Instrumental Variable

Definition

Assume a regression model $y=X\beta +ϵ$ such that $\mathrm{Cov}(X,ϵ)\ne 0$ An instrument variable for the model is a $Z$ such that relevance $\mathrm{Cov}(Z,X)\ne 0$ and validity $\mathrm{Cov}(Z,u)=0$

2 Stage Least Square

If $\mathrm{Cov}(X,ϵ)\ne 0$, the simple estimator $\hat{\beta }$ is not consistent. Therefore, calculate consistent IV estimator using the two stage least square. ${\hat{\beta }}_{IV}=\frac{\widehat{Cov(Z,y)}}{\widehat{\mathrm{Cov}(Z,X)}}\stackrel{P}{\to }\beta$