The generalized instrumental variables estimator

Summary

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 \(k\) .
\(Var\left( ε_i|z_i \right)=σ^2\) .

\[E\left( {z_iε}_i \right)=E\left( z_i\left( y_i-x'_iβ \right) \right)=0\]

\[s\left( β \right)= \frac{1}{n}\sum_{i=1}^{n}{ z_i\left( y_i-x'_iβ \right) }= \frac{1}{n}Z'\left( y-Xβ \right)\]

The method of moments estimator is no longer available since we have more equations than unknowns. Instead, we use generalized method of moments with \(f\left( w_i,β \right)={z_iε}_i=z_i\left( y_i-x'_iβ \right)\) .
Optimal choice for the weighting matrix,

\[W^{opt}={\left( E\left( f\left( w_i,β \right)f{\left( w_i,β \right)}' \right) \right)}^{-1}={\left( σ^2Σ_{zz'} \right)}^{-1}\]

where \(Σ_{zz'}=E\left( z_iz'_i \right)\) .
\(σ^2\) is a scalar and it can be removed. As an estimator of \(Σ_{zz'}\) we can use

\[ \frac{1}{n}\sum_{i=1}^{n}{ z_iz'_i }= \frac{1}{n}Z'Z\]

That is, we use \({\left( Z'Z \right)}^{-1}\) as our weighting matrix and (removing the 1/n)

\[Q\left( β \right)={\left( Z'\left( y-Xβ \right) \right)}'{\left( Z'Z \right)}^{-1}\left( Z'\left( y-Xβ \right) \right)\]

\[b_{GIV}={\left( X'P_ZX \right)}^{-1}X'P_Zy\]

where \(P_Z=Z'{\left( Z'Z \right)}^{-1}Z\) . This is called the generalized IV estimator or sometimes simply the IV estimator.
It is not possible to find \(E\left( b_{GIV} \right)\) or \(E\left( b_{GIV}|X \right)\) or \(Var\left( b_{GIV}|X \right)\) (The generalized IV estimator has no small sample properties).
Result: The generalized IV estimator is consistent and asymptotically normal. As an estimator of \(Var\left( b_{GIV} \right)\) we use

\[s^2{\left( X'P_ZX \right)}^{-1}\]

where \(s^2= \frac{1}{n}\sum{ e_i^2 }\) where \(e_i=y_i-x_i'b_{GIV}\) are the generalized IV residuals.