do individuals “strategically divorce” to maximize marital ... · as the knowledge surrounding...

DoIndividuals“StrategicallyDivorce”ToMaximizeMaritalSettlements?

JamesFanSID20352777

UsingdatatakenfromtheAlamedaCountydistrict,Itestthehypothesisthatindividualsplantheirdivorcefilingsaroundthe10‐yearmarkwhereeconomicallypaymentscanlastalifetime.Ifirstobservegraphicevidencewithindistributionsforthistheoryof“strategicdivorce”,thenuseKolmogorov‐SmirnovTeststoverifytheirdistributionaldifferences.Ithenconducttimeseriesregressionstocheckforthestatisticalsignificanceatthe10‐yearmark,lastlyconstructinghazardratesaftereachyearupuntil20.Iconcludethatthereissignificanceatthe10‐yearmarkbutneithergenderexhibitsaparticularbehaviornorasawholethatwouldindicatetheexistenceof“strategicdivorce”.

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1 Introduction

Astheknowledgesurroundingdivorceanditsprocessbecomesclearer,the

economicssurroundingdivorcelawplaysalargerrolealso.Californiamaritallaw

countsamarriagelasting10yearsorlongerasalong‐termmarriage.Inthesecases

thespouseearninglessincomehastherighttoalimonypaymentsandsupportto

maintaintheirstandardoflivingforaslongastheyneedorforaslongasthepaying

spouseiscapableofpaying.Thetheoryof“strategicdivorce”,orindividualsfiling

unilaterallyfordivorcetoeitherextractoravoidalargesettlement,becomesan

importanteconomicthreat.

Thereisnoliteraturecurrentlypublishedexploringtheincidenceofthis

theory,howevertheconceptssurroundingtheeconomicsofdivorcehavebeen

coveredinparticularbyStevenson(2007).BetseyStevensonhascovereddivorce

lawaswellastheeconomicssurroundingmarriageextensively,andinher2007

work“TheImpactofDivorceLawsonMarriage‐SpecificCapital”shetracksthe

courseofdivorcelawchangesinthe1970sand1980sasamajorinfluenceon

householdbehavior.Stevensonfindsasignificantdecreaseininvestmentmarried

couplesspendtowardsmarriagespecificcapitalsuchasspousaleducation,children,

andhouseholdspecialization,whileinvestmentinhomeownershipitselfvaries

basedonlocalpropertylaws.Herpaperfocusesonthehousehold’sdecisionmaking

pertainingtosecuringindividualsagainstadissolutionthough,andnottheactual

motivationstowardsusingunilateraldivorceasameansoffinancialgain.

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Thispaperexaminesthestatisticssurroundingdivorceratesinvarying

windowscenteredatthe10‐yearmark,wherefollowingthatmilestone,marriage

dissolutionsettlementsmayresultinlifetimesupport.Inthispaper,varying

questionswillbeconsideredtofindevidenceoforagainstthetheoryof“strategic

divorce”.Thepaperfirstexaminesgraphicalrepresentationsoftherawdataacross

demographicsaswellaswithinvaryingtimesegments.Thepaperthenapplies

statisticalinferenceteststocomparethedistributionsacrossgroupsandcheckfor

statisticallysignificantvariations.Usinganyevidenceof“strategicdivorce”present

hereIusetimeseriesteststotestthedataforotherevidenceofsignificantchanges

inthecumulativedivorcefilingobservationsaround10years.Finally,Iestimate

hazardratesandthelikelihoodsofdivorceovertimeandasaproductofthe

numberofkidsfromthecurrentmarriage.

Thepaperisstructuredasfollows:thesecondsectionwillpresentpertinent

literatureonthetopicandotheracademicinsight.Thethirdsectionwilldescribe

thestatisticalandeconometricmethodologyused,aswellashypothesizingwhat

shouldbeexpectedfromthedatacollected.Thefourthsectionwillanalyzethe

empiricalresults.Afinalsectionwillconcludethepaperanddiscussextensionsand

applications.

2 LiteratureReview

Asaforementioned,literatureontheactual10‐yearmarkasaneventof

interestdoesnotexistineconomicwriting.InStevenson’s2007work,“Divorce‐Law

Changes,HouseholdBargaining,andMarriedWomen’sLaborSupplyRevisited”she

observesvaryinghouseholdcapitalvaluesasdecisionstakentohavedifferent

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implicationsforthemarriageasoneunitandthevaluethespouseputsinwerethe

marriagetobedissolved.Thedevelopmentofunilateraldivorceasthekeyevent

Stevensonusesinthispaperallowshertoexaminehowmarriedcoupleswill

actuallyinvestlessincapitalsuchaschildrenortheotherspouse’smarket‐based

skills,asthoseconsumptiongainsarenon‐rivalwhenthemarriagestandsbut

wherethehouseholdtodissolvethereturnsontheinvestmentmayfallsignificantly.

AsavaluablehistoricalreferencepointformyresearchStevenson’sfindingsthatas

investmentlevelsfellforjointcapital,reasoningforthisbecomesmoresignificant

whenexamininganevenmoreextremeconceptsuchas“strategicdivorce”.

Stevensonpositsthattheknowledgethateachspouseislesslikelytoreapthe

benefitsofmarriage‐specificcapitalreducedtheincentivetoinvestjointly.

Stevensoncontinuestoexploretheeffectsofdivorcelawchangesonfemale

laborforceinanother2007work,“Divorce‐LawChanges,HouseholdBargaining,

andMarriedWomen’sLaborSupplyRevisited”.Buildingonherpreviouswork

Stevensonfindsthatwhileinvestmentinjointcapitallargelydecreasedasaresultof

70‐80slawchanges,theincidenceofunilateraldivorceledtoariseinmarriedand

unmarriedwomen’slaborforceparticipation.AnumberofStevenson’spapers

becomerelevantinmyworkaswell,particularlyherexaminationofthemarital

shifttowardslessjointinvestment,andafemalemovementtowardshoningmore

individualskillsaswell.Wherestereotypicallymenhaveinthepastbeenthe

breadwinnersforhouseholds,Stevenson’sfindingsofferinterestinginsightsinto

whetherthisideastillholdsaroundthe10‐yearmarkbasedongendercomposition.

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Friedberg(1998)examinestheactualchangeindivorceratesfollowingthe

introductionofunilateraldivorceasanoption.SynthesizingworkdonebyH.

