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?
- The OLS estimator is always unbiased and consistent
- If the explanatory variable is exogenous then the OLS estimator will certainly be unbiased and consistent
- An estimator is consistent if and only if it is unbiased
- If the explanatory variable is exogenous then \(b_2=β_2\)
- If the explanatory variable is exogenous then \(E\left( b_2 \right)=β_2\)
Solution
- No, it is not. The requirement is exogeneity, \(E\left( ε_i|x_i \right)=0\) (or, equivalently, \(E\left( y_i \mid x_i \right)=β_1+β_2x_i\) )
- Yes, certainly
- 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.
- 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:
- Yes. This is true. Event though surely \(b_2≠β_2\) , \(E\left( b_2 \right)\) is equal to \(β_2\) .