Unobserved Heterogeneity Bias When Estimating
the Economic Model of Crime
Todd L. Cherry*
December 10, 1997 – submitted to AEL
February 12, 1998 – accepted
Abstract
Using unique and unpublished panel data from selected U.S. cities, this paper investigates
the consequences of ignoring unobserved heterogeneity in the unit of observation when
estimating the economic model of crime. Results confirm that neglecting to control for
unobserved heterogeneity overstates the ability of sanctions to deter criminal activity. Further,
this upward bias is found to vary significantly across crime types. Interestingly, heterogeneity is
insignificant in the tightly reported crimes of murder and auto-theft while being significant in
assault, robbery, burglary and larceny where individuals and police have greater discretion in
reporting.
*Department of Economics and Finance, University of Wyoming, Laramie WY 82071-3985;
phone: (307) 766-2178, fax: (307) 766-5090, email: tlcherry@uwyo.edu
1
I. Introduction
The notion that individuals respond to incentives in both legitimate and illegitimate
activity goes back at least to Bentham's Principles of Penal Law (1862). Becker (1968)
formalized the deterrence hypothesis by introducing an economic theory explaining the inherent
uncertainty of criminal behavior. Since Becker’s seminal paper, an immense collection of
empirical work has accumulated with results consistently suggesting deterrent effects from
criminal sanctions.1 However, unsettled caveats raise concerns regarding the results. Due to the
extreme cost of obtaining reliable individual level data, one issue arises from the use of aggregate
data. Unobserved heterogeneity in the unit of observation may lead to spurious relationships that
incorrectly imply or exaggerate deterrent effects. Cornwell and Trumbull (1994) first examined
the consequences of ignoring heterogeneity and found that doing so overstates the deterrent
effects for overall criminal activity.
This study extends their approach by examining unobserved heterogeneity bias for
individual crimes.2 Given individual crimes have varying degrees of discipline in reporting,
heterogeneity and the resulting bias should differ across crimes. For example, reporting practices
of murder should be more homogeneous across jurisdictions than burglary and larceny. Results
suggest that unobserved heterogeneity bias varies significantly across individual crime types.
Further, heterogeneity is insignificant in tightly reported crimes while being significant in the
crimes where police and individuals have greater discretion in reporting.
II. Unobserved Heterogeneity Bias
Variances across jurisdictions introduce noise into the analysis which may bias results
towards a deterrent effect. The most noted concern is the underreporting of crimes by individuals
and police, where it is estimated that approximately 50 percent of crimes are unreported to the
F.B.I. (Myers, 1980). Unreported offenses lead to underestimates of crime rates and
overestimates of police department clearance rates (ratio of arrests to crimes). Intentional nonreporting by police departments magnifies the problem. Often judged by its clearance rate, police
departments may adopt loose reporting practices for political reasons. Departments can improve
2
their "effectiveness" by decreasing the number of reported offenses or increasing the number of
cleared offenses, with the former easier to achieve. Police departments can also inflate their
clearance rate by offering leniency to those arrested if they confess to unsolved crimes.
For example, suppose there are two criminal justice departments. Department A follows
strict accounting practices that report high percentages of crimes, while department B is more
lenient and reports lower percentages. Noting that certainty of sanctions is typically measured by
the clearance rate, this disparity causes a problem when the reported data from the two
jurisdictions are analyzed. Even if both departments clear approximately the same portion of
actual crimes, department B appears more effective with an overestimated clearance rate. Thus,
department B has a greater clearance rate and a lower reported crime rate compared to
department A. This indicates the presence of a deterrent effect when both jurisdictions may
actually have similar actual rates with no obvious deterrent effect. 3
III. Model
The economic model of crime proposes that individuals participating in criminal activity
can be thought of as rational economic beings acting under uncertainty. Therrefore, to curb
criminal activity, the theory suggests decreasing its expected return. Given that marginal utility
of income is positive, the certainty and severity of sanctions are inversely related to expected
utility, and thus, criminal activity (Becker, 1968).
At the aggregate level, the number of offenses can be stated as a function of the
probability and cost of apprehension. The following crime equation is estimated with panel data:
Cit = i+´Dit + ´Lit +´Xit + it,
i=1,2....N; t=1,2....T
(1)
where Cit denotes the crime rate of municipality i at time t, Dit contains deterrent variables, Lit
contains labor market variables, and Xit contains related socio-economic variables. Unobserved
municipality effects are captured by i.4 The disturbance terms follow a normal distribution with
zero mean and constant variance.
