ECON 241B, Final Exam
Winter, 2011
1
(20 points)
You wish to estimate the effect of unemployment on criminal behavior in Gotham City. In your model,
Crime
t
, is a function of the unemployment rate,
Unemp
t
, and the presence of Batman,
Batman
t
. You hy
pothesize that unemployment causes an increase in criminal behavior, as represented by
β
1
in the following
model (assume that all variables are expressed as deviations from means):
Crime
t
=
β
1
Unemp
t
+
β
2
Batman
t
+
ε
t
(1)
You make the following assumptions:
•
E
[
Unemp
t
*
ε
t
] =
E
[
Batman
t
*
ε
t
] =
0
•
E
[
Unemp
t
*
Batman
t
]
>
0 (When unemployment is low, business is good for Bruce Wayne and he is
more often traveling for business)
You gather data on crime rates and unemployment rates in Gotham City, but the presence of Batman is
unobservable, so you proceed with the following model:
Crime
t
=
β
1
Unemp
t
+
v
t
(2)
(a) (4pts)
Derive the OLS estimator of the unemployment effect from (2),
ˆ
β
OLS
1
, and show whether or not
the estimator is consistent for the true parameter from (1),
β
1
.
(b) (2pts)
If inconsistent, will your estimator be too large or too small? Explain.
You also are able to gather data on mass layoffs,
Mass
t
, that happen in Gotham City throughout the time
period of interest. These mass layoffs are not seasonal fluctuations and can be thought of as random shocks.
Batman’s presence is not contemporaneously affected by mass layoffs.
(c) (4pts)
List all required conditions for
Mass
t
to be a valid instrument for the endogenous regressor.
(d) (5pts)
Assuming these conditions hold. Derive the 2SLS estimate of the unemployment effect,
ˆ
β
2
SLS
1
and show whether your estimator is consistent. Write out the equations for both stages. How
can we test one of the requirements for a valid instrument using the firststage results?
You can write the second stage of the 2SLS strategy as:
Crime
t
=
β
1
ˆ
Unemp
t
+(
v
t
+
β
1
(
Unemp
t

ˆ
Unemp
t
))
(3)
Suppose you estimate eq. (3) by OLS and use OLS standard errors for inference, ignoring the fact that
ˆ
Unemp
t
is estimated in the first stage. Assuming conditional homokedasticity, the asymptotic variance of
your estimated standard errors when using OLS on the second stage can be written as:
AVAR
(
ˆ
β
OLS
,
2
ndStage
1
) =
VAR
(
v
t
+
β
1
(
Unemp
t

ˆ
Unemp
t
))
E
[
ˆ
Unemp
2
t
]
(4)
Assume that
v
t
and
(
Unemp
t

ˆ
Unemp
t
)
are independent. For accurate inference, we should use an estimate
of the following asymptotic variance (this would be the asymptotic variance of the IV estimator):
AVAR
(
ˆ
β
IV
1
) =
VAR
(
v
t
)
E
[
ˆ
Unemp
2
t
]
(5)
1
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ECON 241B, Final Exam
Winter, 2011
(e) (2pts)
If we mistakenly use an estimate of (4), would you expect your estimated standard errors to be
too big or too small?
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 Fall '08
 Staff
 Unemployment, Estimation theory, OLS, Unempt

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