Question

I need the second bonus question!

BONUS - Prove algebraically that â1 is a consistent estimator of the true effect ofX1 on Y from the population regression model, Y = 10 + 5X1 + U. (Hint: Recall that the Law of Large Numbers can be applied to sample averages to conclude that these converge in probability to their corresponding population counterparts provided that the underlying data are i.i.d. with finite variance.) BONUS- How do you explain that a, can be a consistent, yet biased, estimator in this xample?

Answer 1

The consistent term for sample estimator is used when the variance of the estimator is constant. Var(a1) =0 in that case the value of the estimator will be consistent, efficient. However, if the expected value of a1 is not equal to the value of true population parameter then it will be biased. Therefore, in simple terms a situation when the variance of an estimator is constant but the mean of the sample term is not equal to population parameter will be known as consistent but biased.