ElizabethPeters(1992)andDouglasW.Allen(1992)whichcoveredacrosssection

ofwomenobservedin1979,Friedbergfindsthatdivorceratewouldhavebeen

about6%lowerin1988ifnotypeofunilateraldivorcehadbeenadoptedinstates

thatswitchedtoitafter1968.Friedbergsuggeststhatthemovetowardsunilateral

divorceaccountedfor17%oftheincreaseindivorceratesbetween1968and1988

witheffectsondivorcebehaviornotbeingashockbutratherpermanent.

Wolfers(2003)alsoexaminestheeffectofunilateraldivorcelawsondivorce

rates,findingthatwhilethedivorcelawchangescausedanoticeableriseindivorce

ratesforaboutadecade,thatchangewassubstantiallyreversedoverthenext

decade.Wolfers(2003)notesthattheseradicalchangesofferlittleexplanationfor

theriseinthegeneraldivorcerateoverthepasthalfcentury.Thepaperfinally

concludesthattheresultsarelikelysuggestiveofspousesbargainingwithin

marriagesaseachpartner’sdivorcethreatisnowapossibility.

Rasul(2005)presentsaninterestingquestiontoexplorewithmydatasetas

well,examiningtheeffectsofunilateraldivorcelawsondivorceratespertainingto

thosealreadymarriedatthetimeoftheirintroduction.WhileRasul(2005)confirms

mostexistingliteraturethatthelargerdivorcerateshavebeenraisedasaresultof

unilateraldivorce,thefindingsofhispaperalsosuggestthatthechangetounilateral

divorcecancausethosemarriedtobebettermatchedthanthosemarried

previouslyundermutualconsentdivorcelaws.Iapplytheconceptspresentedby

Wolfers(2003),Friedberg(1998),andRasul(2005)regardingoveralldivorcerates

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tocheckforchangesindivorceratesbetweentheoverallsampleandthetime

framesofinterestaroundthe10‐yearmark.

Thedirectlypertinentlegalhistoryformingthebasisforthispaper’s

questioncanbefoundasfarbackasMarvinv.Marvinin1976.Thecaseestablished

theframeworkbywhichmarriagelawyerswouldhavetounderstandtherightsof

unmarriedcohabitants,statingthatinthecaseofnomaritalcontractexisting,the

commonlawruleappliedmeaningeachindividualwouldtakebackfromthe

relationshipwhattheyhadbroughttoit.Marvinv.Marvin’sprecedentledeventually

tothe1986case,MarriageofBukatywhichestablishedthatpriorcohabitationand

evenapriormarriagemaynotbefactoredintheawardingofspousalsupport.

Similarly,theevenmoresignificantcaseMarriageofPrietschandCalhounin1986

producedarulingthatifasupportedspouse’sincomeandneedshadnotchanged

sincethepriororderthanitwouldnotbeappropriatetoterminatetheorder.

ThefirstcodedirectlyrelevanttothispaperisCaliforniaFamilyCodeSection

4336.ThecodestatesthatpostJanuary1,1988long‐termmarriagesaresubjectto

thecourtretainingjurisdictionindefinitelyincasesofdissolutionandanychanges

inthespousalsupportstructure,includingtermination.Thesectionthencontinues

tostatethatamarriageof10yearsorlongerwillbetermedalong‐termmarriage

unlessprovenotherwisebasedonfactorssuchastheseparationdate,etc.Thiskey

lawestablishedthedefinitionforalong‐termmarriageinCaliforniaaswellashow

thejudgescanmaketheirrulingsregardingdissolution.Thesecondlawfollows,as

CaliforniaFamilyCodeSection4337.Continuingonlong‐termmarriages,thecode

statesthatunlessotherwiseagreeduponbythepartiesinquestion,theobligationof

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onepartytopaysupporttotheotheronlyterminatesupondeathofeitherpartyor

theotherparty’sremarriage.

3 Statistical/EconometricMethodology

Thepaper’sprimaryfocusistoanalyzesignificantstatisticalquestions

surroundingdistributionsofthelargedissolutionfilingscollectionaswellasthe10‐

yearmarkofinterest.Inparticular,thispaperfocusesontheincidenceofdivorce

ratesaround10‐yearsandthedemographicssurroundingtheprobabilityofdivorce

asahazardrate.

ExtensivelymentionedbyStevenson(2007,2007),Wolfers(2003),Rasul

(2005),andFriedberg(1998)istheuseoftheeconomicprincipleoftheCoase

Theoremwhenlookingatdivorcerates.Theconceptsuggestedbylargeamountsof

empiricalliteraturethatfocusedonthechangesindivorceratestaketheCoase

Theoremtobeanullhypothesis,thatachangeintheallocationoftherightto

divorceshouldhavenoeffectonratesofbothmarriageanddivorce.Inthecontext

ofthispaper,Iconsidertheconceptthatanystatisticalchangeindivorceratemay

notbesignificant.ByusingKolmogorov‐Smirnovtestsonvaryingslicesofthedata,I

lookforstatisticallysignificantvariationsindistributions.

Initially,Ipresentthestatisticsinhistogramandkerneldensityplotformas

wellasvarioustablestooffergraphicalrepresentationsandalookatwhatshapes

thedistributiontakesovertimeaswellasinshorttimeframes.Afterexaminingthe

statisticsinrawform,IuseKolmogorov‐Smirnovtestsasthecomparisonsofthe

distributionsoncescaledshouldaccountforthenon‐parametricnatureofmydata.

TheabilityoftheKolmogorov‐Smirnovtesttotakedistributionfreeobservations

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andbestestimatethedifferencesinempiricaldistributionsbetweentheadjusted

bandsoftimeofferstheconvenienceoftestinganytwodistributionswithinthe

overallsample.

Followingthis,Iusecollapsethedatasettoatimeseriessetcomposedof

divorcefilingsobservedevery146days.Thetimeframeselectedheremadeforan

evenmeasureuponthe10‐yearmarkaswellasperevery2years.Abasictime

seriesregressionisthenruntocheckforthestatisticalsignificanceofthe10‐year

markitselfinthenumberofobservationseithersideof10yearsper146‐day

period.

Lastly,Iestimatehazardratesandtheratesofchangefromoneyeartothe

nextfordivorce.Thehazardratesarecalculatedusingprobabilitydensityfunctions

usingmyobservationalpool,andahypothesized,fixeddistributionfunctionor

survivalfunction.Thehazardratesofferinsightsintowhattimeperiodsindividuals

areatgreatestrisktofilefordivorcewithspecialinterestsurroundingthe10‐year

mark.

