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

Exercise 25

(i) Consider the simple regression model y = ?0 + ?1x + u under the first four Gauss Markov assumptions. For some function g(x), for example g(x) = x2 or g(x) = log(1 + x2), define zt = g(x1). Define a slope estimator as

Show that ?1 is linear and unbiased. Remember, because E(u|x) = 0, you can treat both x. and zt as nonrandom in your derivation.

(ii) Add the homoskedasticity assumption, MLR.5. Show that

(iii) Show directly that, under the Gauss-Markov assumptions, Var(r1) 1 is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Appendix B implies that

notice that we can drop x from the sample covariance.]

Step 1 of 3

(i)

For simplicity, define; this is not quite the sample covariance between z and x because it is not divided by n – 1, but will only be used to simplify the notation. Then write as:

This is clearly a linear function of the y_i. Take the weights to be. To show unbiasedness, as usual plug into this equation, and simplify:

Use the fact that always. Now is a function of the z_i and x_i and the expected value of each u_i is zero conditional on all z_i and x_i in the sample. Therefore, conditional on these values:

Because E(u_i) = 0 for all i.