We noted earlier that it is the researcher's responsibility to specify which variables are endogenous and which are exogenous. This will depend on the problem at hand and the a priori information the researcher has. But is it possible to develop a statistical test of exogeneity, in the manner of Granger's causality test?
The Hausman test discussed in Section 19.4 can be utilized to answer this question. Suppose we have a three-equation model in three endogenous variables, Y1, Y2, and Y3, and suppose there are three exogenous variables, X1, X2, and X3. Further, suppose that the first equation of the model is
Y1i = fa0 + fa2 Y2i + fa3 Y3i + a1 Xh- + uu (19.5.1)
If Y2 and Y3 are truly endogenous, we cannot estimate (19.5.1) by OLS (why?). But how do we find that out? We can proceed as follows. We obtain the reduced-form equations for Y2 and Y3 (Note: the reduced-form equations will have only predetermined variables on the right-hand side). From these reduced-form equations, we obtain Y2i and Y3i, the predicted values of Y2i and Y3i, respectively. Then in the spirit of the Hausman test discussed
14As in footnote 12, the authors use AID rather than AID as the regressor. *Optional.
CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 757
earlier, we can estimate the following equation by OLS:
Yu = Po + PiY2i + P3 Y3i + a1 Xu + + ^i + uu (19.5.2)
Using the F test, we test the hypothesis that k2 = k3 = 0. If this hypothesis is rejected, Y2 and Y3 can be deemed endogenous, but if it is not rejected, they can be treated as exogenous. For a concrete example, see exercise 19.16.
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