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APPENDIX E
SOLUTIONS TO PROBLEMS
E.1 This follows directly from partitioned matrix multiplication in Appendix D. Write
1
x
1
y
As shown in Section E.4, this expression is the basis for the asymptotic analysis of OLS using
matrices.
E.2 (i) Following the hint, we have SSR(b) = (y – Xb)(y – Xb) = [
ˆ
u
+ X(
ˆ
β
– b)][
ˆ
u
+ X(
ˆ
β
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
u
ˆ
β
ˆ
β
ˆ
β
–
(ii) By definition of the fitted values,
ˆt
y
=
ˆ
t
xβ
and
t
y
=
t
zβ
. Plugging zt and
β
y
into the
272
regressions are the same, which means the residuals must be the same for all t. (The dependent
(iv) The
j
β
are obtained from a regression of y on XA, where A is the k k diagonal matrix
(v) From part (iii), the estimated variance matrix of
β
is
2
ˆ
A-1(XX)-1(A-1). But A-1 is a
(vi) The t statistic for
j
is, as usual,
E.4 (i)
ˆ ˆ ˆ
E( | ) E( | ) E( | ) .= = = =δ X Gβ X G β X Gβ δ
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(iv) It is easily seen by matrix multiplication that choosing
(v) Straightforward matrix multiplication shows that, for the suggested choice of G-1,
E.5 (i) By plugging in for y, we can write
(ii) We start from the same representation in part (i):
1
()
−
=+β β Z X Z u
and so
(iii) The estimator
β
is linear in y and, as shown in part (i), it is unbiased (conditional on X).