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1 Endogeneity and Causality
Wooldridge Chapters 1516 & Berndt Chapter 8
1.1 Endogeneity due to Omitted Variables
suppose that
wage
is determined by
log
wage
i
=
&
0
+
&
1
educ
i
+
&
2
abil
i
+
e
i
, for
i
= 1
; :::; n
assume that the error term
e
i
is not correlated with the right hand side variables (
educ
i
and
abil
i
)
that means
e
i
can be considered as pure luck
typically the right hand side variables are correlated each other
in this model, it is likely that
educ
i
and
abil
i
are positively correlated
(more able person gets more education)
in practice, however, suppose that ability is not observed and we decide to estimate
log
wage
i
=
&
0
+
&
1
educ
i
+
u
i
, for
i
= 1
; :::; n
in this case, the error term
u
i
contains
abil
i
(
u
i
=
&
2
abil
i
+
e
i
)
therefore the error term
u
i
is likely to be correlated with
educ
i
when any of the right hand side variables are correlated with the error term,
the least squares estimator is not consistent any more
(consistency: as the sample size tends to in&nity, the estimator gets closer to the true parameter)
to see why the estimator is inconsistent
or to see whether
b
&
1
gets closer to
&
1
as
n
gets large
let±s rewrite the above models by
y
i
=
&
0
+
&
1
x
1
i
+
&
2
x
2
i
+
e
i
, for
i
= 1
; :::; n
and
y
i
=
&
0
+
&
1
x
1
i
+
u
i
, for
i
= 1
; :::; n
we know that when we estimate the latter model by the least squares method, we obtain
b
&
1
=
P
n
i
=1
(
x
1
i
&
x
1
)
y
i
P
n
i
=1
(
x
1
i
&
x
1
)
2
substitute the true model into
b
&
1
formula
b
&
1
=
P
n
i
=1
(
x
1
i
&
x
1
) (
&
0
+
&
1
x
1
i
+
&
2
x
2
i
+
e
i
)
P
n
i
=1
(
x
1
i
&
x
1
)
2
=
&
0
P
n
i
=1
(
x
1
i
&
x
1
)
P
n
i
=1
(
x
1
i
&
x
1
)
2
+
&
1
P
n
i
=1
(
x
1
i
&
x
1
)
x
1
i
P
n
i
=1
(
x
1
i
&
x
1
)
2
+
&
2
P
n
i
=1
(
x
1
i
&
x
1
)
x
2
i
P
n
i
=1
(
x
1
i
&
x
1
)
2
+
P
n
i
=1
(
x
1
i
&
x
1
)
e
i
P
n
i
=1
(
x
1
i
&
x
1
)
2
= 0 +
&
1
+
&
2
c
cov
(
x
1
i
; x
2
i
)
d
var
(
x
1
i
)
+
c
cov
(
x
1
i
; e
i
)
d
var
(
x
1
i
)
by the law of large numbers (plus some theory)
b
&
1
!
p
&
1
+
&
2
cov
(
x
1
i
; x
2
i
)
var
(
x
1
i
)
+ 0
6
=
&
1
what we &nd is that
b
&
1
is consistent only if
&
2
= 0
or
cov
(
x
1
i
; x
2
i
) = 0
1
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View Full Document in this example, we expect
&
2
>
0
(more able person earns higher wage) and
cov
(
x
1
i
; x
2
i
)
>
0
in consequence, there is an upward bias in
b
&
1
therefore, when we do not include
abil
i
in the estimation model,
we are likely to get an estimate that overstates the true e/ect of education on wage
let&s go back to the
b
&
1
=
&
1
+
&
2
c
cov
(
x
1
i
; x
2
i
)
d
var
(
x
1
i
)
+
c
cov
(
x
1
i
; e
i
)
d
var
(
x
1
i
)
we know that
c
cov
(
x
1
i
; x
2
i
)
d
var
(
x
1
i
)
is the least squares estimator of a simple regression of
x
2
i
on
x
1
i
that is if we construct a regression model
x
2
i
=
±
0
+
±
1
x
1
i
+
w
i
,
then,
b
±
1
=
c
cov
(
x
1
i
; x
2
i
)
d
var
(
x
1
i
)
the model
x
2
i
=
±
0
+
±
1
x
1
i
+
w
i
is called the auxiliary equation
and we learn that
b
&
1
!
p
&
1
+
&
2
±
1
another way of calculating the bias, consider
y
i
=
&
0
+
&
1
x
1
i
+
&
2
x
2
i
+
e
i
=
&
0
+
&
1
x
1
i
+
&
2
(
±
0
+
±
1
x
1
i
+
w
i
) +
e
i
= (
&
0
+
&
2
±
0
) + (
&
1
+
&
2
±
1
)
x
1
i
+ (
e
i
+
&
2
w
i
)
which implies that
b
&
1
!
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This note was uploaded on 04/04/2011 for the course ECON 401 taught by Professor Staff during the Spring '08 term at University of Washington.
 Spring '08
 Staff
 Macroeconomics

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