Instrumental variables, several explanatory variables

Summary

Setup, several explanatory variables

  • The linear regression model with random sampling,

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n\]

  • For any endogenous variable \(x_j\) we have an instrument \(z_j\) .
  • If \(x_j\) is exogenous, it will be an instrument for itself, \(z_j=x_j\) .
  • In this way, we have a complete set of instruments \(z_2,z_3,…,z_k\) .

The assumptions on the instruments

  • \(z\) -variables are exogeneous,

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

  • \(z_{i,j}\) is correlated with \(x_{i,j}\) for every \(j\)

\[Cov\left( x_{i,j},z_{i,j} \right)≠0\]

  • \(z_i\) will not have a direct effect on \(y_i\) , \(E\left( z_i,x_i \right)\) does not depend on \(z_i\) .

The IV-estimator and its properties

  • The IV-estimator is generally biased.
  • The bias is generally worse the weaker the instruments are.
  • The IV-estimator is consistent under mild conditions.
  • It is possible to find consistent estimates of the standard errors.
  • Inference based on IV-estimator will be approximately correct in large samples. Generally, the weaker the instruments, the larger sample size you need for reasonable results.