The generalized instrumental variables estimator

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

Spot the mistake.

• True moments: \(E\left( {z_iε}_i \right)=E\left( z_i\left( y_i-x'_iβ \right) \right)=0\)
• Sample moments:

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

• Method of moments estimator:

\[ \frac{1}{n}Z'\left( y-Xb_{MM} \right)=0\]

or

\[Z'y=Z'Xb_{MM}\]

or

\[b_{MM}={\left( Z'X \right)}^{-1}Z'y\]