- Exercise 1 in Wooldridge Chapter 14

Answer:

For each , , where we use the assumptions of no serial correlation in and constant variance. Next, we find the covariance between and . Because these each have a zero mean, the covariance is because of the no serial correlation assumption. Because the variance is constant across , .

- Exercise 2 in Wooldridge Chapter 14

Answer:

(i)

The between estimator is the OLS estimator from the cross-sectional regression of on (including an intercept). Because we just have a single explanatory variable and the error term is , we have

since , because by assumption. Then . Therefore,

(ii)

If is serially uncorrelated with constant variance the , and so .

(iii)

As part (ii) shows, when the are pairwise uncorrelated the magnitude of the inconsistency actually increases linearly with . The sign depends on the covariance between and .

- Exercise 3 in Wooldridge Chapter 14

Answer:

(i)

because for all .

(ii)

. Now, and . Therefore, . Thus, we can collect terms:

denote , where and . Then

(iii)

We need to show that for . Now . The rest of the proof is similar to (ii)

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