Simultaneous Equation Model

Simultaneous Equation Models use Two-Stage Lease Squares Regression to analyze circular causality.

• The two dependent variables with circular causality is $Y_1,Y_2$
• $Y_1,Y_2$ are caused respectively by $Z_1,Z_2$
• $W$ cause both $Y_1,Y_2$. The equation is thus:

$$\begin{array}{rl}{Y}_{1i}& ={\beta }_0+{\beta }_1{Y}_{2i}+{\beta }_2{W}_i+{\beta }_3{Z}_1+{ϵ}_{1i}\\ Y_{2i}& ={\gamma }_0+{\gamma }_1{Y}_{1i}+{\gamma }_2{W}_i+{\gamma }_3{Z}_1+{ϵ}_{1i}\end{array}$$

The process for regression is same as the Two-Stage Lease Squares Regression:

1. For the first equasion $Y_{1i}$:
   1. First stage: regress $Y_2$without $Y_1$:

$$Y_{2i}={\pi }_0+{\pi }_1{W}_i+{\pi }_2{Z}_1+{\nu }_i$$

2. Second stage: regress $Y_{1}$ againt $\hat{Y_{2}}$ (the original equation):

$$Y_{1i}={\beta }_0+{\beta }_1\widehat{Y_{2i}}+{\beta }_2{W}_i+{\beta }_3{Z}_1+{ϵ}_{1i}$$

1. Perform the same 2SLS regression for the second equation as well