Deriving the generalized instrumental variables estimator

Problem

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 GMM quadratic form is given by

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

Find the \(β\) which minimizes \(Q\left( β \right)\) .

Solution

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

so \(Q\left( β \right)\) can expanded to

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

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

\[y'Z{\left( Z'Z \right)}^{-1}Z'y-2{y'Z\left( Z'Z \right)}^{-1}Z'Xβ+β'X'Z{\left( Z'Z \right)}^{-1}Z'Xβ=\]

\[y'P_Zy-2y'P_ZXβ+β'X'P_ZXβ\]

Since the second and the third term are equal and using the convenient \(P_Z=Z{\left( Z'Z \right)}^{-1}Z'\) . Differentiating:

\[ \frac{dQ}{dβ}=-{\left( 2y'P_ZX \right)}'+X'P_ZXβ=0\]

Transposing,

\[X'P_ZXβ=X'P_Zy\]

Solving for \(β\) :

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