Source: Economic Report of the President, 1993, Table B-5, p. 355.

Source: Economic Report of the President, 1993, Table B-5, p. 355.



reason for introducing the lagged value of PPCE in this model is to reflect inertia or habit persistence.

The results of estimating these models separately were as follows:

Model A: PPCEt = —1,299.0536 + 0.9204 PDPIt + 0.0931 PDPIt— 1

Model B: PPCEt = —841.8568 + 0.7117 PDPIt + 0.2954 PPCEt— 1

If one were to choose between these two models on the basis of the discrimination approach, using, say, the highest R2 criterion, one would choose (13.8.10); besides, in (13.8.10) both variables seem to be individually statistically significant, whereas in (13.8.9) only the current PDPI is statistically significant (but beware of the collinearity problem!).

But choosing (13.8.10) over (13.8.9) may not be appropriate because for predictive purposes there is not much difference in the two estimated R2 values.

To apply the J test, suppose we assume Model A is the null hypothesis, that is, the maintained model, and Model B is the alternative hypothesis. Now following the J test steps discussed earlier we use the estimated PPCE values from model (13.8.10) as an additional regressor in Model A, giving the following outcome:

PPCEt = 1,322.7958 — 0.7061PDPIt — 0.4357PDPIt—! + 2.1335PPCEf t = (1.5896) (—1.3958) (—2.1926) (3.3141) (13.8.11)

where PPCEf on the right side of (13.8.11) are the estimated PPCE values from model B, (13.8.10). Since the coefficient of this variable is statistically significant (at the two-tail 0.004 level), following the J test procedure, we have to reject Model A in favor of Model B.

Now assuming Model B as the maintained hypothesis and Model A as the alternative hypothesis, and following exactly the same procedure as before, we obtain the following results:

PPCEt = —6,549.8659 + 5.1176PDPI, + 0.6302PPCEt— — 4.6776PPCEA

t = (—2.4976) (2.5424) (3.4141) (—2.1926) (13.8.12)

where PPCEA on the right side of(13.8.12) is obtained from the Model A, (13.8.9). But in this regression, the coefficient of PPCEtA on the right side is also statistically significant (at the two-tail 0.0425 level). This result would suggest that we should now reject Model B in favor of Model A!

All this tells us is that neither model is particularly useful in explaining the behavior of per capita personal consumption expenditure in the United States over the period 1970-1991.

Of course, we have considered only two competing models. In reality, there may be more than two models. The J test procedure can be extended to multiple model comparisons, although the analysis can become quickly complex.

This example shows very vividly why the CLRM assumes that the regression model used in the analysis is correctly specified. Obviously it is very crucial in developing a model to pay very careful attention to the phenomenon being modeled.


Other Tests of Model Selection. The J test just discussed is only one of a group of tests of model selection. There is the Cox test, the JA test, the P test, Mizon-Richard encompassing test, and variants of these tests. Obviously, we cannot hope to discuss these specialized tests, for which the reader may want to consult the references cited in the various footnotes.37

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