2.4 Endogenous explanatory variables  (Page 5/10)

The identification problem

There is an additional issue that arises with estimating systems of equations—identification. Essentially, identification is an algebraic problem. Consider the reduced form equations given earlier in (4) and (5):

$$q_t=\frac{{\alpha }_0+{\alpha }_1{\beta }_0}{1-{\alpha }_1{\beta }_1}+\frac{{\alpha }_1{\beta }_2}{1-{\alpha }_1{\beta }_1}{W}_t+\frac{{\alpha }_2}{1-{\alpha }_1{\beta }_1}{I}_t+\frac{{\alpha }_3}{1-{\alpha }_1{\beta }_1}{p}_t^c+\frac{{\epsilon }_t+{\alpha }_1{\eta }_t}{1-{\alpha }_1{\beta }_1}$$

and

$$p_t^w=\frac{{\beta }_0+{\beta }_1{\alpha }_0}{1-{\alpha }_1{\beta }_1}+\frac{{\alpha }_2{\beta }_1}{1-{\alpha }_1{\beta }_1}{I}_t+\frac{{\alpha }_3{\beta }_1}{1-{\alpha }_1{\beta }_1}{p}_t^c+\frac{{\beta }_2}{1-{\alpha }_1{\beta }_1}{W}_t+\frac{{\beta }_1{\epsilon }_t+{\eta }_t}{1-{\alpha }_1{\beta }_1}.$$

OLS estimation of both of these equations yields unbiased estimates of the parameters in the reduced form equations. Identification asks if we can retrieve the parameters of the structural equations from the reduced form equations. Say, for instance, that we re-write the reduced form equations as:

and

Table 1 shows each of the parameters in (11) and (12) in terms of the parameters of the two reduced form equations. We can recover the parameters of the structural equations by algebraic manipulation of the relationships in Table 1. (This method of estimation—that is, estimating the reduced form equations of a model using OLS and then solving algebraically for the parameters of the structural equations is referred to in the literature as indirect least squares .) For instance,

$$\frac{{\delta }_{21}}{{\delta }_{12}}=\frac{\left(\frac{{\alpha }_2{\beta }_1}{1-{\alpha }_1{\beta }_1}\right)}{\left(\frac{{\alpha }_2}{1-{\alpha }_1{\beta }_1}\right)}={\beta }_1$$

and

$$\frac{{\delta }_{11}}{{\delta }_{23}}=\frac{\left(\frac{{\alpha }_1{\beta }_2}{1-{\alpha }_1{\beta }_1}\right)}{\left(\frac{{\beta }_2}{1-{\alpha }_1{\beta }_1}\right)}={\alpha }_1.$$

One can continue in a likewise manner to find formulae for other of the structural parameters. However, an interesting problem does arrive in that it is also true that ${\beta }_1=\frac{{\delta }_{22}}{{\delta }_{13}}.$ Since there is no a priori reason to believe that $\frac{{\delta }_{22}}{{\delta }_{13}}=\frac{{\delta }_{21}}{{\delta }_{12}},$ we have two estimates of ${\beta }_1.$ This result illustrates the point that there are three possibilities when calculating the structural parameters from the reduced form equations—first, there may be more than one formula for a structural parameter; second, there may be only one formula for a structural parameter; or third, there may be no formula for a structural parameter. We say in the first case that the equation is over-identified ; is exactly identified in the second case; and is under-identified in the third case. It turns out that in the case of an over-identified equation we can to use TSLS to estimate the structural parameters. However, in the case of an exactly identified equation , the TSLS estimators are equal to the indirect-least-squares estimators that can be calculated using estimates of the reduced form equations. Finally, an under-identified equation cannot be estimated by any technique.

Clearly, we need to know how to identify if an equation is either over-identified, exactly identified, or under-identified. A necessary rule is that the number of exogenous variables in a system of equation that are not included in a particular regression must be greater than or equal to the number of endogenous variables on the right-hand-side of the equation for the equation to be either exactly or over identified. Consider the following three-equation model, where the endogenous variables are $y_1,$ $y_2$ , and $y_3$ and the exogenous variables are represented by $x_1$ with $i=1,\dots ,5:$