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OVB More Formally  I
• Now let’s consider OVB more formally.
• Specifically, lets consider the effects of OVB on our
estimator
β
1
.
Recall our equation for
β
1
:
()
11
1
22
1
ˆ
1
,
nn
ii
i
XXYY
n
XX
n
Cov X Y
Var X
β
==
−−
≈
∑∑
OVB More Formally  II
(
)
(
)
()(
)()
)
(
)
01
1
1
1
,,
ˆ
,
=
0,
,
=
=
,
=
,
i
i
i
i
i
i
i
i
i
i
i
i
i
Cov X Y
Cov X
X
u
Var X
Var X
C
o
vX
C
o
X
C
o
vXu
Var X
Cov X X
Cov X u
Var X
Var X
Cov X u
Var X
Cov X u
SD u
Corr X u
Var X
SD X
ββ
++
≈=
+=
+
OVB More Formally  III
• So we have:
• What does this equation say?
• 1) If
Cov
(
X
i
,
u
i
) = 0,
β
1
≈
β
1
, i.e.
β
1
is an unbiased and
consistent estimator of
β
1
. (i.e. no OVB)
• 2) If
Cov
(
X
i
,
u
i
) > 0,
β
1
will generally be greater than
β
1
, i.e.
β
1
is biased upwards (positive bias).
• 3) If
Cov
(
X
i
,
u
i
) < 0,
β
1
will generally be less than
β
1
,
i.e.
β
1
is biased downwards (negative bias).
( )
,
ˆ
i
Cov X u
Var X
≈+
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View Full DocumentOVB More Formally  IV
• So the direction of OVB bias depends directly
on the sign of
Cov
(
X
i
,
u
i
).
• Often we can come up with a good guess of
whether
Cov
(
X
i
,
u
i
) is positive or negative.
This can be helpful, as then we know whether
our estimate
β
1
is biased positively or
negatively.
• So let’s think about this more.
What
determines the sign of
Cov
(
X
i
,
u
i
)?
OVB More Formally  V
• To fix ideas, consider a specific omitted variable
V
i
.
Since it is an omitted variable,
V
i
is part of
u
i
.
Suppose this relationship is:
• Note that in this equation
,
ω
i
represents the
omitted variables
other than V
i
.
Let’s assume
that
Cov
(
X
i
,
ω
i
) = 0, i.e. the other omitted
variables are not correlated with
X
i
.
ii
i
uV
δ
ω
=+
OVB More Formally  VI
• From the above equation, the sign of
δ
measures whether
the omitted variable
V
i
positively or negatively affects
u
i
.
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 Spring '07
 SandraBlack
 Econometrics

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