2-Stage Least Square

metrics

1. Regress endogeous regressor $X$ on $Z$, get $\hat{X}$

$$X={\pi }_0+{\pi }_{1}Z+V$$

And then we get a proper $\hat{X}$

2. Regress $Y$ on $\hat{X}$

$$Y_i={\beta }_0+{\beta }_1{X}_i+u_i$$ $$={\beta }_0+{\beta }_1({\hat{X}}_i+\hat{V})+u_i$$ $$={\beta }_0+{\beta }_1{\hat{X}}_i+{\beta }_1\hat{V}+u_i$$

For ${\hat{\beta }}_1=\frac{S_{\hat{X}Y}}{S_{\hat{X}}^2}$

=\frac{S_{\hat X, \beta_0+\beta_1 \hat X_i +\beta_1 \hat V + u_i}}{S_\hat X} =\frac{\beta_1 S^2_\hat X}{S^2_\hat X}+\frac{ S_{\hat X,u}}{S^2_\hat X}+\frac{ S_{\hat X,\beta_1,\hat V}}{S^2_\hat X}

Conditions

When we use 2SLS to estimate, we have to verify 2 conditions:

1. Relevance Condition: $Cov(Z,X)\ne 0$
2. Validity Condition: $Cov(Z,u)=0$

where $Z$ is the Instrumental Variable