ThispaperusesmarriagedissolutionsfiledwiththeReneC.Davidson

CourthouselocatedinOakland,CaliforniaintheAlamedaDistrict.Thedataavailable

wasfromaregisterofactionscomputersystemwithintheCourthouse’selectronic

files,meaningtheycouldonlybeaccessedon‐site.Thepapertakesdissolution

filingsfromallfourseasonalperiodsintheyearof2007withobservationsfrom

January,February,June,July,andOctober.Theyear2007wasselectedasitisan

approximatelyevendistance20yearsawayfromanumberofthepivotallate1980’s

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eventsdiscussedinsection2,particularlythepassingoftheofFamilyCode4336as

wellasthe1987casesthatprecededtheneedforsuchacode.

Thedatacollectedincludesthepetitionergender,whetherthefinaljudgment

ofdivorcewaspassed(althoughthe10‐yearstipulationappliesthedateoffillingfor

dissolutionasvalidfor10‐years),whetherthecouplehadchildrenfromthecurrent

relationship,thenumberofchildrenfromthecurrentrelationship,andthetime

frommarriagetofilling,andseparationlengthifany.

3.1 Hypothesis

The“strategicdivorce”hypothesispositedbythispaperspecificallysuggests

thattherearestatisticallysignificantbimodalclustersaroundthe10‐yearmarkof

marriageswheredissolutionshavebeenfiled.Withthefamilycodes

aforementionedinsection2andtheknowledgewithindivorcelawofspousal

supportpaymentsthatcanlastalifetimeonceamarriagebecomes“long‐term”I

hypothesizethatthereshouldbeevidencesupporting“strategicdivorce”.Basedon

gendersplitincomedatatakenfromtheUnitedStatescensus,whereincomegaps

havebeensignificantandconsistentsincetheinceptionofthecensusbythis

parameter,Ianticipateastatisticallysignificantdifferenceinthetwodistributions

bygenderaroundthe10‐yearmarkandingeneral.

Thedifficultyofpredictingwhenindividualsbegintheprocessofconsidering

divorcebeforetheybecomeawareofthe10‐yearmarkisavariablebeyondthe

scopeoftestablecontrol.Essentially,the“strategicdivorce”theoryattemptsto

capturethenotionthatpeoplewilleitherdivorceasapremeditatedprotection

againsttheirspousecollectinglonglastingpaymentsorviceversa,butifthe

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sentimentthistheoryattemptstocapturehasnoopportunitytopresentitselfthe

datamaynotreflectaccuratelythecountofdissolutionsthatactuallywereor

shouldhavebeeninfluencedby“strategicdivorce”.Ifaprimaryearningspouse

findsoutjustpastthe10‐yearmarkabouttheexistenceoffamilycodesthatthere

wastheopportunitytofilefordivorcebeforelonglastingspousalsupportmaybe

awarded,heorshemaysimplyarbitrarilyfileanytimeorevenrightafterthe10‐

yearmark.Thiswouldindeedsupportthe“strategicdivorce”hypothesis,butinthe

contextofthispaper’sfurtherintenttofinddemographicspecificitiesofwhoacts

whenandbasedonwhatfactors,anyspeculationsmaybebiased.

Similarly,akeyquestiontoconsiderthatcannotbeaccountedforwasthe

incomeearnedbythepetitionersinquestionaswellasanypotentialpropertytobe

divided.TheReneC.Davidsoncourthousepapersformarriagedissolutiondidnot

stipulatethisinformationbeprovidedonthedissolutionfilingitself.Additionally,

theregisterofactionsasturnedinbythefilersandpetitionersthemselves

frequentlyfailedtodocumenttheirincomeorinothercasesputapproximations.An

extracomponentwasthatunlikethedissolutionrecordsthemselves,which

containedthelengthofmarriageandpetitionerandrespondentname,thecourt

documentsspecifyingtheincomeandpropertysettlementswereinvaryingcases

lockedfromviewatthepublicaccessworkstation.

4.1 StatisticalDistributions

Thedatacollectedtotaled559observations,butingendersplitcases,only

534observationswereusableduetomissinginformation.Ingendersensitivecases,

thegenderreferredtothroughoutthepaperalwaysreferstothegenderofthe

10

petitionerfilingfordivorce.Referringtothetotaldatasamplefirst,thetabulated

databygenderandsplitbetweenpre‐10andpost‐10yeardivorcesshowsthatthe

numberoffemalesfilingfordivorcefromthesampleof534observationsisnearly

oneandahalftimesthenumberofmales.

[ReferenceFigure1]

Otherwise,nothingparticularlydescriptiveaboutthedifferencebetween

pre‐10andpost‐10yearmarriagesforthetwogendersisreadilyvisiblefromthe

rawcounts,howeverthestatisticaldifferencesbetweenthetwogender

distributionswillstillbetestedaswillthepre‐10andpost‐10distributions.

Examiningasliceofdatawithin2‐yearrangesbelowandabovethe10‐year

markindicatesnearlydoubletheamountoffemalesversusmaleswithinthe4‐year

band.

[ReferenceFigure2]

Thesubsamplesizeofmenwithinthis4‐yearrangeisonly37though,

makingtheinferencesdrawnformensomewhatdifficulttoplaceaccurately.

Fromvisuallyreferencingthehistogramandkerneldensityplotsofthedata

withinthis4‐yearperiod,whilethelargerweightofobservationsappearstobepre‐

10years,thereisnotabimodaldistributioneithersideof10yearsthatmightverify

thestrongestformofthe“strategicdivorce”theory.

[ReferenceGraphs1.1‐1.2]

Totestforthepossibilitythatsuchatheorymaybeinverselyrelatedper

gender,adensityplotisprovidedsplitbygenders.

[ReferenceGraph1.3]

11

Thedistributionsseemvisiblydifferentnow,whiletheplotforwomen

appearssomewhatrandomwithanearlynormallookingdistribution,themen’splot

appearssignificantlypositivelyskewed.Forlackofdescriptiveness,histogramsare

notusedinthissection.Understandingthatthekerneldensityplotwillnotcapture

theactualobservationsrightatcertainmarkers,thedensityplotallowsfor

convenient,directoverlappingoftwodistributionsandtheirtrends.

