Theorem:
Under the conditions of the previous theorem, and where
var(
e
) = var(
y
) =
σ
2
I
, the variance of
λ
T
ˆ
β
is unique, and is given by
var(
λ
T
ˆ
β
) =
σ
2
λ
T
(
X
T
X
)

λ
,
where (
X
T
X
)

is any generalized inverse of
X
T
X
.
Proof:
var(
λ
T
ˆ
β
) =
λ
T
var((
X
T
X
)

X
T
y
)
λ
=
λ
T
(
X
T
X
)

X
T
σ
2
IX
{
(
X
T
X
)

}
T
λ
=
σ
2
λ
T
(
X
T
X
)

X
T
X

{z
}
=
λ
T
{
(
X
T
X
)

}
T
λ
=
σ
2
λ
T
{
(
X
T
X
)

}
T
λ
=
σ
2
a
T
X
{
(
X
T
X
)

}
T
X
T
a
(for some
a
)
=
σ
2
a
T
X
(
X
T
X
)

X
T
a
=
σ
2
λ
T
(
X
T
X
)

λ
.
Uniqueness: since
λ
T
β
is estimable
λ
=
X
T
a
for some
a
. Therefore,
var(
λ
T
ˆ
β
) =
σ
2
λ
T
(
X
T
X
)

λ
=
σ
2
a
T
X
(
X
T
X
)

X
T
a
=
σ
2
a
T
P
C
(
X
)
a
Again, the result follows from the fact that projection matrices are unique.
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 Summer '10
 HALL
 Variance, Mathematical terminology, Xt

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