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Specification Error: Omitted and Extraneous Variables—Page 1
Specification Error: Omitted and Extraneous Variables
Omitted variable bias.
Suppose that the “correct”
model is
y
X
X
1
1
2
2
If we estimate
y
a
b X
b X
e
1
1
2
2
we know that E(b
1
) =
1
and E(b
2
) =
2
i.e. the regression coefficients are unbiased estimators of
the population parameters.
Suppose, however, the researcher mistakenly believes
y
X
*
*
*
1
1
and therefore estimates
y
a
b X
e
*
*
*
1
1
i.e. X2 is mistakenly omitted from the model.
How does b
1
(the regression estimate from the
correctly specified model) compare to b
1
* (the regression estimate from the misspecified
model)?
What is E(b
1
*)?
Is it a biased or unbiased estimator of
1
?
If biased, how is it biased?
Note that b
1
*
(
,
)
(
)
Cov X
Y
V X
1
1
Formula for bivariate regression
coefficient
(
,
)
(
)
Cov X
a
b X
b X
e
V X
1
1
1
2
2
1
Substitute the formula for Y
from the correctly specified
model
(
,
)
(
,
)
(
,
)
(
, )
(
)
Cov X
a
b Cov X
X
b Cov X
X
Cov X
e
V X
1
1
1
1
2
1
2
1
1
Expectations rules:
Cov(a+b,c+d) = Cov(a,c) +
Cov(a,d) + Cov(b,c) + Cov(b,d)
0
0
1
1
2
1
2
1
bV X
b Cov X
X
V X
(
)
(
,
)
(
)
Recall that Cov(variable,
constant) = 0.
Also, X’s are
uncorrelated with the residuals.
b
b
Cov X
X
V X
1
2
1
2
1
(
,
)
(
)
Simplify expression.
Taking expectations, we get
E b
(
*)
1
1
2
12
1
2
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View Full DocumentSpecification Error: Omitted and Extraneous Variables—Page 2
Hence,
b
1
* is a biased estimator of
1
.
Further, this bias will not disappear as sample size gets
larger, so the omission of a variable from a model also leads to an inconsistent estimator.
Note that there are two conditions under which b
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 Spring '11
 RichardWilliams

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