Introduction to Econometrics 3rd Edition by James Stock, James Stock

Exercise 17

Consider the one-variable regression model Y i = 0 + 1 X i + u i and suppose that it satisfies the least squares assumptions in Key Concept 4.3. The regressor X i is missing, but data on a related variable Z i are available, and the value of X i is estimated using
a. Show that is the minimum mean square error estimator of X i using Z i. That is, let be some other guess of X i based on Z i and show that
b. Show that .
c. Suppose that E ( u|Z i ) = 0 and that , is used as the regressor in place of X i. Show that is consistent. Is consistent