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.