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

Exercise 13

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