Estimation of the panel data with standard cross-sectional techniques assumes
homogeneity in the unit of observation by restricting the is to equal each other. This pooled
3
model would be appropriate only when municipality effects are homogeneous; otherwise it yields
inconsistent and biased estimates. An appropriate effects model (fixed or random) controls for
heterogeneity and provides consistent and efficient estimates in the presence of heterogeneity.
The consequences of assuming homogeneity across individual crimes is examined by comparing
the results from the pooled model and the appropriate effects model.
IV. Results
Crime, law enforcement and socioeconomic data from selected U.S. cities provide the
sample.
Following Sjoquist (1973), the sample includes municipalities having populations
between 50,000 and 200,000 and populations greater than 50 percent of their total Standard
Metropolitan Statistical Area (SMSA).5 After eliminating the cities not meeting the criterion or
not having sufficient information, 44 municipalities over the census years of 1970, 1980 and
1990 provided the panel of 132 observations.
Table 1 provides the definitions and means for the variables used in estimating the crime
functions. Given that more than one arrest is possible for a single crime, it is not surprising that
the mean clearance rate for the tightly reported murder category exceeds one (1.09). The
clearance rate represents the probability of arrest, i.e., certainty of sanctions. Sentence length is
measured by the ratio of total prison population to new commitments and indicates the severity
of sanctions. Income per capita and the unemployment rate proxy available returns from the
labor market. Municipal population and the proportion that is minority are included to capture
additional variation across crime rates.
Tables 2 and 3 present the estimates of the pooled and effects models. In addition, table 3
provides the Hausman statistics used to determine the appropriate effects model for each crime.
Except in the cases of murder and rape, Hausman tests fail to reject orthogonality between the
regressors and random effects. Thus, a fixed effects approach is appropriate for murder and rape
while the random effects model is appropriate for the remaining crimes. Following previous
research, each model is specified as log-linear so that the estimated coefficients are elasticities.
Results indicate that the probability of arrest has a significant deterrent effect across all crime
4
types while sentence length fails to provide any significant relationships (as in Avio and Clark,
1978). Police has the typical positive relationship with crime rates, indicating the possible
presence of a simultaneous relationship.6 Generally, results concerning labor market and socioeconomic variables follow previous research.
Given the insignificance of sentence length, the focus of this study rests on comparing the
probability of arrest elasticities across the pooled and effects procedures. In doing so, there is
evidence of a bias in the pooled estimates which fail to account for unobserved heterogeneity.
This result corresponds to Cornwell and Trumbull (1994), however, this study extends the
analysis and finds that the bias varies significantly across crime types. The upward bias in the
probability of arrest elasticities (absolute values) ranged from 70 percent for the burglary model
to almost zero for the robbery model. For example, estimated elasticities for the individual crime
of assault are .289 for the pooled model and .214 for the effects model. In this case, failing to
account for heterogeneity overstates the deterrence from certainty of sanctions by 35 percent. For
overall crime, deterrence is overstated by 20 percent by failing to control for heterogeneity.
Interestingly, F-tests reveal that jurisdiction fixed effects are not significant in the tightly reported
crimes of murder and auto-theft while being significant in the crimes of assault, robbery, burglary
and larceny where individuals and police have greater discretion in reporting. Clearly, results
suggest that heterogeneity bias varies across individual crimes.
V. Summary
This study examines unobserved heterogeneity bias when estimating the economic model
of crime. Previous work has shown the existence of this bias when analyzing overall crime. This
paper confirms that result and extends the analysis by finding significant variation in the bias
across crime categories. Municipality effects are found to be insignificant in tightly reported
crimes and significant in ‘loosely’ reported crimes. Results imply estimates failing to address
unobserved heterogeneity overstate the potential deterrent effects from criminal sanctions, but
that this result is not constant across individual crimes.
5
Acknowledgements
The author would like to thank the Bryan School of Business and Economics at the
University of North Carolina at Greensboro for partial funding of this project. Helpful comments
were provided by John List, Joni Hersch and Kip Viscusi.
6
Notes
1
Examples are Ehrlich (1973), Cloninger (1975), Viscusi (1986), Cornwell and Trumbull (1994)
and Mui and Ali (1997) among others.