Extendingthesubsamplesizeto3‐yearrangestwoways,centeredat10

years(atotalofa6yearselection)revealsakerneldensityplotbothfarmore

skewedrightthantheoverallsamplefromthe4‐yearband.

[ReferenceGraph1.4]

Thedistributionnolongerappearsrandomized,whilethelarger6‐yearband

rangewillbecomparedtootheroverlappingandadjacentdistributionswiththe

Kolmogorov‐Smirnovtestsinsubsection4.2.Again,graphicvisualsareprovided

basedongenderaswelltocheckforevidenceofmorespecifiedpartsofthe

“strategicdivorce”hypothesis.

[ReferenceGraph1.5]

Bothsetsofdistributionsviatheirkerneldensityplotsareskewedrightthis

time,notjustthemales.Thedegreetowhichthemaledensityplotisskewed

howeverdoesseemmoreextremethanforfemalesbutagain,theextenttowhich

thetwodistributionsarethesamewillbetestedlater.Visually,itappearsboth

gendersfilemostlybeforethe10‐yearmarkwhenlookingatan8‐yearbandof

observationscenteredat10.Forvisualconveniencetheoveralldensitiesofmale

andfemalepetitionersisalsoprovided:

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[ReferenceGraph1.6]

Ingeneralthegraphsforfemalepetitionersindicatetherearenorealpeaks

indivorceclustersuntilthe6‐yearmark.Theclusterisroughlyaroundthe2100

dayspoint,approximately6years.Ahigh‐densitysetofobservationsaroundthis

pointmaybestatisticallysignificantandduetosomeactualparameter,however

withinthecontextofthe“strategicdivorce”theory,wouldhavelittlesignificance.

Thedistributionformalepetitionersseesthemostsignificantclusteraroundapoint

justbefore2100daysbutanotheronethatcanbevisiblearoundthe3000days

mark,justunder8¼yearsintoamarriage.Thisobservationmaybeofgreater

significancetowards“strategicdivorce”andwillbeinterpretedindepthin

subsection3.2.Ultimatelyfromavisualglanceattherawstatistics,itappearsthe

observationlevelsfortheoverallsampleaswellasgendersplitspeakaround6

yearsinsteadofcloserto10.

4.2 Kolmogorov­SmirnovTests

UsingKolmogorov‐Smirnovtests,Imeasurethestatisticalvariationsin

distributionsacrosssectionsofdissolutionfilingperiodsrangingfrom4‐yeartime

framesto8years.TheKolmogorov‐Smirnovtestoftheavailablestatisticaltestsis

thestrongestoptiontowardscomparingdistributionalequalities.WheretheKuiper

TestalsousesthelargestmagnitudediscrepanciesD+andD‐betweentwo

cumulativedistributionfunctions,theteststatisticsumsthetwo,makingitmore

sensitivetothetailsofthedistribution.Basedonthedistributionofthesetof

observationsasexaminedinsection3,95%oftheobservationsformalesand

femalesfallwithinanapproximate7.9‐10.7yearand7.4‐9.5yearrange

13

respectively.AdditionaltestsconsideredsuchastheAnderson‐DarlingTestare

moresuitedtotestingfordeparturesfromnormality.TheCramer‐vonMisesTestis

analternativetotheKolmogorov‐SmirnovTestandstronglyconsideredbutfor

functionalandhypothesistestingsimilaritiestotheKolmogorov‐SmirnovTestwas

notused.

Thefirstsegmentofinterestasvisuallypresentedinsubsection4.1,isthe6‐

yearbandcenteredatthe10‐yearmark.Takingtheslicefrom7‐13yearsfirstand

comparingtothe1‐7slice,the“strategicdivorce”theorywouldsuggestthatthetwo

groupsarestatisticallydifferentdistributions.Thetestresultsfiguresarepresented

asfollows:

€

1:H0 :F(x) ≤G(x) vs. H1 :F(x) >G(x)2 :H0 :F(x) ≥G(x) vs. H1 :F(x) <G(x)Combined K − S :H0 :F(x) =G(x) vs. H1 :F(x) ≠G(x)

TheDvalueistheKolmogorov‐Smirnovteststatistic,whenplotted,the

maximumverticaldistancebetweenthetwoempiricaldistributionsbeing

compared.Thenotationalformforwhattheteststatisticsrepresentisasfollows:

€

1:D1+ = sup

x(S1(x) − S2(x))

2 :D1− = sup

x(S2(x) − S1(x))

Combined K − S :D1 = supxS1(x) − S2(x)

Thefirsttestof7‐13and1‐7yearbandsareasfollows:

[ReferenceFigure3]

Thep‐valueat0withamaximumDmagnitudeof1(thehighestpossible)

suggeststhatthetwodistributionsaredifferentata1%level.The13‐19yearrange

testresultsareasfollows:

14

[ReferenceFigure4]

Thep‐valueisagain0withafairlyhighmaximummagnitudeDbetweenthe

twodifferencesof.7413.Wewouldclearlynotexpectthedistributionstobethe

sameacrossthesethreetimeframes;infactevidenceofstatisticallydifferent

distributionswouldbeanticipated.Thesignificanceatthe1%levelthatthese

distributionsaredifferenthoweverisstillstrongevidenceofsomethingparticularly

significantabouthowthedistributionschangeoncethesignificantslicecenteredat

10yearsisreached.

ThenextgroupofKolmogorov‐Smirnovtestsarefocusedon2yearadjacent

slices.Forthedatatosupportthe“strategicdivorce”theory,thetestsshouldreturn

highp‐valuesforthefirstslicestested,earlyoninthemarriagetermlengthswith

thep‐valuesexpectedtodecreaseuntilapointofpotentiallystatisticalsignificance

atthe8‐10and10‐12comparisonwherethetwodistributionsshouldbehighly

dissimilar.Beginningfromthestartofthesample,thecomparisonsforthe0‐2and

2‐4yearmarriagetermsareasfollows:

[ReferenceFigure5]

Thep‐valueismoderatelyhigh;toohighforthenullhypothesisofthetwo

distributionsbeingstatisticallythesametoberejected.Wethusfailtorejectthe

hypothesisthatthedistributionsfrom0‐2and2‐4yearsarestatisticallydrawnfrom

thesameempiricaldistribution;inessencethearestatisticallythesame.Thep‐

valueandteststatisticarebothlowerthananideallevelfortheveryfirstbandtobe

testedfrom0‐2and2‐4.Wewouldhaveexpectedstrongerevidencethatthetwo

distributionswouldbestatisticallythesame,butwithvaryinglevelsofnoisetobe

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somewhatexpectedintheearlybandsofmarriagedissolutions,thefollowingtests

maycloserfollowthepatternhypothesized.Theresultsforthe2‐4and4‐6year

bandcomparisonsareasfollows:

