Unbiasedness and consistency in the LRM

Problem

Given a LRM \(y_i=β_1+β_2x_i+ε_i\) . \(b_2\) is the OLS estimator of \(β_2\) . What is true about unbiasedness and consistency?

  1. The OLS estimator is always unbiased and consistent
  2. If the explanatory variable is exogenous then the OLS estimator will certainly be unbiased and consistent
  3. An estimator is consistent if and only if it is unbiased
  4. If the explanatory variable is exogenous then \(b_2=β_2\)
  5. If the explanatory variable is exogenous then \(E\left( b_2 \right)=β_2\)

Solution

  1. No, it is not. The requirement is exogeneity, \(E\left( ε_i|x_i \right)=0\) (or, equivalently, \(E\left( y_i \mid x_i \right)=β_1+β_2x_i\) )
  2. Yes, certainly
  3. No, unbiasedness and consistency are two different nice properties that an estimator may have. Unbiasedness mean that the estimator is “correct on average”, \(E\left( b_2 \right)=β_2\) . Consistency means that \(b_2\) will become equal to \(β_2\) as the sample size goes to infinity. An estimator may be unbiased but not consistent and consistent but not unbiased.
  4. No, at least not with a finite sample size. We will never get the exact correct value for \(β_2\) . However, we have in the next part:
  5. Yes. This is true. Event though surely \(b_2≠β_2\) , \(E\left( b_2 \right)\) is equal to \(β_2\) .