Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 3

Let denote the sample average from a random sample with mean ? and variance ?2. Consider two alternative estimators of ?: W1 = [(n-1)/n] and W2= /2.

(i) Show that W1 and W2 are both biased estimators of ? and find the biases. What happens to the biases as n ??? Comment on any important differences in bias for the two estimators as the sample size gets large.

(ii) Find the probability limits of W1 and W2. {Hint: Use Properties PLIM.1 and PLIM.2; for W1, note that plim [(n-1)/n] = 1.} Which estimator is consistent?

(iii) Find Var(W1) and Var(W2).

(iv) Argue that W1 is a better estimator than if ? is gcloseh to zero. (Consider both bias and variance.)

Step 1 of 5

Has a random sample mean µ and variance s². The two estimators are as follows: