Take the linear model with restrictions Y X0e with E[Xe] 0 and R0 c. Consider three estimators for : b the

Take the linear model with restrictions Y Æ X0¯Åe with E[Xe] Æ 0 and R0¯ Æ

c. Consider three estimators for ¯:

• b¯ the unconstrained least squares estimator

• e¯ the constrained least squares estimator

• ¯ the constrained efficient minimum distance estimator For the three estimator define the residuals bei Æ Yi ¡ X0 i

b¯, eei Æ Yi ¡ X0 i

e¯, ei Æ Yi ¡ X0 i¯, and variance estimators b¾

2 Æ n¡1Pni

Æ1 be2 i , e¾

2 Æ n¡1Pni

Æ1 ee2 i , and ¾2 Æ n¡1Pni

Æ1 e2i

.

(a) As ¯ is the most efficient estimator and b¯ the least, do you expect ¾2 Çe¾

2 Çb¾

2 in large samples?

(b) Consider the statistic Tn Æb¾

¡2 Xn iÆ1

(bei ¡ eei )2 .

Find the asymptotic distribution for Tn when R0¯ Æ c is true.

(c) Does the result of the previous question simplify when the error ei is homoskedastic?