Lecture Notes 10
Econ 410 – Introduction to Econometrics
1
Instrumental Variables Regression
Instrumental variables (IV) regression is a way to obtain consistent estimators of the
regression coefficients when one or more of the regressors are correlated with the error
term. This correlation might arise for different reasons: omitted variables, errorsin
variable, simultaneous causality….
In IV regression, the variables in the regression model are called:
•
endogenous variables, if they are correlated with the error term;
•
exogenous variables, if they are not correlated with the error term.
Let
X
be an endogenous variable. The idea of IV regression is to think of
X
as
composed of two parts: one that is correlated with
u
and the other one that is not. If we
had information that allowed us to isolate this second part, than we could focus on the
variations of
X
that are not correlated with the error term and disregard the part of
X
that causes problems to the OLS estimator.
The information about the movements in
X
that are uncorrelated with
u
is obtained
from one or more additional variables that are called instrumental variables
(or
instruments
). IV regression uses these instruments to isolate the movements in
X
that are
uncorrelated with
u
which in turns permits to estimate consistent OLS coefficients.
Regression model with one regressor
Consider the model:
i
i
i
u
X
Y
+
+
=
1
0
β
β
If
(
)
0
,
≠
i
i
u
X
corr
, then
(
)
0

≠
i
i
X
u
E
and the OLS estimators
0
ˆ
β
and
1
ˆ
β
are biased and
inconsistent.
Let
Z
be the instrument that we want to use to isolate the movements in
X
that are
uncorrelated with
u
. For
Z
to be a valid instrument, it must satisfy two conditions:
1)
Instrument relevance
:
(
)
0
,
≠
i
i
X
Z
corr
This condition ensures that the movements in
Z
are related to the movements in
X
.
2)
Instrument exogeneity
:
(
)
0
,
=
i
i
u
Z
corr
This condition ensures that the part of the movements in
X
that is captured by
Z
is
exogenous (that is, not correlated with the error term).
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Lecture Notes 10
Econ 410 – Introduction to Econometrics
2
Two Stage Least Squares Estimator
The Two Stage Least Squares (TSLS) estimator is a common IV estimator that can be used
when one or more of the regressors are correlated with the error term. The TSLS estimator
is obtained in two stages.
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 Fall '08
 Staff
 Econometrics, Regression Analysis, Yi, error term, TSLS

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