1. UNIVERSITY OF CINCINNATI, CARL H LINDNER COLLEGE OF BUSINESS, DEPARTMENT OF
ECONOMICS
An Empirical Examination
of the Economic Model of
Crime Using a Time Series
Approach
The Effects of Prison Sentencing Policy Changes
in Virginia on Criminal Activity
Scott Littrell
July, 2015

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1. Introduction
In Gary Becker’sinnovative magnumopus, Crimeand Punishment:An EconomicApproach,he
positedthatindividualsengagingincriminalactivityare rational economicagentsseekingto
maximize theirprofit.Since then,scholarsacrossmultiple disciplineshave usedthe premisesof
thismodel oncriminal behaviortogaina more robustunderstandingbehind the motivationof
criminalsandthe role punishmentplaysinenforcinglawsand deterringcriminalbehavior.Much
of thisresearchhasfocuseduponvalidatingBecker’smodelthroughempirical research.Utilizing
the economictheorypresentedby Beckerwhichstates thatcriminalsare rational economic
agentsthat respondtoincentives andchangestocostsstructures,publicpolicyhasemerged
aimed,inpart,at deterringcriminalbehavior.Namely,muchpolicyisbuiltuponthe theorythat
greatereconomiccostswill reduce criminal behaviors;forinstance,increasingpolicingpatrolsor
creatingstrictersentencing(i.e.the costsof gettingcaught) will leadtoreductionsincrimes.
To testBecker’stheory,thispaperbuildsuponthe researchconductedinthree previous
academicstudies:Testing theEconomicModel of Crime: The NationalHockey League’sTwo-
Referee Experiment(Levitt2002), Intervention TimeSeries Analysisof Crime Rates (Sridharan,et.
Al.,2003), andTime Series Analysisof Crime Rates (Greenberg,2001). Levittexaminesthe NHL’s
attemptto decrease the number of penalties byincreasingthe numberof refereesfromone to
two. The approachesutilizedcompare the numberof penaltieswithone refereetothe number
of penaltieswithtworeferees. Sridharan,et.al.utilizedatime seriesapproachtoexamine the
effectsof Virginia’sabolishmentof parole in1995 by usingdata from1984 through1998. The
modelsusedwere anARIMA and a structural VARto compare the pre-interventionperiodtothe
post-interventionperiod.GreenbergtestedBecker’stheory byexaminingthe relationship

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between homicideandrobbery ratesandunemploymentrates overtime byusinganerror
correctionmodel (ECM).
The goal of thispaper isto make a contributiontothe previousresearchby empiricallytesting
Becker’stheory bywayof a time seriesapproach. Byexpandinguponthe workof Sridharanet
al.(2003), thisstudywill analyze the Virginialegislature’sdecisiontoabolishparole forfelony
offendersin1995. Thisstrictersentencingpolicyisanattempttodetercriminalsfrom
committingcrimesbyincreasingthe costof gettingcaught,intermsof time servedinprison.
The economictheorywill be tolookat criminal behaviorthroughanexpectedutilityfunction.
The empirical frameworkwill be tomake a predictionfromthe beginningof the intervention
and compare it tothe actual trend. ARIMA modelswill be usedalongwithastandard ordinary
leastsquares (OLS) approachforpreliminaryanalysis.Afterthe statistical analysiswe will revisit
the expectedutilityfunctiontojustifythe empirical resultsthroughacriminal’sresponseto
economicincentives.
2. Data
The data utilizedinthispaperisdrawnfromthe UniformCrime ReportingStatistics(UCRData
Online) whichcollectsandreportstime seriesdataona monthlybasis.The analysiswillutilize
the followingcrime rates asthe dependentvariables:burglary,rape,theft,larceny,autotheft,
murder,assault,andpropertycrime. The unemploymentrate forVirginiaandthe Consumer
Price Index (CPI) statisticswere collectedfromthe Federal Reserve EconomicData(Economic
ResearchDivision). See Table 1fora full descriptionof the variables.

