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

Exercise 19

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.