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PolicyResearchWorkingPaper11059

DesignofPartialPopulationExperimentswithanApplicationtoSpilloversinTaxCompliance

GuillermoCruces

DarioTortarolo

GonzaloVazquez–Bare

WORLDBANKGROUP

DevelopmentEconomics

DevelopmentResearchGroupFebruary2025

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PolicyResearchWorkingPaper11059

Abstract

Thispaperdevelopsaframeworktoanalyzepartialpopula–tionexperiments,ageneralizationoftheclusterexperimentaldesignwhereclustersareassignedtodifferenttreatmentintensities.Theframeworkallowsforheterogeneityinclus–tersizesandoutcomedistributions.Thepaperstudiesthelarge–samplebehaviorofOLSestimatorsandcluster–ro–bustvarianceestimatorsandshowsthat(i)ignoringclusterheterogeneitymayresultinseverelyunderpoweredexper–imentsand(ii)thecluster–robustvarianceestimatormaybeupward–biasedwhenclustersareheterogeneous.Thepaperderivesformulasforpower,minimumdetectable

effects,andoptimalclusterassignmentprobabilities.Alltheresultsapplytoclusterexperiments,aparticularcaseoftheframework.ThepapersetsupapotentialoutcomesframeworktointerprettheOLSestimandsascausaleffects.Itimplementsthemethodsinalarge–scaleexperimenttoestimatethedirectandspillovereffectsofacommunicationcampaignonpropertytaxcompliance.Theanalysisrevealsanincreaseintaxcomplianceamongindividualsdirectlytargetedwiththemailing,aswellascompliancespilloversonuntreatedindividualsinclusterswithahighproportionoftreatedtaxpayers.

ThispaperisaproductoftheDevelopmentResearchGroup,DevelopmentEconomic.ItispartofalargereffortbytheWorldBanktoprovideopenaccesstoitsresearchandmakeacontributiontodevelopmentpolicydiscussionsaroundtheworld.PolicyResearchWorkingPapersarealsopostedontheWebat

/prwp.Theauthors

maybecontactedatdtortarolo@.Averifiedreproducibilitypackageforthispaperisavailableat

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fordirectaccess.

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ThePolicyResearchWorkingPaperSeriesdisseminatesthefindingsofworkinprogresstoencouragetheexchangeofideasaboutdevelopmentissues.Anobjectiveoftheseriesistogetthefindingsoutquickly,evenifthepresentationsarelessthanfullypolished.Thepaperscarrythenamesoftheauthorsandshouldbecitedaccordingly.Thefindings,interpretations,andconclusionsexpressedinthispaperareentirelythoseoftheauthors.TheydonotnecessarilyrepresenttheviewsoftheInternationalBankforReconstructionandDevelopment/WorldBankanditsaffiliatedorganizations,orthoseoftheExecutiveDirectorsoftheWorldBankorthegovernmentstheyrepresent.

ProducedbytheResearchSupportTeam

DesignofPartialPopulationExperiments

withanApplicationtoSpilloversinTaxComplianc

e*

GuillermoCruces,U.ofNottingham&CONICET-CEDLAS-UNLP

DarioTortarolo,WorldBankDECRG

GonzaloVazquez-Bare,UCSantaBarbara

JELCODES:C01,C93,H71,H26,H21,O23.

KEYWORDS:partialpopulationexperiments,spillovers,randomizedcontrolledtrials,clusterex-periments,two-stagedesigns,propertytax,taxcompliance.

*WethankYuehaoBai,YoussefBenzarti,AugustinBergeron,JavierBirchenall,MatiasCattaneo,MaxFarrell,KelseyJack,HeatherRoyer,DougSteigerwaldandAlisaTazhitdinovaforvaluablediscussionsandsuggestions,andseminarpartici-pantsatthe2021NationalTaxAssociationconference,IFS,CEDLAS-UNLP,andthe2022AdvanceswithFieldExperimentsconference.WethankJulianAmendolaggineandJuanLuisSchiavonifortheirinvaluablesupportthroughouttheproject.WethankBrunoCrponandRolandRathelotfortheirhelpinobtainingtheirdata.Theviewsexpressedinthispaperareentirelythoseoftheauthors.TheydonotnecessarilyrepresenttheviewsoftheInternationalBankforReconstructionandDevelop-ment/WorldBankanditsaf?liatedorganizations,orthoseoftheExecutiveDirectorsoftheWorldBankorthegovernmentstheyrepresent.Correspondingauthor:DarioTortarolo,E-mail:

dtortarolo@.

ThisprojectwasreviewedandapprovedinadvancebytheInstitutionalReviewBoardattheUniversityofNottingham.ThedesignforthisexperimentwaspreregisteredintheAEARCTRegistry(RCTID:AEARCTR-0006569).Allremainingerrorsareourown.