2
The seven individual crimes examined are the index crimes of murder, rape, robbery, assault,
burglary, larceny and auto theft. Index is the aggregate of the individual crimes.
3
Similarly, the issue of heterogeneity bias exists when the certainty of sanctions is measured by
the ratio of convictions to reported crimes. Given the discretion of conviction falls upon a
different branch of the criminal justice system than the discretion of arrest, the resulting bias of
the two sources of unobserved heterogeneity likely varies.
4
Time effects were controlled for in all estimations leaving the analysis to focus on heterogeneity
bias related to the unit of observation.
5
This criteria was to reduce spillover effects from neighboring communities.
6
A simultaneous analysis revealed similar findings and was excluded from the presentation for
succinctness.
7
References
Avio, K.L. and C.S. Clark (1978) The Supply of Property Offenses in Ontario: Evidence on
the Deterrent Effect of Punishment, Canadian Journal of Economics, 11, 1-19.
Becker, G. S. (1968) Crime and Punishment: An Economic Approach, Journal of Political
Economy, 76, 169-217.
Cornwell, C. and W.N. Trumbull (1994) Estimating the Economic Model of Crime with Panel
Data, The Review of Economic and Statistics, 76, 360-366.
Ehrlich, I. (1973) Participation in Illegitimate Activities: A Theoretical and Empirical
Investigation, Journal of Political Economy, 81, 521-65.
Mui, H.W. and M.M. Ali (1997) Economic Analysis of Crime and Punishment: An Asian
Case, Applied Economic Letters, 4, 261-265.
Myers, S. (1980) Why are Crime Rates Underreported? What is the True Crime; Does it
Matter?, Social Science Quarterly, 61, 23-43.
Nagin, D. (1978) General Deterrence: A Review of the Empirical Evidence in Deterrence and
Incapacitation: Estimating the Effects of Criminal Sanctions on Crime Rates (Ed.)
Blumstein, Nagin, and Cohen, National Academy of Sciences, Washington D.C.
Sjoquist, D.L. (1973) Property Crime and Economic Behavior: Some Empirical Results,
American Economic Review, 63, 439-446.
United States Bureau of Census, 1970 (1980) (1990) Census of Population, General
Population Characteristics, Washington: GPO, 1972 (1982) (1992).
United States Bureau of Census, 1970 (1980) (1990) Census of Population, Social and
Economic Characteristics, Washington: GPO, 1972 (1982) (1992).
United States Department of Justice, Uniform Crime Reports 1970 (1980) (1990): Crime in the
United States, 1971 (1981) (1991).
Viscusi, W.K. (1986) The Risks and Rewards of Criminal Activity: A Comprehensive Test of
Criminal Deterrence, Journal of Labor Economics, 4, 317-40.
8
Table 1 - Variable Definitions and Sample Means
Variable
Definition
Mean
crime ratea
index
murder
rape
assault
robbery
burglary
larceny
auto theft
offenses per 10,000 persons
total offenses per 10,000 persons
murders per 10,000 persons
rapes per 10,000 persons
assaults per 10,000 persons
robberies per 10,000 persons
burglaries per 10,000 persons
larcenies per 10,000 persons
auto thefts per 10,000 persons
probability of arrestb
index
murder
rape
assault
robbery
burglary
larceny
auto theft
clearance rate (arrests per offenses)
% total crimes cleared
% murders cleared
% rapes cleared
% assaults cleared
% robberies cleared
% burglaries cleared
% larcenies cleared
% auto thefts cleared
.316
1.09
.521
.489
.622
.190
.262
.230
sentence lengthc
