Empirical
Example
Walter Sosa Escudero
(wsosa@udesa.edu.ar)
Universidad de San Andres - UNLP
Panel Data Models
In this exercise, we will replicate
the results in “Estimating the
Economic Model of Crime with
Panel Data”, by C. Cornwell and
W. Trumbull (1994).
The Data
• Cornwell and Trumbull estimate an economic
model of crime.
• Panel dataset of North Carolina counties.
• They use single and simultaneous equations
panel data estimators to address two sources
of endogeneity: unobserved heterogeneity
and conventional simultaneity.
• The data are county level, so there is a
relatively low level of aggregation.
Model and Alternative
Estimators
• The basic assumption is:
An individual´s participation in the
criminal sector depends on the relative
monetary return to illegal activities and
the degree to which the criminal justice
system is able to affect the probabilities
of apprehension and punishment.
• Cornwell and Trumbull specify the following
crime equation:
Rit  X ´it   P´it   i  eit
i  1,..., N
t  1,...,T
(1)
• where: Rit is the crime rate.
X´it contains variables which control
for the relative return to legal opportunities.
(wcon, wtuc, wtrd, wfir, wser, wser, wmfg, wfed, wsta,
wloc, polpc, urban, density, west, central, pctymle,
pctmin)
P´it contains a set of deterrent
variables. (prbarr, prbconv, prbpris, avgsen)
i are fixed effects (may be correlated
with (X´it, P´it)).
eit are typical disturbance terms.
Summary of variables
Dependant
variable
Probability
of arrest
Probability of
conviction
Probability
of prison
Sanction
severity
• The “between” transformation of (1) is:
Ri  X ´i   P´i   i  ei
(2)
1
The data are expressed in county means: Ri  T  Rit
t
• The “within” transformation of (1) is:
Rit  X ´it   P´it   eit
(3)
The data are in deviations from means: Rit  Rit  Ri .
(3) Does NOT depend on the county effects.
• The authors adopt a log-linear specification so that their
estimated coefficients are interpretable as elasticities.
“Between” Model:
• (2) leads to cross-section estimators
which neglect unobserved county
heterogeneity.
• If unobserved characteristics are
correlated with (X´it, P´it), such
procedure will produce inconsistent
estimates.
“Within” Model:
• By using (3), both sources of
endogeneity may be addressed.
• If the only problem is correlation
between (X´it, P´it) and unobserved
heterogeneity, then consistent
estimation is possible by performing
least squared on (3).
• Conventional simultaneity can be
accounted for by using 2SLS to (3).
“Between” Model
Balanced
Panel:
N = 90
T=7
Test F: Joint
significance, it
rejects the
null.
• With the exception of PP, the elements of ˆ
have the correct NEGATIVE signs.
• Only the estimated coefficient of PA and PC
are statistically significant at the usual
significance levels.
• The estimated arrest and conviction
elasticities are, respectively, -0.65 and –0.53.
• For the rest of the variables, only lpolpc,
ldensity, west, central and lpctmin are
statistically significant at 5%.
• For example, if the number of police per
capita increases 1%, the crime rate increases
in 0.36%.
• The “between” estimator is consistent
only if (X´it, P´it) is orthogonal to both i
and eit.
• The “within” estimator is a simple
solution to the violation of the
orthogonality condition that (X´it, P´it) is
uncorrelated with unobserved
heterogeneity.
Fixed Effects Estimation
Balanced
Panel:
N = 90
T=7
Test F: Joint
significance, it
rejects the
null.
Region and urban
dummies and
percentage minority
variable do not vary
over time, they are
eliminated by the
within transformation.
Fixed Effects Test: it rejects the null.
So, the fixed effects are significative.
• Now, the estimated coefficient of PP has the
correct (negative) sign and is statistically
significant.
• The within estimate of the deterrent effect of
S is small and statistically insignificant.