[ReferenceFigure6]

Theresultsforthe2‐4and4‐6yeartimeframesarenearlyexactlyas

anticipated,withthemoresignificantp‐valuedroppingveryslightly.Thereshould

beincreasingevidencethatthetwodistributionsarenotthesame,butthep‐valueis

stilltoohightorejectthenullthatthedistributionsweredrawnfromthesame

empiricaldistribution.Theteststatistichasfallenabitasthenumberofdegreesof

freedomhaschanged.Themoreimportantvalueagainisthatthep‐valuehasfallen

aswell.Thefollowingtestwasconductedoverthe4‐6and6‐8yearbands,withthe

correspondingresults:

[ReferenceFigure7]

Thep‐valueisincrediblyandunexpectedlylowforthistest.Werejectthe

nullhypothesisthenthatthesedistributionshavebeendrawnfromthesame

empiricaldistributionat5%.Thep‐valuehasdroppedfromtheprevioustestof2‐4

and4‐6yearbandsbutagain,perhapsfarmorethanwemighthaveexpectedtosee.

Ifresultsbestsupportingthe“strategicdivorce”aretofollow,thep‐valuewill

continuetodecrease,whiletheteststatisticshouldrise.Theevidencethoughatthis

stageisalreadyverystrongthatthedistributionofdivorcefilingsingeneralis

beginningtorapidlychangeinshape.Whilethisdoesnotsupportastrongcaseof

“strategicdivorce”theresultsfromthistestdofollowwhatwasvisuallyseenin

subsection4.1,thatthemostsignificantpeakbeforeachangeinshapewasatthe6‐

16

yearmarkinsteadof10.Followingthistest,theresultsofa6‐8and8‐10yearband

comparisonare:

[ReferenceFigure8]

Theresultsherearecompletelyunexpected,withthehighestobservedp‐

valueyet,aswellasthelowestteststatistic.Withsuchstrongevidencewefailto

rejectthenullthatthesetwodistributionsareactuallydrawnfromthesame

empiricaldistribution.

Again,thisinferenceisthefirstofthefourtestsconductedthusfartoprovide

evidenceagainst“strategicdivorce”,suggestingthatthemovementofthe

dissolutionfilingsisbeginningtofollowanapproximatelymarkabletrendas

opposedtostatisticallysignificantshifts.Thetestforthe8‐10and10‐12bandsthat

maybethemostcriticaltoexamineisthenperformedwiththefollowingresults:

[ReferenceFigure9]

Theresultsareagainintheoppositedirectionfromwhatwashypothesized.

Wherethep‐valuewasexpectedtosteadilydroptowardshighlysignificant

evidenceagainstthenulltosuggesttheincidenceof“strategicdivorce”,the

distributionsarestatisticallydrawnfromthesameempiricaldistribution.

Thecolossalp‐valueindicatesthatatthemostcriticaltestjunctureof8‐10

and10‐12yearrangesforevidenceof“strategicdivorce”thebandseithersideare

statisticallylikelytobethesame.Havingacolossalp‐valueheresuggeststhatthe

two2keyframeseithersideof10yearsdonothavequitethebimodalclustering

expectedrightaround10years.Theconceptthatthe10‐yearmarkisastatistically

significanteventaffectingthenumberofdissolutionsobservedistestedusinga

17

collapsedtimeseriesregressioninsubsection4.3.Atthispointtheideathatthetwo

distributionsarestatisticallydrawnfromdifferentdistributionsdoesnothold.The

10‐12and12‐14yearframesaretestedwithresultsasfollows:

[ReferenceFigure10]

Thep‐valueissmallerbutstillfartoolargetorejectthenullatanyimportant

levelofsignificance.Giventhatthe10‐12yearbandshowedageneraldownward

trendindissolutionfilingsobserved,the“strategicdivorce”hypothesiswouldbe

supportedbydownwardmovementinthe12‐14bandaswell.Suchasimilartrend

wouldbecapturedbythetwodistributionsbeingdrawnfromthesameempirical

distributionfunction.The“strategicdivorce”theoryisnotnecessarilyprovenor

disprovenbydistributionalsimilaritiesbetweentheseadjacentbands,howeverin

thecaseofthecriticalpointat10years,evenforthe8‐10and10‐12rangesshowing

similardistributions,ifthe6‐8yearsand8‐10yearsaswellasthistestshowedhigh

p‐values,wemightstillseeevidenceof“strategicdivorce”,simplynotinthebimodal

shapeexpected.Essentiallyforthosetwotestsboundedateithertheupperorlower

limitbythe10‐yearmark,mayindicatestatisticallysimilardistributionsthatrise

aroundthe10‐yearmarkandfallafterwards.Kolmogorov‐Smirnovtestswouldfind

thattheseindividualtestsusehighlysimilardistributionsbutnotnecessarilythe

factthatthereisakeytrendwithinthe4yearpre‐10andpost‐10observationrates

indissolutionfilings.TotestthisasthelastKolmogorov‐Smirnovdistribution

comparisonofthetotalsample,the6‐10and10‐14bandsaretestedwiththe

followingresults:

[ReferenceFigure11]

18

Thep‐valueof0meansata1%levelofsignificancethetwo4yearrangesare

indeedstatisticallydrawnfromdifferentempiricaldistributionfunctions.

AggregatingtheKolmogorov‐Smirnovtestsdonefor6yearbandswithonecritical

onecenteredat10,testsdonefor2yearadjacentbands,andthelasttestdirectly

comparing4yeardistributionseithersideof10years,afewobservationscanbe

maderegardingwhatproofthereappearsfororagainst“strategicdivorce”evident

inthesampledobservations.