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Table 1: Variable definitions
Definition of Variables
Murder Rate Monthlymurderrate inVirginiabetween1980 and2013
Rape Rate Monthlyrate of rapesin Virginiabetween1980 and 2013
Robbery Rate Monthlyrobberyrate in Virginiabetween1980 and 2013
Assault Rate Monthlyassaultrate in Virginiabetween 1980 and 2013
Property Crime Rate Monthlypropertycrime rate in Virginiabetween1980 and 2013
Burglary Rate Monthlyburglaryrate inVirginiabetween1980 and2013
Larceny Rate Monthlylarcenyrate in Virginiabetween1980 and 2013
GTA Rate Monthly rate of auto theftsinVirginiabetween1980 and 2013
UnemploymentRate Virginianunemploymentrate
CPI National ConsumerPrice Index
3. EconomicFramework
Thispaperis builtfromthe theoretical frameworkof Becker(1968),specificallythe expected
utilityfunction asitrelatestocriminal behavior. A utility-maximizingcriminal isexpectedto
commitcriminal acts until theirmarginal benefit(the probabilityof profitingoff of the crime)
equalshismarginal cost(the probabilityof gettingcaughtmultipliedbythe expectedcostof the
penalty).Thisisshowninequation1.
Equation (1) pC = B
Where p isthe probabilityof gettingcaught,Cisthe expectedcostof the penalty,andBis the
immediate benefitof committingthe crime whethertheyare monetaryornon-monetary. From
this,a criminal’sexpectedutilitycanbe derivedasthe probabilityof benefitingplusthe
probabilityof gettingcaughtandpayingthe penalty.Thisisillustratedinequation2.
Equation(2) EU = pU(W+ B) + pU(W + B – C)

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Where U isthe criminal’sutilityfunctionandWisthe criminal’sinitialwealth. Thismeansthatif
the payoff of committingthe crime isgreaterthanthe criminal’sinitial wealth minusthe costof
gettingcaught, thanit isworth committingthe crime.
A shortcomingof thisapproachis thatnot all criminalsare rational whenweighingtheir
marginal benefitversustheirmarginal cost.However,intheirmindtheymaybe behaving
perfectlyrational.If theyperceivethat theyare notlikely togetcaught thentheywill commit
the crime eventhoughthere are increasedpolice patrols.If theyperceive thatevenif theyget
caught the penaltywill notbe toosevere,thentheyare likelytocommitthe crime eventhough
the penaltymay be severe. Differencesinhow acriminal definestheirutilityfunctioncompared
withhowpolicymakersmightexpectutilityfunctionstobe definedamongcriminalslikely
affectsthe effectivenessof suchpolicy.The keyprincipletoconsideris how one definesutility
and weightstheirexpectedutility.If the resultsof this paperdonotsupporta stricter
sentencingpolicy, itmeansthe criminalsdonotperceivethispolicyasbeingmajorfactorin
whethertheycommitacrime or not. Perhapstheyare unaware of the increasedsentencing
policywhenmakingtheirdecisiontocommitacriminal act.
Anotherpotential drawbackof Becker(1968) liesinthe satisfactionsome criminalsreceive by
committingcrimesandare not doingitfor any monetarygain.Rape,murder,andassaultare
examplesof crimesthatcriminalsmayindulge evenwhenthere isnoincrease inmonetary
wealth.The utilitytheygainis implicitandtheircompulsionsmaybe an evenbiggerincentive
than increasingmonetarywealth. Assumingthata percentage of the criminal populationfeelsa
sense of enjoymentfrom committingcrimes,itcanbe difficulttocontrol because of the
compulsive nature of the criminals.

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4. Empirical Methodology
To empiricallyexamine the impactof Virginia’sabolitionof parole on criminal activity,the
utilizationof various timeseriesapproachesare necessary.Greenberg(2001) proposedthatthe
rate at whichindividual i commitscrimes yi isa functionof motivationandopportunityattime t.
Thisrelationshipcanbe showninthe formof a regressioninequation3.
Equation (3) yit = α + β1(opportunity) +β2(motivationit) +eit
Greenberg(2001) recognizedthatpeople have savings,unemploymentbenefits andcanreceive
governmentassistancewhichwill affectthe motivation.Additionally,if peopleare home more,
thenthat takesawaythe opportunityof apotential burglary,whichcandecrease crime.
Greenbergsuggeststhe motivationeffectof unemploymentis laggedforayear at t – 1 whichis
representedbyUt-1.Thisisrepresentedinequation4.
Equation (4) yit = α + β1Ut + β2Ut-1 + et
Sridharan etal. (2003) contributedtothe methodological debate of testingcrime ratesover
time throughthe contextof interventionanalysis.Aninterventiondummyvariable wasutilized
to testthe impact of the policy,represented by Ιt,whichequalszerobefore the fixedtime
periodand1 after.The regressioncoefficient δmeasuresthe change inthe meanof crime rates
afterthe intervention. The variable yt isthe time seriesof crime rates, x̕t isa k Χ 1 vectorof
explanatoryvariablesandβisa k Χ 1 vectorof the regression coefficients.Thisisillustratedin
equation 5.
Equation (5) yt = α + x̕t β + δ Ιt + et
Since time series datatendtohave serial correlationproblems, the ARIMA methodandthe VAR
methodwasutilized totestthe crime rate trendsbefore andafterthe intervention. Forthe

