Simultaneity and instrumental variables

Summary

• Setup:
• Random sample \(\left( y_i,x_i \right)\) for \(i=1, \ldots ,n\) where \(x_i\) is a scalar
• A linear regression model \(y_i=β_1+β_2x_i+ε_i\)
• Simultaneity, \(x_i=y_i+z_i\) where \(z_i\) is an exogenous variable, \(E\left( ε_i \mid z_i \right)=0\) .
• \(β\) can then be consistently estimated using the IV estimator with \(z\) as an instrument for \(x\) .