Exogeneity in the linear regression model
Problem
Setup: a linear regression model with a random sample
\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+ \ldots +β_kx_{i,k}+ε_i\]
What is true about exogeneity?
- \(E\left( ε_i \mid x_i \right)=0\) for all \(i=1, \ldots ,n\)
- \(E\left( ε_i \right)=0\) for all \(i=1, \ldots ,n\)
- If any of the \(x\) -variables are correlated with the error term, \(cov\left( x_{i,j},ε_i \right)≠0\) for some \(j\) , then the \(x\) -variables cannot be exogenous.
- If the \(x\) -variables are independent of the error terms and \(E\left( ε_i \right)=0\) , then the \(x\) -variables must be exogenous.
- \(E\left( y_i \mid x_i \right)=β_1+β_2x_{i,2}+β_3x_{i,3}+ \ldots +β_kx_{i,k}\) will hold even if the \(x\) -variables are not exogenous.
Solution
- Yes, this is true. This is the definition of exogeneity. The error term must have an expected value of zero no matter which x-value we condition on.
- Yes, this follows from the law of iterated expectations. \(E\left( ε_i \right)=E\left( E\left( ε_i \mid x_i \right) \right)=E\left( 0 \right)=0\) .
- This is true. If \(cov\left( x_{i,j},ε_i \right)≠0\) then \(E\left( ε_i \mid x_i \right)\) cannot be zero since \(E\left( ε_i \mid x_i \right)=0 \implies cov\left( x_{i,j},ε_i \right)=0\) . Basically, if \(cov\left( x_{i,j},ε_i \right)>0\) then \(E\left( ε_i \mid x_i \right)\) will tend to depend positively on \(x_i\) .
- This is true. If the \(x\) -variables are independent of the error terms, then it must follow that \(E\left( ε_i \mid x_i \right)=E\left( ε_i \right)\) .
- This is false. In this case, \(E\left( ε_i \mid x_i \right)\) is not zero. Assuming exogeneity in the LRM is the same as assuming that \(E\left( y_i \mid x_i \right)=β_1+β_2x_{i,2}+β_3x_{i,3}+ \ldots +β_kx_{i,k}\) .