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purposesof this paper,the OLS andthe ARIMA techniquesusedby Sridharanetal.(2003) will
be utilized before the legislationandaftertotest the effectiveness. Before anytechniqueisto
be implemented,the orderof integrationmustbe discovered. Bylookingata line graphof the
time series,itisapparentthatthere isa downwardtrendforeverycrime variable. Thiswill
require differencingtomake the variablesstationary. Afterthe variableshave beendifferenced
appropriately,the modelscanbe built andthe significance of the policyinterventioncanbe
examined.Sirdharanetal.(2003) onlyuseddata up until 1999 because of availability,sothis
study will use datathrough2013.
5. Descriptive Analysis
To beginitis useful tocompare the meansof the pre-interventionperiod(1980-1995) to the
post-interventionperiod(1995-2013). Fromthe results,all of the crime rates have declined
post-intervention,exceptforthe assaultrate,robberyrate,andthe autotheftrate whichhad
small gains.The magnitude of the interventioncoefficient,whetherpositiveornegative,is
small,meaningthateffectsare notmajor. The strongestreductionappearstobe burglary and
propertycrime.Table 2 showsthe descriptive statisticsof the entire timeseries,the pre-
interventionperiod,andthe post-interventionperiod.

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Table 2: Descriptive Statistics
Descriptive Statistics
Variable Period N Mean St. Dev. Min. Max.
Murder Rate
1980 to 2013 408 0.55 0.17 0.21 1.08
1980 to 1995 181 0.67 0.14 0.33 1.08
1995 to 2013 229 0.65 0.13 0.33 1.08
Rape Rate
1980 to 2013 408 2.13 0.48 0.91 3.25
1980 to 1995 181 2.32 0.50 0.91 3.25
1995 to 2013 229 2.30 0.50 0.91 3.25
Robbery Rate
1980 to 2013 408 8.74 2.15 3.49 14.34
1980 to 1995 181 9.90 1.78 6.28 14.34
1995 to 2013 229 9.97 1.70 6.28 14.34
Assault Rate
1980 to 2013 408 13.32 2.66 7.00 21.00
1980 to 1995 181 14.02 2.26 9.00 21.00
1995 to 2013 229 14.40 2.28 9.00 21.00
Property Crime Rate
1980 to 2013 408 269.70 58.86 137.80 420.50
1980 to 1995 181 318.70 32.50 256.50 420.50
1995 to 2013 229 313.80 32.81 246.00 420.50
Burglary Rate
1980 to 2013 408 51.30 19.29 21.91 118.60
1980 to 1995 181 69.36 13.52 44.22 118.60
1995 to 2013 229 64.95 14.88 39.77 118.60
Larceny Rate
1980 to 2013 408 199.40 39.15 109.70 303.50
1980 to 1995 181 228.40 24.95 174.80 303.50
1995 to 2013 229 227.40 24.40 174.80 303.50
GTA Rate
1980 to 2013 408 19.04 5.74 6.25 33.68
1980 to 1995 181 20.90 5.44 11.96 33.68
1995 to 2013 229 21.47 5.08 11.96 33.68
Consumer Price Index 1980 to 2013 408 158.40 43.99 78.00 234.70
Virginai Unemployment Rate 1980 to 2013 408 4.75 1.34 2.10 7.90
*UniformCrime ReportingStatistics - UCRData Online
*Federal Reserve EconomicData - EconomicResearchDivision
6. Regression Analysis
To understandthe impactof the intervention,itisnecessarytomodifyequation(5) toaccount
for seasonalityandmultiple explanatoryvariablestoforma regression model withseasonal
effects.Thisis illustratedin equation6:

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Equation(6) yt = α + δ Ι t + ϒ st + β1 x̕t + β2 x̕t + et
The dependentvariable yt will be representedbythe particularcrime rate beingexamined.
Seasonalityisrepresentedby st andϒ representsthe coefficientsof the seasonaldummy
variables. The explanatoryvariablesrepresentedby x̕t will be representedbyak Χ 1 vectorof
the unemploymentratesinVirginiaanda k Χ1 vectorof the CPI foreach monthfrom1980 to
2013. The interventionbeginninginJanuary1995 will be representedby Ιt.
Since the variablesare notstationary,there couldbe spuriousregressionproblems whichcan
overstate the results.Eachvariable wasdifferencedonce andbecame stationary. The approach
utilizedinthisanalysisistobeginwithalinearregression model of the crime rate againstthe
interventionvariable andseasonal dummyvariables totestthe significance of the intervention.
Next,unemploymentisaddedtothe equationandthenthe CPIisaddedto see if otherfactors
affectcrime and whatimpacttheyhave on the significance of the intervention. The coefficients
for the interventionhadasmall negative relationshipto assaultrate,robberyrate,andthe auto
theftrate,the remainingcrime rateshave a positive relationship. The p-valueswere not
significantin anyof the crime rates. Whenunemploymentandthe CPIwere addedtothe
model,the intervention remainedstatisticallyinsignificantineachof the crime rates.The
coefficientsandthe R-square onlyslightlychangedasmore explanatoryvariableswere added.
In summary, the resultsdonotindicate thatany of the crime ratesare significantlyaffectedby
the intervention.The additionof the unemploymentrate andthe CPIalsohad no effectonthe
intervention.Table 3illustrates the regressionresultsasmore explanatoryvariablesare added.
Figure 1 showsa graphical illustrationof the regressionresults.Inthe graphs,notice how the
murderrate hada spike atthe interventioninsteadof the expecteddrop.Eachof the other
crime ratesdroppedsharplyinlieuof the start of the interventionandthensharplyincreases

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immediatelyafterthe initial shock.Despitethe initial shock, all of the crime ratesremained
stable andlargelyunaffectedbythe stiffersentencingpolicy.Serialcorrelationissues canmake
OLS results lookmore significantthantheyactuallyare andtherefore these findings maybe
misleading.
Table 3: Regressionresults
Figure 1: Graphic representationof the effectivenessof the intervention.
Coefficient Std. Error t-stat p-value R2
int + seas 0.005 0.036 0.132 0.896 0.346
int + seas + unemp -0.007 0.039 -0.184 0.855 0.359
int + seas + unemp + cpi -0.008 0.040 -0.188 0.852 0.359
int + seas 0.026 0.132 0.195 0.847 0.376
int + seas + umemp 0.035 0.145 0.244 0.809 0.376
int + seas + unemp + cpi -0.008 0.146 -0.054 0.957 0.414
int + seas -0.065 0.290 -0.223 0.825 0.576
int + seas + umemp -0.102 0.319 -0.319 0.752 0.577
int + seas + unemp + cpi -0.097 0.331 -0.293 0.771 0.577
int + seas -0.006 0.025 -0.248 0.805 0.642
int + seas + umemp -0.017 0.027 -0.620 0.540 0.653
int + seas + unemp + cpi -0.019 0.028 -0.684 0.499 0.655
int + seas 0.363 3.004 0.121 0.905 0.885
int + seas + umemp -0.017 3.303 -0.005 0.996 0.886
int + seas + unemp + cpi -0.576 3.397 -0.170 0.867 0.888
int + seas 0.339 0.873 0.389 0.700 0.721
int + seas + umemp 0.361 0.961 0.376 0.710 0.721
int + seas + unemp + cpi 0.137 0.979 0.140 0.889 0.731
int + seas 0.285 2.288 0.125 0.902 0.889
int + seas + umemp 0.152 2.518 0.060 0.952 0.889
int + seas + unemp + cpi -0.260 2.592 -0.100 0.921 0.891
int + seas -0.262 0.432 -0.606 0.549 0.739
int + seas + umemp -0.530 0.460 -1.153 0.257 0.756
int + seas + unemp + cpi -0.453 0.473 -0.959 0.345 0.761
Burglary Rate
Larceny Rate
Auto Theft Rate
Estimated Interventions for Regression Models
Murder Rate
Rape Rate
Robbery Rate
Assault Rate
Property Crime Rate

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7. Seasonal ARIMA model
The ARIMA model utilizedisdevelopedbygoingbackto the start of the interventionperiodto
make a predictionandthencomparingitto the actual resultsof the time seriesmodel. Since
seasonalityisincludedinthisstudy,the ARIMA (p,d,q)(P,D,Q) model wasutilized. The firstthing
to do whenbuildinganARIMA model isto checkthe orderof integration.Everycrime rate was
integratedatorderI(1) exceptforthe murderrate, whichwasstationarypriorto the
intervention. The ACFandPACFtestswere usedtodeterminethe appropriate ARandMA
terms,whichservedasa startingpoint. Sridharanetal.(2003) usedthe “airline model”whichis
ARIMA(0,0,1)(0,0,1) foreverycrime rate. The approach for thisanalysiswastopickthe best
model foreach of the crime rates.Using the resultsof the ACFand PACF testsas a starting
point, the lagselectionprocesswasotherwise donebyatrial anderror method.Several
variationsof (p,d,q)(P,D,Q) weretestedandthe bestmodel waschosenbylookingatthe AIC
and BIC.
Once the appropriate ARIMA modelswere built,the timeperiod wasnarrowedto between
1990 and 2000 inorder to examine the trendandpredictionmore closely.A predictionof 36
periods(3years) wasimplemented atthe startof the interventionandcomparedtothe actual
crime rate trend.Thisprocesswasrepeatedforeachof the crime rates.
The resultsshowthat the murderrate was the onlycrime that beganto decrease significantly at
the start of the intervention. The trendsof the remainingcrime rates are unaffectedbythe
intervention. Withthe exceptionof the rape rate and the assault rate,each of the crime rates
appearto be alreadyona downwardtrendbefore the interventionbegan. The murderrate is
the onlyvariable significantlyaffected atthe startof the intervention.There isnoevidence of
the interventionbeingeffective for anyof the othercrime rates. Whenthe interventiondummy

