Instrumental variables, one explanatory variable

Summary

Setup

• The linear regression model with random sampling,

\[y_i=β_1+β_2x_i+ε_i, i=1,…,n\]

• \(x_i\) is endogenous
• OLS will not give us unbiased or consistent estimators, standard errors are inconsistent and inference is invalid.

Instrumental variable

A variable \(z\) is called an instrumental variable (IV) for the endogenous variable \(x\) in the LRM if the following are satisfied:

• \(z\) is exogeneous,

\[E\left( ε_i|z_i \right)=0, i=1,…,n\]

• \(z_i\) is correlated with \(x_i\)

\[Cor\left( x_i,z_i \right)≠0\]

• \(z_i\) will not have a direct effect on \(y_i\) , \(E\left( z_i,x_i \right)\) does not depend on \(z_i\) .
• We say that the instrument is weak/strong if \(Cor\left( x_i,z_i \right)\) is small/large.

The IV-estimator

• The IV-estimator of \(β_2\) is given by

\[b_2= \frac{\sum_{i=1}^{n}{ \left( z_i-\bar{z} \right)\left( y_i-\bar{y} \right) }}{\sum_{i=1}^{n}{ \left( z_i-\bar{z} \right)\left( x_i-\bar{x} \right) }}\]

• The IV-estimator of \(β_1\) is given by

\[b_1=\bar{y}-b_2\bar{x}\]