• Conditioning on the county effects causes the
(absolute value of the) estimated deterrent
elasticities associated with PA and Pc
decrease by 41% and 43%, respectively.
Variable
lprbarr
lprbconv
lprbpris
lavgsen
(PA)
(PC)
(PP)
(S)
Between
Within
% Variation
Coefficient Coefficient
-0.6475095
-0.5282029
0.2965068
-0.235888
-0.3849533
-0.3006001
-0.1913185
0.0261132
-41
-43
-35
-89
• In the Fixed Effects model, both sources of
endogeneity may be addressed.
• First, if the only problem is correlation
between (X´it, P´it) and unobserved
heterogeneity, then consistent estimation is
possible by performing OLS on (3). This
within estimator can be viewed as an
instrumental variables estimator with
instruments (deviations from means) that are
orthogonal to the effects by construction.
• Conventional simultaneity can be accounted
for by using 2SLS to estimate (3).
• If the constant terms specific for each
county were randomly distributed,
between counties, we can estimate a
Random Effects Model.
• In order to estimate a Random Effects
Model, it´s necessary to assume that
the explanatory variables are
uncorrelated to the specific term for
each county.
• A Hausman test can be constructed to
evaluate FE / RE estimates.
Hausman Test
• RE estimators:
INCONSISTENT
• FE estimators:
CONSISTENT
It rejects the null, so there are systematic
differences between FE and RE coefficients.
Random Effects and Serial Correlation
• Bera-Yoon-Sosa Escudero (2001):
– BP Test for random effects implicitly assume no
autocorrelation.
– The presence of random effects confuse the BP
test, inducing to reject Ho, even though it is
correct.
– The same thing happens with the autocorrelation
test.
– BYS: modified tests.
• Joint Test Baltagi-Li (1991)
– Test for the joint null of no autocorrelation and no
random effects (low power, less informative).
• Sosa Escudero (2001):
– Joint Test for random effects and positive serial
correlation (one-sided, one-directional).
Results of the Tests
• In the Random Effects tests: the null is
H0 : 
2

0
in the Random Effects model.
• The test rejects this null, so the OLS estimators are
NOT BLUE.
• In the Serial Correlation tests: the null is
H0 :   0
• The test rejects this null
In all tests,
we reject
the null.
• Note that the statistics decrease in all the
adjusted versions of the tests:
• LM Test for random effects, assuming no serial
correlation: 672.89.
• Adjusted LM test for random effects, which works
even under serial correlation: 340.20.
• LM test for first order serial correlation, assuming no
random effects: 375.04.
• Adjusted test for first order serial correlation, which
works even under random effects: 42.36.
• LM Joint test for random effects and serial
correlation: 715.24. This Joint Test rejects the joint
null, but is NOT informative about the direction of the
misspecification.
Instrumental Variables
• Conventional simultaneity may exist between
the crime rate, the probability of arrest and
the number of police per capita.
• Counties experiencing rising crime rates,
holding police resources constant, would see
probabilities of arrest fall.
• But, increases in crime may motivate a
county to increase policing resources which
would increase the probability of arrest.
• Now, we also allow for the possibility that PA
and the number of police per capita may be
correlated with eit.
• Applying 2SLS to the “within” model, we
address simultaneity as well as unobserved
heterogeneity.
• We need at least two exogenous instruments
(uncorrelated with e and the effects).
• We use as instruments a mix of different
offense types and per capita tax revenue.
2SLS with
Fixed
Effects
• PA, PC and PP are
NOT statistically
significant.
• Only lwfed and
lwloc are
statistically
significant.
• The Fixed Effects
are statistically
significant.
2SLS to
Between
Model
• PA and PC are
statistically
significant and
have the correct
signs.
• PP is NOT
statistically
significant.
• PA is 30% lower
in 2SLS to
“between” than to
2SLS to “within”
model.
• The statistical consequences of
neglecting unobserved heterogeneity in
our sample are serious whether single
or simultaneous equations estimators
are used!