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17.5 Weighted Least Squares 743
Exercises
17.1 Consider the regression model without an intercept term, Yi = b1Xi + ui
(so the true value of the intercept, b0, is zero).
a. Derive the least squares estimator of b1 for the restricted regression
model Yi = b1Xi + ui. This is called the restricted least squares esti-
mator (bn1RLS) of b1 because it is estimated under a restriction, which in
this case is b0 = 0.
b. Derive the asymptotic distribution of bnR1 LS under Assumptions #1
through #3 of Key Concept 17.1.
c. Show that bnR1 LS is linear [Equation (5.24)] and, under Assumptions #1
and #2 of Key Concept 17.1, conditionally unbiased [Equation (5.25)].
d. Derive the conditional variance of bn1RLS under the Gauss–Markov
conditions (Assumptions #1 through #4 of Key Concept 17.1).
e. Compare the conditional variance of bnR1 LS in (d) to the conditional
variance of the OLS estimator bn1 (from the regression including an
intercept) under the Gauss–Markov conditions. Which estimator is
more efficient? Use the formulas for the variances to explain why.
f. Derive the exact sampling distribution of bnR1 LS under Assumptions #1
through #5 of Key Concept 17.1.
g. Now consider the estimator b1 = g n 1Yi > g n 1Xi. Derive an
i= i=
expression for var(b1 ͉ X1, c, Xn) - var(bn1RLS 0 X1, c, Xn) under
the Gauss–Markov conditions and use this expression to show that
var( b1 0 X1, c, Xn) Ú var(bnR1 LS 0 X1, c, Xn).
17.2 Suppose that (Xi,Yi) are i.i.d. with finite fourth moments. Prove that the
sample covariance is a consistent estimator of the population covariance—
that is, sXY ¡p sXY, where sXY is defined in Equation (3.24). (Hint: Use
the strategy outlined in Appendix 3.3 and the Cauchy–Schwarz inequality.)
17.3. This exercise fills in the details of the derivation of the asymptotic distribu-
tion of bn1 given in Appendix 4.3.
a. Use Equation (17.19) to derive the expression
A 1 n vi (X - mX) A 1 n
n - n
2n(bn1 - b1) = ia= 1 - 1 n ia= 1ui,
n n
1 - X )2 a (Xi X )2
n a (Xi
i=1
i=1
where vi = (Xi - mX)ui.
