Endogenous variables

Summary

Typo at 0.44 in lecture: the final \(β\) should be \(β_4\)

Example 1

• True model:

\[E\left( x_{i,2},x_{i,3},x_{i,4} \right)=β_1+β_2x_{i,2}+β_3x_{i,3}+β_4x_{i,4} , i=1,…,n\]

• We assume that

\[E\left( x_{i,2},x_{i,3},x_{i,4} \right)=β_1+β_2x_{i,2}+β_3x_{i,3}, i=1,…,n\]

• and specify

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+ε_i, i=1,…,n\]

• If an included variable ( \(x_2\) or \(x_3)\) is correlated with the missing variable ( \(x_4\) ), then it is endogenous
• If an included variable is independent of the missing variable, then it is exogenous

Example 2

• True model: \(y_{i,1},y_{i,2}\) are determined jointly from:

\[y_{i,1}= β_1+β_2x_{i,2}+β_3y_{i,2}+ε_i, i=1,…,n\]

\[y_{i,2}= γ_1+γ_2x_{i,2}+γ_3y_{i,1}+ν_i, i=1,…,n\]

• Then

\[E\left( x_{i,2},y_{i,2} \right)=β_1+β_2x_{i,2}+β_3y_{i,2}, i=1,…,n\]

• cannot hold in general.
• If we consider the first equation on its own, then \(y_{i,2}\) is endogenous.
• \(x_{i,2}\) will be exogenous if \(E\left( ε_i|x_i \right)=E\left( ν_i|x_i \right)=0\)

Example 3

• True model:

\[E\left( z_i,x_i \right)=β_1+β_2z_{i,2}+β_3z_{i,3}\]

• where \(x_{i,2}\) is a measurement of true \(z_{i,2}\) and \(x_{i,3}\) is a measurement of true \(z_{i,3}\) .
• We run \(y\) on \(x_2\) and \(x_3\) .
• If a variable has measurement errors, it will be endogenous.
• If a variable does not have measurement errors, it will be exogenous.