Instrumental variables

Summary

• Setup
• Random sample \(\left( y_i,x_i,z_i \right)\) for \(i=1, \ldots ,n\) where \(x_i\) is \(k×1\) and \(z_i\) is \(r×1\)
• A linear regression model, \(y_i=x'_iβ+ε_i\)
• Exogeneity fails, \(E\left( ε_i \mid x_i \right)≠0\)
• Definition: instrumental variables. We say that \(z_i\) \(\left( r×1 \right)\) are instruments for \(x_i\) if

\[E\left( ε_i|z_i \right)=0\]

• and

\[E\left( z_ix'_i \right)=Σ_{zx'}\]

• has rank = \(k\) . We need \(r\) to be at least equal to \(k\) .
• If x-variable \(j\) is exogenous, \(E\left( ε_i \mid x_{i,j} \right)=0\) then we can use this variable as an instrument (we say that it is an instrument for itself ).