Editing Simultaneous equations model (section)

===Using cross-equation restrictions to achieve identification===

In simultaneous equations models, the most common method to achieve [[Parameter identification problem|identification]] is by imposing within-equation parameter restrictions.<ref name= "Woolridge">Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.</ref>  Yet, identification is also possible using cross equation restrictions.

To illustrate how cross equation restrictions can be used for identification, consider the following example from Wooldridge<ref name= "Woolridge" />

:<math>\begin{align}
y_1 &= \gamma_{12} y_2 + \delta_{11} z_1 + \delta_{12} z_2 + \delta_{13} z_3 + u_1 \\
y_2 &= \gamma_{21} y_1 + \delta_{21} z_1 + \delta_{22} z_2 + u_2
\end{align}</math>

where z's are uncorrelated with u's and y's are [[endogenous variable|endogenous]] variables.  Without further restrictions, the first equation is not identified because there is no excluded exogenous variable.  The second equation is just identified if {{math|''δ''<sub>13</sub>≠0}}, which is assumed to be true for the rest of discussion.

Now we impose the cross equation restriction of {{math|''δ''<sub>12</sub>{{=}}''δ''<sub>22</sub>}}. Since the second equation is identified, we can treat {{math|''δ''<sub>12</sub>}} as known for the purpose of identification. Then, the first equation becomes:

:<math>y_1 - \delta_{12} z_2 = \gamma_{12} y_2 + \delta_{11} z_1 + \delta_{13} z_3 + u_1</math>

Then, we can use {{math|(''z''<sub>1</sub>, ''z''<sub>2</sub>, ''z''<sub>3</sub>)}} as [[instrumental variable|instruments]] to estimate the coefficients in the above equation since there are one endogenous variable ({{math|''y''<sub>2</sub>}}) and one excluded exogenous variable ({{math|''z''<sub>2</sub>}}) on the right hand side.  Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification.