Unobserved Heterogeneity Bias when Estimating



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Using unique and unpublished panel data from selected U.S. cities, this paper investigates the consequences of ignoring unobserved heterogeneity in the unit of observation when estimating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

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.[i] 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.[ii] 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 non-reporting 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 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. [iii]

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 = αi+ β´Dit + θ´Lit + ψ´Xit + εit, 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 αi.[iv] 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 αis to equal each other. This pooled 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).[v] 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 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.[vi] Generally, results concerning labor market and socio-economic 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.

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.

Notes

1Examples are Ehrlich (1973), Cloninger (1975), Viscusi (1986), Cornwell and Trumbull (1994) and Mui and Ali (1997) among others.

2The 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.

3Similarly, 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.

4Time effects were controlled for in all estimations leaving the analysis to focus on heterogeneity bias related to the unit of observation.

5This criteria was to reduce spillover effects from neighboring communities.

6A simultaneous analysis revealed similar findings and was excluded from the presentation for succinctness.

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.

Table 1 - Variable Definitions and Sample Means

|Variable |Definition |Mean |

| | | |

|crime ratea |offenses per 10,000 persons | |

| index |total offenses per 10,000 persons |596.595 |

| murder |murders per 10,000 persons |.794 |

| rape |rapes per 10,000 persons |3.698 |

| assault |assaults per 10,000 persons |13.446 |

| robbery |robberies per 10,000 persons |27.980 |

| burglary |burglaries per 10,000 persons |156.096 |

| larceny |larcenies per 10,000 persons |422.043 |

| auto theft |auto thefts per 10,000 persons |39.041 |

| | | |

|probability of arrestb |clearance rate (arrests per offenses) | |

| index |% total crimes cleared |.316 |

| murder |% murders cleared |1.09 |

| rape |% rapes cleared |.521 |

| assault |% assaults cleared |.489 |

| robbery |% robberies cleared |.622 |

| burglary |% burglaries cleared |.190 |

| larceny |% larcenies cleared |.262 |

| auto theft |% auto thefts cleared |.230 |

| | | |

|sentence lengthc |prison population/new commitments |2.093 |

| | | |

|policea |officers per 10,000 persons |16.634 |

| | | |

|incomed |real income per capita |3417.652 |

| | | |

|unemploymentd |% population not employed |5.487 |

| | | |

|populationd |municipality’s population/10,000 |9.01176 |

| | | |

|minorityd |% population that is non-white |14.926 |

| | | |

Sources:

aFBI’s Uniform Crime Report (UCR)

bunpublished, obtained through personal contact with the FBI

cStatistical Abstract

dU.S. Census of the Population

Table 2 - Pooled Results of Crime Function

| |Index |Murder |Rape |Assault |Robbery |Burglary |Larceny |Auto |

|Constant |-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) |

| | | | | | | | | |

|P(arrest) |-.203‡ |-.279‡ |-.190‡ |-.289‡ |-.284‡ |-.202‡ |-.098† |-.150‡ |

| |(.043) |(.108) |(.061) |(.069) |(.082) |(.050) |(.045) |(.043) |

| | | | | | | | | |

|Sentence |.103 |-.102 |-.100 |.112 |.068 |-.072 |.188 |.208 |

| |(.108) |(.235) |(.248) |(.243) |(.341) |(.146) |(.119) |(.165) |

| | | | | | | | | |

|Police |.322‡ |-.451* |.176 |1.318‡ |.680* |.566‡ |-.055 |.849‡ |

| |(.118) |(.258) |(.274) |(.268) |(.371) |(.160) |(.130) |(.180) |

| | | | | | | | | |

|Income |.401* |-.282 |.094 |1.492‡ |.109 |.265 |.582† |.469 |

| |(.234) |(.512) |(.544) |(.530) |(.731) |(.318) |(.259) |(.357) |

| | | | | | | | | |

|Unemploy |.197† |.086 |.135 |.624‡ |-.086 |.292† |.270‡ |.082 |

| |(.102) |(.223) |(.236) |(.231) |(.318) |(.139) |(.113) |(.154) |

| | | | | | | | | |

|Population |.175‡ |.417‡ |.234* |.558‡ |.165 |.287‡ |.129† |.247‡ |

| |(.054) |(.120) |(.125) |(.123) |(.169) |(.074) |(.060) |(.082) |

| | | | | | | | | |

|Minority |.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) |

| | | | | | | | | |

|[pic] |.803 |.559 |.603 |.669 |.526 |.582 |.394 |.381 |

| | | | | | | | | |

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 |.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) |

| | | | | | | | | |

|P(arrest) |-.169‡ |-.221† |-.128† |-.214‡ |-.297‡ |-.119‡ |-.069† |-.118‡ |

| |(.036) |(.096) |(.057) |(.055) |(.055) |(.037) |(.031) |(.036) |

| | | | | | | | | |

|Sentence |.048 |.032 |.035 |-.108 |.062 |-.069 |.090 |.268† |

| |(.098) |(.229) |(.244) |(.200) |(.269) |(.116) |(.089) |(.139) |

| | | | | | | | | |

|Police |.470‡ |.640† |.309 |1.336‡ |1.292‡ |.786‡ |.346‡ |.878‡ |

| |(.126) |(.282) |(.299) |(.272) |(.369) |(.160) |(.125) |(.185) |

| | | | | | | | | |

|Income |.255 |-.663 |.036 |1.178† |-.131 |.056 |.269 |.681* |

| |(.255) |(.566) |(.598) |(.553) |(.747) |(.327) |(.257) |(.375) |

| | | | | | | | | |

|Unemploy |.115 |.011 |.074 |.456† |-.103 |.230† |.205‡ |.008 |

| |(.090) |(.214) |(.228) |(.184) |(.242) |(.107) |(.082) |(.126) |

| | | | | | | | | |

|Population |.120* |.365† |.144 |.509‡ |.078 |.203† |.039 |.197* |

| |(.070) |(.146) |(.152) |(.162) |(.224) |(.097) |(.079) |(.108) |

| | | | | | | | | |

|Minority |.070† |.458‡ |.239‡ |.399‡ |.405‡ |.124‡ |-.004 |-.028 |

| |(.032) |(.068) |(.071) |(.074) |(.102) |(.044) |(.036) |(.049) |

| | | | | | | | | |

|[pic] |.810 |.581 |.624 |.684 |.543 |.586 |.365 |.412 |

| | | | | | | | | |

|Ho: var(u)=0 (ols vs. random-effects) |

|LM(3) |14.99‡ |5.39† |4.58† |24.48‡ |28.47‡ |27.39‡ |32.14‡ |22.26‡ |

|Ho: municipality effects are equal (ols vs. fixed-effects) |

|F(43,79) |1.39 |0.94 |0.80 |1.78† |1.78† |2.27‡ |2.98‡ |1.39 |

|Ho: effects and regressor are orthogonal (random vs. fixed-effects) |

|χ2(9) |12.66 |19.81† |16.97† |8.20 |2.55 |11.55 |12.99 |2.46 |

standard errors in parentheses unless otherwise noted

* significance at the 10% level; † at the 5% level; ‡ at the 1% level

( fixed-effects estimates; otherwise random-effects

-----------------------

[i]

[ii]

[iii]

[iv]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download