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

Exercise 14

Consider the regression model, where for simplicity the intercept is omitted and all variables are assumed to have a mean of zero. Suppose that Xi is distributed independently of ( w i, u i) but Wi, and ui, might be correlated and let and be the OLS estimators for this model. Show that
a. Whether or not wi, and ui are correlated,
b. If Wi and u i are correlated, then is inconsistent.
c. Let be the OLS estimator from the regression of Y on X (the restricted regression that excludes IT). Provide conditions under which has a smaller asymptotic variance than , allowing for the possibility thatWi, and u i are correlated.