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

Exercise 5

Assume that the model y = X? + u satisfies the Gauss-Markov assumptions and let be the OLS estimator of ?. Let Z = G(X) be an n × (k + 1) matrix function of × and assume that Z?X [a (k + 1) × (k + 1) matrix] is nonsingular. Define a new estimator of ? by = (Z?X)1Z?y.

(i) Show that E( X) , so that is also unbiased conditional on X.

(ii) Find Var( X). Make sure this is a symmetric, (k + 1) × (k + 1) matrix that depends on Z, X, and ?2.

(iii) Which estimator do you prefer, or ? Explain

Step 1 of 3

Given:

1) satisfies the Gauss-Markov assumptions

2) is the OLS estimator of

3) is the matrix function of

4) is nonsingular matrix

5)

(i)

Since,

This implies, satisfies the Gauss-Markov assumptions

Since, due to Gauss-Markov assumption