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?

  1. \(E\left( ε_i \mid x_i \right)=0\) for all \(i=1, \ldots ,n\)
  2. \(E\left( ε_i \right)=0\) for all \(i=1, \ldots ,n\)
  3. 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.
  4. If the \(x\) -variables are independent of the error terms and \(E\left( ε_i \right)=0\) , then the \(x\) -variables must be exogenous.
  5. \(E\left( y_i \mid x_i \right)=β_1+β_2x_{i,2}+β_3x_{i,3}+ \ldots +β_kx_{i,k}\) will hold even if the \(x\) -variables are not exogenous.

Solution

  1. 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.
  2. Yes, this follows from the law of iterated expectations. \(E\left( ε_i \right)=E\left( E\left( ε_i \mid x_i \right) \right)=E\left( 0 \right)=0\) .
  3. 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 \(x_i\) .
  4. This is true. If the \(x\) -variables are independent of the error terms, then it must follow that \(E\left( ε_i \mid x_i \right)=E\left( ε_i \right)\) .
  5. 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 \(E\left( y_i \mid x_i \right)=β_1+β_2x_{i,2}+β_3x_{i,3}+ \ldots +β_kx_{i,k}\) .