GIV and the optimal weighting matrix
Problem
- Setup: more instruments than explanatory variables
- 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 \(r×1\) with \(r>k\)
- A linear regression model, \(y_i=x'_iβ+ε_i\)
- Exogeneity fails, \(E\left( ε_i \mid x_i \right)≠0\)
- z-variables are instruments, \(E\left( ε_i|z_i \right)=0\) and \(E\left( z_ix'_i \right)=Σ_{zx'}\) is \(r×k\) with rank \(k\) .
- \(Var\left( ε_i|z_i \right)=σ^2\) .
The optimal weighting matrix in gmm is \(W^{opt}={\left( E\left( f\left( w_i,β \right)f{\left( w_i,β \right)}' \right) \right)}^{-1}\) . Show that in our setup,
\[E\left( f\left( w_i,β \right)f{\left( w_i,β \right)}' \right)=σ^2Σ_{zz'}\]
Solution
\(f\left( w_i,β \right)=z_iε_i\) and \(f{\left( w_i,β \right)}'=ε_iz'_i\) ( \(ε_i\) is a scalar and you can put it wherever you want). Thus,
\[f\left( w_i,β \right)f{\left( w_i,β \right)}'=ε_i^2z_iz'_i\]
Now,
\[E\left( f\left( w_i,β \right)f{\left( w_i,β \right)}' \right)=E\left( ε_i^2z_iz'_i \right)=E\left( E\left( ε_i^2z_iz'_i|z_i \right) \right)=E\left( z_iz'_iE\left( ε_i^2|z_i \right) \right)=\]
\[=E\left( z_iz'_iVar\left( ε_i|z_i \right) \right)=E\left( z_iz'_iσ^2 \right)=σ^2E\left( z_iz'_i \right)=σ^2Σ_{zz'}\]
where \(Σ_{zz'}=E\left( z_iz'_i \right)\) .