9.12 Consider the one-variable regression model Yi = b0 + b1Xi + ui, and suppose 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 = E1Xi Zi2. 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 = g1Zi2 be some other guess of Xi based on Zi, and show that E31X n

i - Xi2 24 Ú E31X

i - Xi2 24.(Hint: Review Exercise 2.27.)

b. Show that E1wi X

i2 = 0.

c. Suppose that E1ui Zi2 = 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?