized as follows:

Hnnth..;.- Hypothesis: a = 0

"See C. W. J. Granger and P. Newbold, Forecasting Economic Time Series, 2nd ed. (New York: Academic Press, 1986), pp. 266-267 on combination of forecasts.

"McAleer and Pesaran, "Statistical Inference." G. R. Fisher and M. McAleer, "Alternative Procedures and Associated Tests of Significance for Non-nested Hypotheses," Journal of Econometrics, Vol. 16, 1981, pp. 103-119, suggest what is known as the JA test. Unlike the y-test, which is asymptotic, the JA test is a small-sample test (if x and z are fixed in repeated samples).

"A good review and discussion of the limitations of nonnested tests is L. G. Godfrey, "On the Use of Misspecification Checks and Tests of Non-nested Hypotheses in Empirical Econometrics," Economic Journal, Supplement, 1985, pp. 69-81.

and MacKinnon show that we can substitute y, = yz for -yz in (12.29) and then test a = 0. They show that under , a from (12.27) is asymptotically N(0, 1). They call this the J-test (because a and are estimated jointly). Note that the J-tests are one degree of freedom tests irrespective of the number of explanatory variables in and ,.

The relationship between the J-test and optimum combination of forecasts is as follows: We have two forecasts of from the two models (12.25) and (12.26). Call these and y„ respectively. The two forecasts can be combined to produce a forecast with a smaller forecast error.* The combination can be written as

= (1 - ) + ay. The value of a that gives the minimum forecast error is

" ~ \2 + \i - 2p\,\2

\i and \l are, respectively, the variances of the forecast errors (y - ) and (y ~ Si) and p is the correlation between these forecast errors. Also,

ago if and only if § p

Corresponding to the optimum combination of forecasts, equation (12.29) can also be written as

= (1 - ) + ay, + V

where v is the error term. This can also be written as

- = ( 1 - ) + V

If we estimate this equation and the estimate of a is not significantly different from zero, we can say that , does not add anything to explaining over . If a is significantly different from zero, , explains over and above .

We will not pursue the extensions of the J-test here." We will, however, discuss a few of the limitations. One limitation of the test is that sometimes it rejects both Hq and , or accepts and ,. Although this conclusion suggests

12.12 TESTS FOR NONNESTED HYPOTHESES 517

that we should go back and examine both models, in many cases we would like to have some ranking of the models.

An alternative procedure is to embed the two models given by and , into a comprehensive model

= fix + yz + * (12.30)

and testing „ by testing 7 = 0 and testing , by testing p = 0. When there are more than one variable in x and z, these tests will be F-tests. This is contrast to the J-test, which is a one-degree-of-freedom test whatever the number of explanatory variables in and ,.

The Encompassing Test

At first sight it would appear that there is no relationship between the F-tests and the J-test. This is not so. Mizon and Richard suggest a more general test called the "encompassing test" of which the F-test and J-test are special cases. The encompassing principle is based on the idea that a model builder should analyze whether his model can account for salient features of rival models.*" The encompassing test is a formulation of this principle. If your model is specified by and the rival model by ,, a formal test of against , is to compare 7 and obtained under , from equation (12.26), with the probability limits of these parameters under your hypothesis - Comparing 7 with plim 71 gives the mean encompassing test. Comparing with plim df gives the variance encompassing test. Mizon and Richard show that the F-test is a mean encompassing test and the J-test is a variance encompassing test. This explains why the -test is a one-degree-of-freedom test no matter how many explanatory variables there are in the models given by and ,. The complete encompassing test (CET) suggested by Mizon and Richard is a joint test that compares 7, and a] with their probability limits under . A discussion of this joint test is beyond the scope of this book. Further, there is not much empirical evidence on it. The F-test and F-test, on the other hand, can be very easily implemented.

In summary, to test the nonnested hypothesis against ,, we need to apply two tests:

1. The J-test testing a = 0 based on equation (12.27).

2. The F-test testing 7 = 0 in the comprehensive model (12.30).

As to how these tests perform in small samples, some experimental evidence suggests that the one-degree-of-freedom F-test is better than the F-test at re-

"G. E. Mizon and J. F. Richard, "The Encompassing Principle and Its Application to Testing Non-nested Hypotheses," Econometrica, Vol. 54, 1986, pp. 657-678.

"J. E. H. Davidson and D. F. Hendry, "Interpreting Econometric Evidence: The Behavior of Consumers Expenditures in the U.K.," European Economic Review, Vol. 16, 1981, pp. 179-198.