The6yearbandtestsprovidedtheunderlyingfoundationforthereeven

beingacriticalpointcenteredbetween7and13years.Thenotionbehindthose

testswasthattheincidenceofanypeakorclusterofhighfrequencydissolutions

wastobefoundinthatrange.The2‐yearbandtestsweretotryandprovea

somewhatambitiouslinearitytrendindistributionslicesthatshouldhaveclimbed

steadilyuntilthe10‐yearmarkwhereaverylowp‐valuewouldhaveprovidedfairly

strongevidenceofsomeformofsharpstatisticaldiscontinuityordiscontinuities

withinthe8‐12yearframe.Thefactthattherewasinsteadstrongevidencethatthe

twobandscomparedwithinthe8‐12yearframewerestatisticallydrawnfromthe

samedistributionfunctionledtothe4‐yearbandtest.Basedonthegraphical

representationspresentedthroughoutsubsection4.1andevenfromthe8‐12year

frame,the4‐yearbandtestwasperformedtotestaweaker“strategicdivorce”

hypothesis,thatwhiletheremaynotbestrongbimodalevidencethe10‐yearmark

isstillstatisticallysignificant.Thetwodistributionsbeingdirectlysplitbythe10‐

yearmarkwereshownindeedata1%leveltobedrawnfromdifferentempirical

19

distributionssuggestingthatthesubsequentpaneldatatestmayprovidemore

conclusiveevidencesupportingorrefuting“strategicdivorce”.

Theoverallsamplethusdoesnotprovidestrongevidenceofdistributional

differencescenteredat10years.Theotherdistributionalquestionofinterest

pertainstothegendersplitswithtestsfirstrunfortheoverallgenderdistributions,

then2‐yearsplitsbeforeandafter10years,then4yearsbeforeandafter.The

resultsfortheoverallsampleareasfollows:

[ReferenceFigure12]

Thefairlylargep‐valueat.486suggeststhatthemaleandfemale

distributionsforthecollectivesamplesetarestatisticallydrawnfromthesame

empiricaldistribution.Beingthatthereare534observationsinplayherethatmay

notbeunexpected.Thetestssurroundingthecriticalpointat10yearsmayprovide

moreevidenceofonegender’stendencytofileatacertainpointoranother.The

resultsforthe8‐10yearframeareasfollows:

[ReferenceFigure13]

Thedistributionsagainshowafairlyhighp‐valueat.269indicatingthatfor

this2‐yearbandbefore10yearsneithergendersignificantlyfilesfordivorcesata

higherrate.Thisagainsupportswhatthevisualevidencefromsubsection4.1

suggested.Thetestfor10‐12yearsyieldsthefollowingresults:

[ReferenceFigure14]

Thep‐valueisincrediblyhighforthethirdstraighttest,at.872.Theevidence

isverystrongagainthattherearenostatisticallysignificantdeviationsbetweenthe

maleandfemaledistributions.Again,thevisualrepresentationsfromsubsection4.1

20

showedverylittlevariationbetweenthetwogendersthroughthis8‐12yearperiod

withthestatisticalsimilaritiesbetweenthetwodistributionsnowproven.Thelast

twotestsrunforthetwogenderdistributionscoverthelargerrangeof6‐10and

then10‐14years,whereevidencefromthelargersamplesuggestedthattherewasa

statisticallysignificantdifferencebetweenthosetwodistributions.Theresultsfor

the6‐10yearrangeformaleandfemalepetitionersisasfollows:

[ReferenceFigure15]

Againthep‐valueisverylargeat.744,withteststatisticmaximumsthatdo

notcomeclosetothedifferencesobservedwherethedistributionswouldbe

statisticallydifferent.Thefinaltestforthe10‐14yearrangeisrunwiththe

followingresults:

[ReferenceFigure16]

Optimistically,thisisthefifthstraighttestofthetwogendersthathas

suggestedneithersidefilesfordivorcesignificantlymorethantheotherduringthe

timeframesofinterest.Compoundingthevisualevidencefromsubsection4.1and

thelargep‐valuesfordistributionsobservedheresuggeststhatforthetimeseries

regressionthecollapseddataneednotaccountforgenderdifferences.

4.3 TimeSeriesRegression

Totesttheevenweakerhypothesisfrom“strategicdivorce”thatthe10‐year

markisastatisticallysignificantevent,Iuseabasictimeseriesregression.Icollapse

thedatasoitisatime‐seriesdataset–whereeachobservationisatimeperiod

insteadofadivorce.Thecollapseddatasetreorderedintoatimeseriessetisthen

bunchedinto146‐dayclustersofobservationstocapturethevelocityorrateat

21

whichdissolutionsarefiled.Thefirsttestisrunforthe6‐14yearbandwiththe

followingresults:

[ReferenceFigure17]

Thewaythedataisbunched,thecoefficientonthepost‐10yearevent

variablesuggeststhatper146‐daygroupobservedafterthe10‐yearmark,thereare

1.1778lessmarriagesobservedthanbeforethatsamemark.Thehighp‐valueat

.718aswellasaconfidenceintervaloftheestimatethatrangesfrom(‐7.977,5.621)

suggestthecoefficientislikelytobestatisticallyzero.Theadditionaltimetrend

variablehasap‐valueof.101meaningevenata10%significancelevel,thereis

almostnoeffectonthedataovertime.Anothertestisthenrunfortheoverall

samplewiththefollowingresults:

[ReferenceFigure18]

Theregressionthistimewithoutthe6‐14yeartimeconstraintreturnsap‐

valueof.001forthecoefficientonthepost‐10dummy.Thecoefficientis

significantlynegativethistimeat‐7.879,whichsuggeststhatafterthe10‐yearmark,

within146daygroupingstheaveragenumberofdivorcesfiledhasfallenbynearly

8fromwhattheobservationlevelswereatbefore10years.Theestimateis

significantat1%andprovidessomeevidencethatoverthetotalsamplesizethereis

astatisticallysignificantshiftatthe10‐yearmark.

Intermsofevidenceof“strategicdivorce”thetimeseriestestcontrolsforthe

growthovertimeofdivorcefilingssothesignificanceofthepost‐10variableis

meaningful.Proofhereofthatmarkbeingstatisticallysignificantexistsbutsuggests

onlytheweakestformofthe“strategicdivorce”hypothesis.