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variable wasremovedfromthe model,the predictionsdidnotsignificantlychange. Fromthis,it
can be concluded thatthe interventionisnot highly significant.Figure 2illustratesthe prediction
versusthe actual trendfor each of the crime rates.
Figure 2: ARIMA predictions

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8. Conclusion
Whenexaminingthe change inmeansfromthe periodbeforethe intervention(1980-1995) to
the periodafterthe intervention(1995-2013), the policyseemstobe effective.Withthe
exceptionof the assaultrate,eachof the crime rates decreasedafterthe intervention. When
studyingthe effectivenessof the interventiononthe crime rates,aregressionmodel with
seasonal effectswasapplied.The interventionhad nosignificanteffecton anyof the crime
rates. Afteraddingthe CPIand the Virginiaunemploymentrate asexplanatoryvariables,the
interventionstill wasnotsignificant.Fromthese findings,itcanbe concluded thatthere are
otherfactors that affectreportedcrimesmore thanthe intervention.Since timeserieshasa
tendencytohave serial correlation, the OLSmethodmayhave misleadingresults.Soa seasonal
ARIMA model isanappropriate methodtoevaluate the policyintervention.Aftermakinga
predictionfromthe pre-interventionperiodandcomparingtothe actual trends,the murder
rate appearsto be the onlytype of crime that was significantlyaffectedbythe intervention.
Whenremovingthe interventiondummyvariable,the resultsremainedalmostidentical.
In lightof the findingsinthisanalysis,itappearsthat Virginia’sdecisionto eliminate paroledid
not have a significanteffectonreducingreported criminal activity.The only crime rate thatwas
significantwasthe murderrate inthe ARIMA model.Whenexaminingacriminal’sexpected
utilityfunction, itcanbe concluded thatthe cost of gettingcaughtis notenoughof a deterrent
for criminals tostopcommittingcrimes orthe criminalsdonothave enoughinformationto
change theirbehavior.Hence acriminal’smarginal benefitisgreaterthanthe marginal costs.
The evidence of thispolicy analysisleadstothe conclusionthat strictersentencingisnotan
effectivestrategy forreduce criminal activity.

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Becker,Gary S. "Crime andPunishment:AnEconomicApproach." Journalof PoliticalEconomy J POLIT
ECON 76.2 (1968): 169. NationalBureau of EconomicResearch.Web.8 June 2015.
Greenberg,DavidF."Time SeriesAnalysisof Crime Rates." Journalof QuantitativeCriminology 4thser.17
(2001): 291-327. ResearchGate.Web.8 June 2015.
Sridharan,Sanjeev, SuncicaVujic,andS.j.Koopman."InterventionTime SeriesAnalysisof Crime
Rates."Tinbergen InstituteDiscussion Paper 2003-040/4 (2003): 1-33. SSRN JournalSSRN Electronic
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Levitt,StevenD."Testingthe EconomicModel of Crime:The National HockeyLeague'sTwo-Referee
Experiment."Contributionsin EconomicAnalysis& Policy 1.1 (2002): n.
pag. Http://bfi.uchicago.edu/price-theory.GaryBeckerMiltonFriedmanInstituteforResearchin
Economicsat the Universityof Chicago,2002. Web.26 July2015.
US. Bureauof Labor Statistics, UnemploymentRatein Virginia [VAURN],retrievedfromFRED,Federal Reserve
Bank of St. Louishttps://research.stlouisfed.org/fred2/series/VAURN/,August2,2015.
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"UniformCrime ReportingStatistics." UniformCrimeReporting Statistics.N.p.,n.d.Web.02 Aug.2015.
<http://www.ucrdatatool.gov/Search/Crime/State/StatebyState.cfm>.