Simultaneity

Summary

• Setup: random sample \(\left( y_i,x_i \right)\) for \(i=1, \ldots ,n\) where \(x_i\) is a scalar
• We believe that:

\[E\left( y_i \mid x_i \right)=β_1+β_2x_i\]

• such that \(x_i\) is exogenous in the model

\[y_i=β_1+β_2x_i+ε_i\]

• However, the reality is that

\[x_i=y_i+z_i\]

• where \(z_i\) is an exogenous variable, \(E\left( ε_i \mid z_i \right)=0\) .
• Solving for \(x_i\) :

\[x_i= \frac{β_1}{1-β_2}+ \frac{z_i}{1-β_2}+ \frac{ε_i}{1-β_2}\]

• which demonstrates that

\[cov\left( x_i,ε_i \right)= \frac{Var\left( ε_i \right)}{1-β_2}\]

• Thus, \(x_i\) must be endogenous and the OLS estimator is biased and inconsistent.
• Result,

\[plim b_2=β_2+\left( 1-β_2 \right) \frac{σ_ε^2}{σ_z^2+σ_ε^2}\]

• where \(σ_ε^2=Var\left( ε_i \right)\) (unconditional variance) and \(σ_z^2=Var\left( z_i \right)\) . The second term is called the simultaneity bias .