The instrumental variables (IV) estimator

Summary

• Setup: same number of instruments as 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 \(k×1\)
• 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 \(k×k\) and invertible
• Suppose further that \(Var\left( ε_i|z_i \right)=σ^2\) .
• True moments:

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

• Sample moments

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

• The method of moments estimator for this model is called the instrumental variables (IV) estimator :

\[b_{IV}={\left( \sum_{i=1}^{n}{ z_ix'_i } \right)}^{-1}\sum_{i=1}^{n}{ z_iy_i }={\left( Z'X \right)}^{-1}Z'y\]

• It is not possible to find \(E\left( b_{IV} \right)\) or \(E\left( b_{IV}|X \right)\) or \(Var\left( b_{IV}|X \right)\) (The IV estimator has no small sample properties).
• Result: The IV estimator is consistent and asymptotically normal. As an estimator of \(Var\left( b_{IV} \right)\) we use

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

• where \(P_Z=Z{\left( Z'Z \right)}^{-1}Z'\) and \(s^2= \frac{1}{n}\sum{ e_i^2 }\) where \(e_i=y_i-x_i'b_{IV}\) are the IV residuals.