prison population/new commitments
2.093
policea
officers per 10,000 persons
16.634
incomed
real income per capita
unemploymentd
% population not employed
populationd
municipality’s population/10,000
9.01176
minorityd
% population that is non-white
14.926
Sources:
a
FBI’s Uniform Crime Report (UCR)
b
unpublished, obtained through personal contact with the FBI
c
Statistical Abstract
d
U.S. Census of the Population
596.595
.794
3.698
13.446
27.980
156.096
422.043
39.041
3417.652
5.487
9
Table 2 - Pooled Results of Crime Function
Constant
P(arrest)
Sentence
Index
Murder
Rape
Assault
Robbery
Burglary
Larceny
Auto
-1.169
-5.263
-4.538
-21.99‡
-3.439
-3.280
-.995
-5.847†
(1.884)
(4.125)
(4.353)
(4.254)
(5.871)
(2.554)
(2.088)
(2.871)
-.203‡
-.279‡
-.190‡
-.289‡
-.284‡
-.202‡
-.098†
-.150‡
(.043)
(.108)
(.061)
(.069)
(.082)
(.050)
(.045)
(.043)
.103
-.102
-.100
.112
.068
-.072
.188
.208
(.108)
(.235)
(.248)
(.243)
(.341)
(.146)
(.119)
(.165)
.322‡
-.451*
.176
1.318‡
.680*
.566‡
-.055
.849‡
(.118)
(.258)
(.274)
(.268)
(.371)
(.160)
(.130)
(.180)
.401*
-.282
.094
1.492‡
.109
.265
.582†
.469
(.234)
(.512)
(.544)
(.530)
(.731)
(.318)
(.259)
(.357)
.197†
.086
.135
.624‡
-.086
.292†
.270‡
.082
(.102)
(.223)
(.236)
(.231)
(.318)
(.139)
(.113)
(.154)
.175‡
.417‡
.234*
.558‡
.165
.287‡
.129†
.247‡
(.054)
(.120)
(.125)
(.123)
(.169)
(.074)
(.060)
(.082)
.059†
.513‡
.254‡
.406‡
.481‡
.151‡
.015
-.063
(.026)
(.056)
(.059)
(.059)
(.082)
(.035)
(.028)
(.039)
F(9,122)
60.17
19.44
23.15
30.43
17.15
21.23
10.45
9.94
(p-value)
(.000)
(.000)
(.000)
(.000)
(.000)
(.000)
(.000)
(.000)
R2
.803
.559
.603
.669
.526
.582
.394
.381
Police
Income
Unemploy
Population
Minority
standard errors in parentheses unless otherwise noted
* indicates significance at the 10% level, † at the 5% level, ‡ at the 1% level
Table 3 - Random/Fixed-Effects Results of Crime Function
Index
Murder
Rape
Assault
Robbery
Burglary
Larceny
Auto

Constant
P(arrest)
Sentence
Police
Income
Unemploy
Population
Minority
R2
.389
-2.014
-3.357
-18.508‡
-2.028
-.965
1.683
-6.997†
(2.100)
(4.617)
(4.860)
(4.602)
(6.247)
(2.727)
(2.156)
(3.118)
-.169‡
-.221†
-.128†
-.214‡
-.297‡
-.119‡
-.069†
-.118‡
(.036)
(.096)
(.057)
(.055)
(.055)
(.037)
(.031)
(.036)
.048
.032
.035
-.108
.062
-.069
.090
.268†
(.098)
(.229)
(.244)
(.200)
(.269)
(.116)
(.089)
(.139)
.470‡
.640†
.309
1.336‡
1.292‡
.786‡
.346‡
.878‡
(.126)
(.282)
(.299)
(.272)
(.369)
(.160)
(.125)
(.185)
.255
-.663
.036
1.178†
-.131
.056
.269
.681*
(.255)
(.566)
(.598)
(.553)
(.747)
(.327)
(.257)
(.375)
.115
.011
.074
.456†
-.103
.230†
.205‡
.008
(.090)
(.214)
(.228)
(.184)
(.242)
(.107)
(.082)
(.126)
.120*
.365†
.144
.509‡
.078
.203†
.039
.197*
(.070)
(.146)
(.152)
(.162)
(.224)
(.097)
(.079)
(.108)
.070†
.458‡
.239‡
.399‡
.405‡
.124‡
-.004
-.028
(.032)
(.068)
(.071)
(.074)
(.102)
(.044)
(.036)
(.049)
.810
.581
.624
.684
.543
.586
.365
.412
24.48‡
28.47‡
27.39‡
32.14‡
22.26‡
2.27‡
2.98‡
1.39
11.55
12.99
2.46
Ho: var(u)=0 (ols vs. random-effects)
LM(3)
14.99‡
5.39†
4.58†
Ho: municipality effects are equal (ols vs. fixed-effects)
F(43,79)
1.39
0.94
0.80
1.78†
1.78†
Ho: effects and regressor are orthogonal (random vs. fixed-effects)
2(9)
12.66
19.81†
16.97†
8.20
standard errors in parentheses unless otherwise noted
* significance at the 10% level; † at the 5% level; ‡ at the 1% level
2.55
1

fixed-effects estimates; otherwise random-effects
1
4
5
6