## Nested Versus Nonnested Models

In carrying out specification testing, it is useful to distinguish between nested and non-nested models. To distinguish between the two, consider the following models:

Model A: Y = 01 + 02 X2i + 03 X3i + 04 X4i + 05 X5i + u-Model B: Yj = 01 + 02 X2j + 03 X3j + uj

We say that Model B is nested in Model A because it is a special case of Model A: If we estimate Model A and test the hypothesis that 04 = 05 = 0 and do not reject it on the basis of, say, the F test,31 Model A reduces to Model B. If we add variable X4 to Model B, then Model A will reduce to Model B if 05 is zero; here we will use the t test to test the hypothesis that the coefficient of X5 is zero.

Without calling them such, the specification error tests that we have discussed previously and the restricted F test that we discussed in Chapter 8 are essentially tests of nested hypothesis.

31More generally, one can use the likelihood ratio test, or the Wald test or the Lagrange Multiplier test, which were discussed briefly in Chap. 8.

530 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

Now consider the following models:

Model C: Y = al + a + a + u Model D: Y = ßi + fa ^i + ß3 Zx + Vi where the X's and Z's are different variables. We say that Models C and D are non-nested because one cannot be derived as a special case of the other. In economics, as in other sciences, more than one competing theory may explain a phenomenon. Thus, the monetarists would emphasize the role of money in explaining changes in GDP, whereas the Keynesians may explain them by changes in government expenditure.

It may be noted here that one can allow Models C and D to contain re-gressors that are common to both. For example, X3 could be included in Model D and Z2 could be included in Model C. Even then these are nonnested models, because Model C does not contain Z3 and Model D does not contain X2.

Even if the same variables enter the model, the functional form may make two models non-nested. For example, consider the model:

Models D and E are non-nested, as one cannot be derived as a special case of the other.

Since we already have looked at tests of nested models (t and F tests), in the following section we discuss some of the tests of non-nested models, which earlier we called model mis-specification errors.

According to Harvey,32 there are two approaches to testing non-nested hypotheses: (1) the discrimination approach, where given two or more competing models, one chooses a model based on some criteria of goodness of fit, and (2) the discerning approach (my terminology) where, in investigating one model, we take into account information provided by other models. We consider these approaches briefly.

Consider Models C and D above. Since both models involve the same dependent variable, we can choose between two (or more) models based on some goodness-of-fit criterion, such as R2 or adjusted R2, which we have already discussed. But keep in mind that in comparing two or more models,