Economics 241B
Finite Sample Properties of OLS Estimators
We deal in turn with the estimator
B
and the estimator
S
2
.
Linear Estimators
From
B
= (
X
0
X
)
1
X
0
Y
we see that
B
is a linear estimator.
Unbiased Estimators
To verify that the OLS estimators are unbiased, note
B
= (
X
0
X
)
1
X
0
U:
Thus
E
(
B
j
X
) = (
X
0
X
)
1
X
0
E
(
U
j
X
) = 0
:
Because
E
(
B
j
X
) = 0
whenever
E
(
B
j
X
) =
we have shown that
B
is conditionally unbiased.
Further, by the law of total
expectations
E
[
E
(
B
j
X
)] =
so that the estimator is (unconditionally) unbiased.
Variance of the OLS Estimators of
Let
A
= (
X
0
X
)
1
X
0
, so
AA
0
= (
X
0
X
)
1
. Because
is not random
V
(
B
j
X
) =
V
(
B
j
X
)
=
V
(
AU
j
X
) =
AV
(
U
j
X
)
A
0
:
Yet
V
(
U
j
X
) =
E
(
UU
0
j
X
) =
±
2
I
n
, so
V
(
B
j
X
) =
±
2
(
X
0
X
)
1
:
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For the classic regression model, the OLS estimator is the best linear unbiased
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 Fall '08
 Staff
 Economics

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