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Imbens, Lecture Notes 20, ARE213 Spring ’06
1
ARE213
Econometrics
Spring 2006 UC Berkeley Department of Agricultural and Resource Economics
Endogeneity IV:
Heterogeneity and Local Average Treatment Effects
Here we investigate the interpretation of instrumental variables estimators given hetero
geneity in the eﬀect of the endogenous regressor. We shall see that instrumental variables
estimates estimate diﬀerent averages depending on the instruments. We focus on the case
where the endogenous regressor is binary. The example we will use is based on work by
Angrist on estimating the eﬀect of veteran status on earnings.
1. Potential Outcomes
First we set up the problem in a slightly diﬀerent way, using potential outcomes. Let
Y
i
(0) and
Y
i
(1) be two potential outcomes for unit
i
, one for each value of the endogenous
regressor or treatment. The ±rst potential outcome
Y
i
(0) gives the outcome if person
i
were
to serve in the military, irrespective of whether this person served or not. The second gives
the potential outcome given military service, again irrespective of whether the person served
or not. We are interested in the causal eﬀect of military service,
Y
i
(1)

Y
i
(0). We cannot
directly observe this since we can only observe either
Y
i
(0) or
Y
i
(1), but not both. Let
D
i
be the actual value of the endogenous regressor, equal to zero or one. We observe
D
i
and
Y
i
=
Y
i
(
D
i
)=
±
Y
i
(1)
if
D
i
=1
Y
i
(0)
if
D
i
=0
.
Now we introduce the instrumental variable set up by de±ning similar potential outcomes
for the treatment. We focus on the case with a binary instrument
Z
i
. De±ne two potential
outcomes
D
i
(0) and
D
i
(1), representing the value of the endogenous regressor given the two
values for the instrument. The actual value of the endogenous variable is
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This note was uploaded on 08/01/2008 for the course ARE 213 taught by Professor Imbens during the Spring '06 term at University of California, Berkeley.
 Spring '06
 IMBENS

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