9.12 Consider the one-variable regression model Yi = b0 + b1Xi + ui and suppose that it satisfies the least squares assumptions in Key Concept 4.3. The regressor Xi is missing, but data on a related variable, Zi, are available, and the value of Xi is estimated usingX

i = E(Xi 0Zi). Let wi = X

i - Xi.

a. Show that X

i is the minimum mean square error estimator of Xi using Zi. That is, let X n

i = g(Zi) be some other guess of Xi based on Zi, and show that E3(X n

i - Xi)24 Ú E3(X

i - Xi)24. (Hint: Review Exercise 2.27.)

b. Show that E(wi 0 X

i) = 0.

c. Suppose that E(ui 0Zi) = 0 and that X

i is used as the regressor in place of Xi. Show that b n

1 is consistent. Is b n

0 consistent?