The generalized IV when r = k
Problem
Show that the generalized IV estimator
\[b_{GIV}={\left( X'P_ZX \right)}^{-1}X'P_Zy\]
becomes equal to the IV estimator
\[b_{IV}={\left( Z'X \right)}^{-1}Z'y\]
if \(r=k.\) \(P_Z=Z{\left( Z'Z \right)}^{-1}Z'\) .
Hint: Substitute \(P_Z\) into the formula of \(b_{GIV}\) and use the matrix problem from before.
Solution
\[b_{GIV}={\left( X'Z{\left( Z'Z \right)}^{-1}Z'X \right)}^{-1}X'Z{\left( Z'Z \right)}^{-1}Z'y\]
If \(r=k\) then \(X'Z\) and \(Z'X\) are square and invertible (else \(b_{IV}\) does not exist)
We have
\[{\left( X'Z{\left( Z'Z \right)}^{-1}Z'X \right)}^{-1}={\left( Z'X \right)}^{-1}Z'Z{\left( X'Z \right)}^{-1}\]
Therefore,
\[b_{GIV}={\left( Z'X \right)}^{-1}Z'Z{\left( X'Z \right)}^{-1}X'Z{\left( Z'Z \right)}^{-1}Z'y\]
We have
\[{\left( X'Z \right)}^{-1}X'Z=I\]
and
\[b_{GIV}={\left( Z'X \right)}^{-1}Z'Z{\left( Z'Z \right)}^{-1}Z'y\]
We have
\[Z'Z{\left( Z'Z \right)}^{-1}=I\]
and
\[b_{GIV}={\left( Z'X \right)}^{-1}Z'y=b_{IV}\]