Example 194

Demand function: Qt = a0 + a1 Pt + a2It + a3Rt + u-t t (19.2.28) Supply function: Qt = 0O + 0t Pt + 02Pt—1 + u2t (19.2.22)

In this model Ptand Qtare endogenous and It, Rt, and Pt—1 are predetermined. The demand function excludes exactly one variable Pt—1, and hence by the order condition it is exactly identified. But the supply function excludes two variables It and Rt, and hence it is overiden-tified. As noted before, in this case there are two ways of estimating 01, the coefficient of the price variable.

Notice a slight complication here. By the order condition the demand function is identified. But if we try to estimate the parameters of this equation from the reduced-form coefficients given in (19.2.31), the estimates will not be unique because 01, which enters into the computations, takes two values and we shall have to decide which of these values is appropriate. But this complication can be obviated because it is shown in Chapter 20 that in cases of overidentification the method of indirect least squares is not appropriate and should be discarded in favor of other methods. One such method is two-stage least squares, which we shall discuss fully in Chapter 20.

Models

Problem

750 PART FOUR: SIMULTANEOUS-EQUATION MODELS

As the previous examples show, identification of an equation in a model of simultaneous equations is possible if that equation excludes one or more variables that are present elsewhere in the model. This situation is known as the exclusion (of variables) criterion, or zero restrictions criterion (the coefficients of variables not appearing in an equation are assumed to have zero values). This criterion is by far the most commonly used method of securing or determining identification of an equation. But notice that the zero restrictions criterion is based on a priori or theoretical expectations that certain variables do not appear in a given equation. It is up to the researcher to spell out clearly why he or she does expect certain variables to appear in some equations and not in others.

The order condition discussed previously is a necessary bUt not sUfficient condition for identification; that is, even if it is satisfied, it may happen that an equation is not identified. Thus, in Example 19.2, the supply equation was identified by the order condition because it excluded the income variable It, which appeared in the demand function. But identification is accomplished only if a2, the coefficient of It in the demand function, is not zero, that is, if the income variable not only probably but actually does enter the demand function.

More generally, even if the order condition K — k > m — 1 is satisfied by an equation, it may be unidentified because the predetermined variables excluded from this equation but present in the model may not all be independent so that there may not be one-to-one correspondence between the structural coefficients (the P's) and the reduced-form coefficients (the n's). That is, we may not be able to estimate the structural parameters from the reduced-form coefficients, as we shall show shortly. Therefore, we need both a necessary and sufficient condition for identification. This is provided by the rank condition of identification, which may be stated as follows:

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