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1
Introduction to the Instrumental
Variable (IV)
Method
1
Instrumental Variables Regression
(SW Chapter 12)
Three important threats to internal validity are:
mitted variable bias from a variable that is correlated with
omitted variable bias from a variable that is correlated with
X
but is unobserved, so cannot be included in the regression;
simultaneous causality bias (
X
causes
Y
,
Y
causes
X
);
errorsinvariables bias (
X
is measured with error)
2
Instrumental variables regression can eliminate bias when
E
(
u

X
)
¹
0 – using an
instrumental variable
,
Z
IV Regression with One Regressor
and One Instrument
(SW Section 12.1)
Y
i
=
0
+
1
X
i
+
u
i
IV regression breaks
X
into two parts:
a part that might be
correlated with
u
, and a part that is not.
By isolating the part
that is not correlated with
u
, it is possible to estimate
1
.
This is done using an
instrumental variable
,
Z
i
, which is
3
uncorrelated with
u
i
.
The instrumental variable detects movements in
X
i
that are
uncorrelated with
u
i
, and uses these to estimate
1
.
Terminology:
endogeneity and
exogeneity
An
endogenous
variable is one that is correlated with
u
n
ogen us
ariable is one that is uncorrelated with
An
exogenous
variable is one that is uncorrelated with
u
Historical note:
“Endogenous” literally means “determined
within the system,” that is, a variable that is jointly
determined with Y, that is, a variable subject to simultaneous
salit
Ho e er this de nition is narro
and IV
4
causality.
However, this definition is narrow and IV
regression can be used to address OV bias and errorsin
variable bias, not just to simultaneous causality bias.
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Three conditions for a valid
instrument
Y
i
=
0
+
1
X
i
+
u
i
For an instrumental variable (an “
instrument
”)
Z
to be valid, it
must satisfy two conditions:
1.
Instrument relevance
: corr(
Z
i
,
X
i
)
¹
0
2.
Instrument exogeneity
: corr(
Z
i
,
u
i
) = 0
3.
Exclusion
:
Z
cannot be an omitted variable in model
5
Suppose for now that you have such a
Z
i
(we’ll discuss how to
find instrumental variables later).
How can you use
Z
i
to estimate
1
?
The IV Estimator, one
X
and one
Z
6
Two Stage Least Squares, ctd.
7
8
3
Two Stage Least Squares, ctd.
Suppose you have a valid instrument,
Z
i
.
Stage 1:
Regress
X
i
on
Z
i
, obtain the predicted values
ˆ
i
X
Stage 2:
Regress
Y
i
on
ˆ
i
X
; the coefficient on
ˆ
i
X
is
the TSLS estimator,
1
ˆ
TSLS
.
9
1
ˆ
TSLS
is a consistent estimator of
1
.
10
The IV Estimator, one X and one Z, ctd.
Explanation #2: a little algebra…
Y
i
=
0
+
1
X
i
+
u
i
Thus,
cov(
Y
i
,
Z
i
) = cov((
0
+
1
X
i
+
u
i
)
,
Z
i
)
= cov(
0
,
Z
i
) + cov(
1
X
i
,
Z
i
) + cov(
u
i
,
Z
i
)
=
0
+ cov(
1
X
i
,
Z
i
) +
0
=
1
cov(
X
i
,
Z
i
)
11
where cov(
u
i
,
Z
i
) = 0 (instrument exogeneity); thus
1
=
cov( ,
)
cov(
,
)
ii
YZ
X
Z
The IV Estimator, one X and one Z, ctd.
1
=
cov( ,
)
ov(
)
Z
cov(
,
XZ
The IV estimator replaces these population covariances with
sample covariances:
1
ˆ
TSLS
=
YZ
s
,
12
XZ
s
s
YZ
and
s
XZ
are the sample covariances.
This is the TSLS
estimator – just a different derivation!
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Consistency of the TSLS estimator
1
ˆ
TSLS
=
YZ
s
XZ
s
The sample covariances are consistent:
s
YZ
p
cov(
Y
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This note was uploaded on 11/28/2010 for the course ECON Economics taught by Professor Davidbrownstone during the Spring '10 term at UC Irvine.
 Spring '10
 DAVIDBROWNSTONE

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