Simultaneous equations

Summary

If you have two dependent variables \(y_1\) and \(y_2\) and each can be modelled using a linear regression model, we say that we have simultaneous equations .
The LRM for \(y_1\) may contain \(y_2\) as well as additional explanatory variables and vice versa.

Example

\[y_{i,1}=β_1+β_2y_{i,2}+β_3x_{i,2}+β_4x_{i,3}+ε_{i,1} i=1,…,n\]

\[y_{i,2}=γ_1+γ_2y_{i,1}+γ_3x_{i,2}+γ_4x_{i,4}+ε_{i,2} i=1,…,n\]

These equations are also called structural equations as they are meant to describe the actual structure of the model.
If

\[E\left( ε_{i,1}|x_i \right)=E\left( ε_{i,2}|x_i \right)=0\]

then \(x_{i,2},x_{i,3},x_{i,4}\) are called exogenous variables (determined outside the system). \(y_{i,1}\) , \(y_{i,2}\) are called endogenous variables (determined within the system).
If we solve the structural equations for \(y_{i,1}\) , \(y_{i,2}\) then we get what is called the reduced form equations . These will, in general, demonstrate that

\[E\left( ε_{i,1}|y_{i,2} \right)≠0, E\left( ε_{i,2}|y_{i,1} \right)≠0\]

Thus, \(y_2\) is not exogenous in the first equation and \(y_1\) is not exogenous in the second equation.

Main point

The structural equations in a simultaneous equations system cannot, in general, be estimated individually with OLS as they contain endogenous variables.
The bias introduced from estimating a single structural equation is called the simultaneous equations bias .
Whenever in a single LRM, you can argue that not only does an \(x\) -variable partly explain \(y\) but it is also the case that the \(x\) -variable is partly explained by \(y\) , then you actually have simultaneous equations. Your \(x\) -variable will not be exogenous and you model will suffer from simultaneous equations bias.