ECON 321
2013
A.J. Rogers

Assignment 2
Due October 17, 4:00pm

1. Consider the model y = w( + z( + u, i.e., y = X( + u with X = [w : z],
(‘ = (( : (). Here w and z are non-stochastic n ( 1 vectors so X is n ( 2. y is n ( 1, as is u. Assume that E[u] = 0, E [uu’] = (2In. The OLS estimator of ( is
b = (X’X)-1X’y, with b’ = (a : d), so a, d are the OLS estimators of a, d respectively.

(a) Someone believes that a + d = 0, and notes that IF this is true we can write the model as
y = wa – za + u

and we could use this to get an estimator, a*, of ? by a single simple OLS regression. Write down the formula for a*.

(b) Find the expectation and variance of the estimator, a*, of ? you obtained in part (a) but for the case where the restriction ? + ? = 0 is not necessarily true. [You may find it convenient to rewrite y in the formula for a* as y = (w – z)? + z(? + ?) + u.]

(c) Find E[a*], E[a] and var[a*], var[a], assuming that w’w = 1, z’z = 1, w’z = r, with
-1 0, 0