22

4.4 HazardRate

Thefinalapplicationtakenwiththesampledobservationsinvolves

calculatingahazardrateformarriagesbasedoneachpassingyear.Usingamodified

formulafromthetraditionalhazardrateformulathehazardrateformarriagesuses

anassumedstandardsetofmarriagesfromtime0andobserveddatainplaceofthe

probabilitydensityfunctionandthesurvivalfunction.Theformulaisasfollows:

€

Divt = probability of being divorced after time tMarrt = probability of being married after time tNt = instantaneous number of divorces in time tTt = remaining number of divorces by time t

Pr(Div1 | Marr0) =N1

(T1 − N0)

Pr(Div2 | Marr1) =N2

(T2 − N1)(1−D1)

Pr(Divt | Marrt−x ) =Nt

(Tt − Nt−x )((1−Dt−x ) × (1−Dt−x−1) ×…(1−D1))

Basedonthisformulawhichcontinuallyaccountsforthesurvivingratioof

marriages,thehazardratescalculatedareasfollows:

[ReferenceTable1]

Thehazardratesclimbtowards10years,butactuallypeakat7yearsandnot

at10.Thisfindingisactuallyexactlyinlinewiththegraphicalrepresentations

showninsubsection4.1.Thehazardratereachespeaksat7yearsand16years,

whilethenumberofdivorcesobservedpeaksinyears6and7.Again,thisismostly

inaccordancetowhatwasexpectedfromthestatisticaldistributions,wherepeaks

inobservationclustersdidnotsurround10years.

23

Thehazardrateprovidesaninformativelookatthelikelihoodofdivorce

followingacertainyearconditionalonhavingsurvivedtheyearsprior,basedonthe

instantaneouslychangingpopulationoftotalmarriagesfromtime0.Astheyears

progress,thedecreasingratioofpopulationofsurvivingmarriagesgenerallyfollows

adecreasingquantityofdivorcesobserved.Thatthebandofhazardratesnever

exceedsapeakof7.78%andaminimumof2%(withthevalleyappearinginyear1)

suggestsoptimisticallythattheremaybenoevidenceof“strategicdivorce”.The

hazardratesfrom8‐14yearsinparticularnevervarybeyonda1%rangewhichis

againpositiveevidencethatindividualsarenotmorepronetodivorcewithinthose

ranges.Theobservationlevelsofdivorcefilingsgenerallydropsduringthattime

frameaswellasfallingpastthe14‐yeartimeaswell.

Ingeneralthelowvarianceoftheobservedhazardratesandabsenceofany

realtrendwithinthe20‐yeartermcalculatedprovidesnoevidencewhatsoeverof

“strategicdivorce”.

5. Conclusions

Thecentralquestionof“strategicdivorce”intheinstanceofmarriages

observedinAlamedaCountyintheyear2007isaddressedfirstbytakinggraphical

representationsandstatisticalorderingsofthedataforvisualevidenceofany

skewnessorpeaksinthedistributionsbasedaround10years.Thelargestpeakfor

theoverallsamplewasactuallydetectedaroundthe2100daysorapproximately

the6yearmark.Uponsplittingthedistributionbygendertherewaslittlevisual

evidencethatthetwodistributionsweresignificantlydivergent,whilebothsubsets

ofobservationsfollowedtheoverallsample’sshapewithapeakaround6years.The

24

overallandindividualgenderplotsallshowedprecipitousdropsindivorce

observationsjustfollowing10yearshowever,aswellasacontinuingdownward

trendinobservations.

Usingthevisualrepresentationsindicatingthemostevidenceinfavorof

“strategicdivorce”,Kolmogorov‐Smirnovtestsareusedtocheckforthe

distributionalequalitiesofkeysegmentsofthedata.Forastrongcaseof“strategic

divorce”,itishypothesizedthatobserving2yearbandsbeginningfrom0‐2and2‐4

upthroughandjustpastthe10‐yearmark,thep‐valuesshouldsteadilyfalltowards

10yearsandthenriseagain.Suchfindingswouldindicatethatwheretheearly

movementofthesetofobservationsisconsistentandtheincidenceofdivorce

filingsshouldbeincreasing,around10yearsthereshouldbeasignificantshiftin

thedistributionofobserveddivorces.Thefindingsinsteadsuggestedthatatthe

mostcriticalpointof10years,distributions2yearsaboveandbelowthismarkare

statisticallydrawnfromthesameempiricaldistributionorthattheyarefunctionally

notdifferent.Thetestofthesetwobandsat8‐10and10‐12yearsshowedan

incrediblylargep‐value,confirmingwhatwasobservedfromthegraphical

representations.Thestrongestevidenceofdistributionsbeingstatisticallydifferent

wasthetestofthe4‐6and6‐8mark,againobservedfromthedensityplotsand

histogramsashavingapeakat6years.

Extendingtoasingulartestofdistributions4yearsbelowandabovethis10‐

yearmark,therewassignificantevidencethatthe6‐10and10‐14bandswere

distributionalverydifferent.Withthisstrongevidenceofdistributionaldifferences,

atimeseriesregressionwasrunforthis6‐14rangeandthelargersample,usingthe

25

10‐yearmarkasaneventdummy.Thetimeseriesregressionreturnedacoefficient

withtheexpectednegativesignbutwithap‐valuesuggestingitwasstillstatistically

zero.Atimeseriesregressionrunfortheentireobservationsetsuggestedthatat

the1%levelthepost‐10markwasastatisticallysignificantevent.

Thelastapplicationwasincalculatinghazardratestotryanddetermine

whencoupleswereathighestriskfordivorcingaswellastheconditional

probabilitylevelchangesfromyeartoyear.Thehighesthazardratescalculated

followedthe6th,7th,and16thyears,withthehighestobserveddivorcefilings

followingyears6and7.

Fromthegraphicalrepresentationspresentedinsubsection4.1throughthe

distributionalcomparisonsfromsubsection4.2tothedirecttimeserieseventtests

fromsubsection4.3andlastlythroughthehazardratescalculatedinsubsection4.4,

atbestonlyaweakcaseof“strategicdivorce”canbefound.Thelargestpeakas

showninallsubsectionsisactuallyat6years.Thefindingsoptimisticallysuggest

thatnotonly“strategicdivorce”likelynotpresent,inparticular,thereisno

differencecross‐genderindivorcefilingtimes.

Cruciallyhowever,itmustbeacknowledgedthatthedataset’simperfections

maynotlenditselftostrongerinferences.Thevalueofsuchdatapointsasthe

numberofkidswithinmarriagesdidnotproveusefulinhelpingjudgetheincidence

ofdivorceorsuggestanythinginterestingaroundthe10‐yearmark.Additionallyas

notedearlier,datathatcouldhaveprovenincrediblydescriptivepertainingto

maritalincome,sharedproperty,orwhatchildsupportpaymentswereset.Thedata

setalsoasawholedidnotfeatureaproportionallylargesampleofobservations

26

particularlyclosetothe10‐yearmark,meaningmosttestsofinterestdonefocused

atthatpointusedanywherefrom99‐253observations.Lastly,althoughoveralarge

enoughunifiedamountofcountiesthereshouldnotbesignificantlydifferent

findings,basedonmedianincomelevels,theremaystillbevariationslimitingthe

abilityofthesefindingstobeextended.Inareaswithhighmedianincomessayin

MarinCountyifthereisoneprimaryearner,theremaybestrongerevidenceof

“strategicdivorce”,versusalowerincomecounty.

