9.13 Assume that the regression model Yi = b0 + b1Xi + ui satisfies the least squares assumptions in Key Concept 4.3. You and a friend collect a random sample of 300 observations on Y and X.

a. Your friend reports that he inadvertently scrambled the X observations for 20% of the sample. For these scrambled observations, the value of X does not correspond to Xi for the ith observation; rather, it corresponds to the value of X for some other observation. In the notation of Section 9.2, the measured value of the regressor, X

i, is equal to Xi for 80% of the observations, but it is equal to a randomly selected Xj for the remaining 20% of the observations. You regress Yi on X

i. Show that E1b n

12 = 0.8b1.

b. Explain how you could construct an unbiased estimate of b1 using the OLS estimator in (a).

c. Suppose now your friend tells you that the X’s were scrambled for the first 60 observations but that the remaining 240 observations are correct.

You estimate b1 by regressing Y on X, using only the correctly measured 240 observations. Is this estimator of b1 better than the estimator you proposed in (b)? Explain.