sosaescudero-ExampleCornwell

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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:
• 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)
αi are fixed effects (may be correlated 
with (X´it, P´it)).
eit are typical disturbance terms. 
itiititit
ePXR +++= αγβ ´´
Tt
Ni
,...,1
,...,1
=
= ( 1 )
Dependant 
variable
Probability 
of arrest
Probability of 
conviction
Probability 
of prison
Sanction 
severity
Summary of variables
• The “between” transformation of (1) is:
The data are expressed in county means: 
• The “within” transformation of (1) is:
The data are in deviations from means: . 
(3) Does NOT depend on the county effects.
iiiii
ePXR +++= αγβ ´´ ( 2 )
∑
−=
t
iti
RTR
1
iitit
RRR −=
itititit
ePXR ++= γβ ´´ ( 3 )
• 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 
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%.
γ̂
• The “between” estimator is consistent 
only if (X´it, P´it) is orthogonal to both αi
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. 
Balanced 
Panel:
N = 90
T = 7
Fixed Effects Estimation
Test F: Joint 
significance, it 
rejects the 
null.
Fixed Effects Test: it rejects the null. 
So, the fixed effects are significative.
Region and urban 
dummies and 
percentage minority 
variable do not vary 
over time, they are 
eliminated by the 
within transformation.
lprbarr (PA) -0.6475095 -0.3849533 -41
lprbconv (PC) -0.5282029 -0.3006001 -43
lprbpris (PP) 0.2965068 -0.1913185 -35
lavgsen (S) -0.235888 0.0261132 -89
Between 
Coefficient
Within 
Coefficient
Variable % Variation
• 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.
• 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.
It rejects the null, so there are systematic 
differences between FE and RE coefficients.
• RE estimators: 
INCONSISTENT
• FE estimators: 
CONSISTENT
Hausman Test
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
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
• The test rejects this null
0:
2
0
=µσH
In all tests, 
we reject 
the null.
0:
0
=ρH
• 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.
• TheFixed 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!