InthecaseofAlamedaCountyhowever,theevidenceispromisingthat

economicdriversdonotplayasignificantroleinindividuals’decisionstofilefor

divorce.Inparticularwhereitmighthaveseemedlogicalfortheretobegender

differencesaswellinfilingtimes,thefindingssuggestthatthetwodistributionsalso

donotvary.

6. Acknowledgements

IwouldliketothankmyadvisorAaronEdlinforhishelpfulcommentsand

suggestionstowardsthedirectionofmyresearch.Iamalsoverygratefulto

MatthewBotschforhisinvaluableadviceandinsightintheprocessoftransitioning

myresearchintotractablehypothesesandtests.

Anyviewsexpressedareminealoneandnotnecessarilythoseofmyadvisor

ormydepartment.

27

7. References

[1]Blair,Sandra,BobPickus,andCarolAmyx.CaliforniaMaritalSettlement&Other

FamilyLawAgreements.Berkeley,CA:ContinuingEducationoftheBar,

California,1997.Print.

[2]Friedberg,Leora."DidUnilateralDivorceRaiseDivorceRates?EvidenceFrom

PanelData."TheAmericanEconomicReview88.3(1998):608‐27.

Www.jstor.org.AmericanEconomicAssociation,19Sept.2005.Web.24Apr.

2012.

[3]Laskin,Jared."AChronologicalSummaryofCalifornia"Palimony"LawSince

Marvin."UnmarriedCouplesandtheLaw.Goldman&KagonLawCorporation.

Web.25Apr.2012.<http://www.palimony.com/>.

[4]Lurvey,Ira,KathrynKirkland,andDianaRichmond.CaliforniaFamilyLaw:

PracticeandProcedure.2nded.Vol.2.MatthewBender,1994.Print.

[5]Rasul,Imran.2006.Marriagemarketsanddivorcelaws.JournalofLaw,

Economics,andOrganization22,no.1:30–69.

[6]Stevenson,Betsey."TheImpactofDivorceLawsonMarriage‐SpecificCapital."

JournalofLaborEconomics25.1(2007):75‐94.Print.

[7]Stevenson,Betsey."Divorce‐LawChanges,HouseholdBargaining,andMarried

Women'sLaborSupplyRevisited."PSCWorkingPaperSeries(2007):1‐24.

UniversityofPennsylvaniaScholarlyCommons.PopulationStudiesCenter,10

May2007.Web.25Apr.2012.<http://repository.upenn.edu>.

[8]Wolfers,Justin."DidUnilateralDivorceLawsRaiseDivorceRates?A

ReconciliationandNewResults."NBERWorkingPapersSeries(2003):1‐29.

NationalBureauofEconomicResearch,Oct.2003.Web.25Apr.2012.

<http://www.nber.org/papers/w10014>.

28

Figure1 Pre-10 Years Post-10 Years Total Female 184 132 316 Male 118 100 218 Total 302 232 534

Figure2

Pre-10 Years Post-10 Years Total Female 35 27 62 Male 21 16 37 Total 56 43 99

Graph1.1 Graph1.2

Graph1.3

29

Graph1.4

Graph1.5

30

Graph1.6

Figure3Smaller Group D P-Value Corrected

1 0.0000 1.0000 2 -1.0000 0.0000

Combined K-S 1.0000 0.0000 0.0000 Figure4Smaller Group D P-Value Corrected

1 0.7413 1.0000 2 -1.0000 0.0000

Combined K-S 0.7413 0.0000 0.0000 Figure5Smaller D P-Value Corrected

31

Group 1 0.0952 0.607 2 -0.1973 0.117

Combined K-S 0.1973 0.234 0.176 Figure6Smaller Group D P-Value Corrected

1 0.1797 0.110 2 -0.0025 1.000

Combined K-S 0.1797 0.219 0.168 Figure7Smaller Group D P-Value Corrected

1 0.0007 1.000 2 -0.2978 0.001

Combined K-S 0.2978 0.002 0.001 Figure8Smaller Group D P-Value Corrected

1 0.1299 0.335 2 -0.0503 0.849

Combined K-S 0.1299 0.645 0.572 Figure9Smaller Group D P-Value Corrected

1 0.0864 0.696 2 -0.0743 0.764

Combined K-S 0.0864 0.993 0.988 Figure10Smaller Group D P-Value Corrected

1 0.2062 0.184 2 -0.1018 0.662

Combined K-S 0.2062 0.367 0.288 Figure11Smaller Group D P-Value Corrected

1 0.6015 0.000 2 0.000 1.000

Combined K-S 0.6015 0.000 0.000 Figure12Smaller D P-Value Corrected

32

Group 1 0.0736 0.247 2 -0.0081 .983

Combined K-S 0.0736 0.486 0.444 Figure13Smaller Group D P-Value Corrected

1 0.0667 0.890 2 -0.2762 0.135

Combined K-S 0.2762 0.268 0.190 Figure14Smaller Group D P-Value Corrected

1 0.1690 0.563 2 -0.1875 .493

Combined K-S 0.1875 0.872 0.803 Figure15Smaller Group D P-Value Corrected

1 0.0894 0.620 2 -0.1245 0.396

Combined K-S 0.1245 0.744 0.676 Figure16Smaller Group D P-Value Corrected

1 0.1731 0.320 2 -0.2028 0.210

Combined K-S 0.2028 0.416 0.328 Figure17

Figure18

33

Table1Year Pr(Div|Marr) Nt Tt0 0 0 10001 .02 20 10002 .0302 29 9803 .0321 34 9514 .0462 29 9175 .0347 27 8886 .0658 48 8617 .0731 47 8138 .0534 30 7669 .0587 30 73610 .0564 26 70611 .0572 24 68012 .0498 12 65613 .0597 21 63714 .0500 16 61615 .0608 18 60016 .0778 21 58217 .0292 7 56118 .0479 11 55419 .0466 10 54320 .0598 12 533