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Economics 326
Methods of Empirical Research in Economics
Lecture 12: Properties of OLS in the multiple
regression model
Vadim Marmer
University of British Columbia
March 3, 2009
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View Full Document Multiple regression and OLS
I
Consider the multiple regression model with
k
regressors:
Y
i
=
β
0
+
β
1
X
1
,
i
+
β
2
X
2
,
i
+
. . .
+
β
k
X
k
,
i
+
U
i
.
I
Let
ˆ
β
0
,
ˆ
β
1
,
. . .
,
ˆ
β
k
be the OLS estimators: if
ˆ
U
i
=
Y
i
ˆ
β
0
ˆ
β
1
X
1
,
i
ˆ
β
2
X
2
,
i
. . .
ˆ
β
k
X
k
,
i
,
then
n
∑
i
=
1
ˆ
U
i
=
n
∑
i
=
1
X
1
,
i
ˆ
U
i
=
. . .
=
n
∑
i
=
1
X
k
,
i
ˆ
U
i
=
0
.
1/16
Multiple regression and OLS
I
As in Lecture 10, we can write
ˆ
β
1
as
ˆ
β
1
=
∑
n
i
=
1
˜
X
1
,
i
Y
i
∑
n
i
=
1
˜
X
2
1
,
i
,
where
I
˜
X
1
,
i
˜
X
1
,
i
=
X
1
,
i
ˆ
γ
0
ˆ
γ
2
X
2
,
i
. . .
ˆ
γ
k
X
k
,
i
.
I
ˆ
γ
0
,
ˆ
γ
2
,
. . .
,
ˆ
γ
k
are the OLS coe¢ cients:
∑
n
i
=
1
˜
X
1
,
i
=
∑
n
i
=
1
˜
X
1
,
i
X
2
,
i
=
. . .
=
∑
n
i
=
1
˜
X
1
,
i
X
k
,
i
=
0
.
I
Similarly, we can write
ˆ
β
2
as
ˆ
β
2
=
∑
n
i
=
1
˜
X
2
,
i
Y
i
∑
n
i
=
1
˜
X
2
2
,
i
,
where
I
˜
X
2
,
i
˜
X
2
,
i
=
X
2
,
i
ˆ
δ
0
ˆ
δ
1
X
1
,
i
ˆ
δ
3
X
3
,
i
. . .
ˆ
δ
k
X
k
,
i
.
I
ˆ
δ
0
,
ˆ
δ
1
,
ˆ
δ
3
,
. . .
,
ˆ
δ
k
are the OLS coe¢ cients:
∑
n
i
=
1
˜
X
2
,
i
=
∑
n
i
=
1
˜
X
2
,
i
X
1
,
i
=
∑
n
i
=
1
˜
X
2
,
i
X
3
,
i
=
. . .
=
∑
n
i
=
1
˜
X
2
,
i
X
k
,
i
=
0.
2/16
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View Full Document The OLS estimators are linear
I
Consider
ˆ
β
1
:
ˆ
β
1
=
∑
n
i
=
1
˜
X
1
,
i
Y
i
∑
n
i
=
1
˜
X
2
1
,
i
=
n
∑
i
=
1
˜
X
1
,
i
∑
n
l
=
1
˜
X
2
1
,
l
Y
i
=
n
∑
i
=
1
w
1
,
i
Y
i
,
where
w
1
,
i
=
˜
X
1
,
i
∑
n
l
=
1
˜
X
2
1
,
l
.
I
Recall that
˜
X
1
are the residuals from a regression of
X
1
against
X
2
,
. . .
,
X
k
and a constant, and therefore
w
1
,
i
depends
only on
X
3/16
Unbiasedness
I
Suppose that
1.
Y
i
=
β
0
+
β
1
X
1
,
i
+
β
2
X
2
,
i
+
. . .
+
β
k
X
k
,
i
+
U
i
.
2. Conditional on
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This note was uploaded on 03/24/2012 for the course ECON 326 taught by Professor Whisler during the Spring '10 term at The University of British Columbia.
 Spring '10
 whisler
 Economics

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