34See Keith M. Carlson, "Does the St. Louis Equation Now Believe in Fiscal Policy?" Review, Federal Reserve Bank of St. Louis, vol. 60, no. 2, February 1978, p. 17, table IV.
CHAPTER THIRTEEN: ECONOMETRIC MODELING 533
Davidson-MacKinnon J Test.35 Because of the problems just listed in the non-nested F testing procedure, alternatives have been suggested. One is the Davidson-MacKinnon J test. To illustrate this test, suppose we want to compare hypothesis or Model C with hypothesis or Model D. The J test proceeds as follows:
1. We estimate Model D and from it we obtain the estimated Yvalues, YD.
2. We add the predicted Y value in Step 1 as an additional regressor to Model C and estimate the following model:
where the YD values are obtained from Step 1. This model is an example of the encompassing principle, as in the Hendry methodology.
3. Using the t test, test the hypothesis that a4 = 0.
4. If the hypothesis that a4 = 0 is not rejected, we can accept (i.e., not reject) Model C as the true model because YiD included in (13.8.5), which represent the influence of variables not included in Model C, have no additional explanatory power beyond that contributed by Model C. In other words, Model C encompasses Model D in the sense that the latter model does not contain any additional information that will improve the performance of Model C. By the same token, if the null hypothesis is rejected, Model C cannot be the true model (why?).
5. Now we reverse the roles of hypotheses, or Models C and D. We now estimate Model C first, use the estimated Y values from this model as re-gressor in (13.8.5), repeat Step 4, and decide whether to accept Model D over Model C. More specifically, we estimate the following model:
where Yf are the estimated Y values from Model C. We now test the hypothesis that 34 = 0. If this hypothesis is not rejected, we choose Model D over C. If the hypothesis that 34 = 0 is rejected, choose C over D, as the latter does not improve over the performance of C.
Although it is intuitively appealing, the J test has some problems. Since the tests given in (13.8.5) and (13.8.6) are performed independently, we have the following likely outcomes:
Hypothesis: a4 = 0
Hypothesis: fi4 = 0 Do not reject Reject
Do not reject Accept both C and D Accept D, reject C
Reject Accept C, reject D Reject both C and D
35R. Davidson and J. G. MacKinnon, "Several Tests for Model Specification in the Presence of Alternative Hypotheses," Econometrica, vol. 49, 1981, pp. 781-793.
534 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
As this table shows, we will not be able to get a clear answer if the J testing procedure leads to the acceptance or rejection of both models. In case both models are rejected, neither model helps us to explain the behavior of Y. Similarly, if both models are accepted, as Kmenta notes, "the data are apparently not rich enough to discriminate between the two hypotheses [models]."36
Another problem with the J test is that when we use the t statistic to test the significance of the estimated Y variable in models (13.8.5) and (13.8.6), the t statistic has the standard normal distribution only asymptotically, that is, in large samples. Therefore, the J test may not be very powerful (in the statistical sense) in small samples because it tends to reject the true hypothesis or model more frequently than it ought to.
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