2

1Introduction

Randomizedcontrolledtrials(RCTs)areextensivelyusedineconomics.Alargefractionoftheseexperi-mentsarebasedontheassumptionthatthetreatmentassignmentofoneunitorsubjectdoesnotin?uencetheoutcomesofothers.Theassumptionofnointerference,however,maybeviolatedinmanysettings.Insuchcases,identifyingandmeasuringspilloversbetweenunitsiscrucialforunderstandingthenatureandmagnitudeofinteractionsbetweensubjects,aswellasforaccuratelyassessingthedirectimpactofthetreatment.

Whiletheearlyexperimentalliteratureconsideredtheimpactonuntreatedunitsinanex-postmanner(e.g.

MiguelandKremer,

2004

),?eldexperimentsincorporatingspillovereffectsintotheirdesignhavegainedtractioninappliedresearch.Insettingswhereunitsaregroupedintoindependentclusters,suchasschools,villages,or?rms,acommondesignisthepartialpopulationdesign.Partialpopulationdesignsareageneralizationoftheclustereddesignwhereinclustersassignedtodifferenttreatmentintensitiesorsaturationsarecomparedtopurecontrolclusterswithnotreatedunits(

Mof?t,

2001;

Du?oandSaez,

2003;

HudgensandHalloran,

2008;

HiranoandHahn,

2010;

Bairdetal.,

2018

).Thevariationintreatmentintensityallowsresearcherstodisentanglethedirectandindirecteffectsofatreatment.Inthispaper,weprovideaframeworktoanalyzethistypeofexperimentwhenclustersareheterogeneous.

Weconsidertwodimensionsofclusterheterogeneitythathaveimportantpracticalimplications:het-erogeneityinclustersizesandheterogeneityinoutcomedistributionsacrossclusters(distributionalhet-erogeneity)

1.

Whenanalyzinganexperimentwithheterogeneousclusters,correctlyaccountingforthisheterogeneityiscrucialforseveralreasons.Ontheonehand,varianceformulashavetobeadjustedac-cordingly,andfailingtodosomayresultinseverelyunderpoweredexperiments.Ontheotherhand,clusterheterogeneitycanaffecttheaccuracyofthelargesamplenormalapproximation,andinferencebasedonthisapproximationcanbemisleadingwhenclustersareveryheterogeneous(

Carter,Schnepel

andSteigerwald,

2017;

Djogbenou,MacKinnonand?rregaardNielsen,

2019;

HansenandLee,

2019;

SasakiandWang,

2022;

Chiang,SasakiandWang,

2023

).

Withthesechallengesinmind,ourpaperprovides?vecontributions.First,inTheorem

1

,wederiveanasymptoticdistributionalapproximationforOLSregressionestimatorsinasettingwithbetween-clusterheterogeneity.Weconsideradouble-arrayasymptoticsettingwhereclustersizesareallowed,butnotre-quired,togrowwiththesamplesize.WeprovideconditionsunderwhichOLSestimatorsareconsistentforcluster-size-weightedaveragesofwithin-clusterdifferencesinmeans,andareasymptoticallynormal.Wealsoshowthat,inthepresenceofdistributionalheterogeneity,theusualcluster-robustvarianceestima-torisgenerallyupward-biased,andhenceinferencebasedonthisestimatorisconservative(Proposition

1

).Whilesimilarresultshavebeenobtainedindesign-basedsettingswithnon-randompotentialoutcomes(seee.g.

HudgensandHalloran,

2008;

BasseandFeller,

2018;

Abadieetal.,

2022;

Jiang,ImaiandMalani,

1Wenotethatourframeworkallowsforgeneralformsofbetween-clusterheterogeneity,butassumesthatoutcomesareiden-ticallydistributedwithineachcluster.Thegeneralizationofourresultstothecasewhereoutcomedistributionsareheteroge-neouswithinaclusterisleftforfutureresearch.

3

2023

),toourknowledgewearethe?rsttoshowthisresultinasuperpopulationsettingunderdistributionalheterogeneity.

Oursecondcontributionistoderiveexplicit,closed-formformulastoconductpowerandminimumdetectableeffect(MDE)calculationsunderthetwoaforementionedsourcesofclusterheterogeneity.Wethenconsideranintermediatesettingwhereclustersdifferinsizebutnotintheiroutcomedistributions,whichsimpli?espowerandminimumdetectableeffectscalculationsandcanbeappliedmoreeasilywhenbaselineoutcomedataisnotavailable.Weshowhowourformulasgeneralizethoseavailableintheexistingmethodologicalliteratureonexperimentaldesign(

Du?o,GlennersterandKremer,

2007;

Hirano

andHahn,

2010;

Bairdetal.,

2018

)byallowingformultipletreatmentintensities,clusterheterogeneity,heteroskedasticityandgeneralformsofintraclustercorrelationinoutcomesandtreatments.

Ourthirdcontributionistoderiveoptimalassignmentprobabilitiesdeterminingtheproportionofclus-terstobeassignedtoeachtreatmentsaturation(Theorem

2

).Weprovideatractable,closed-formsolutiontotheoptimalchoiceproblemofminimizingaweightedaverageofestimators’variances.Wealsodiscusshowalternativeoptimalitycriteriamaybeusedincombinationwithourvarianceformulasusingnumericalmethods.

Ourfourthcontributionistosetupapotentialoutcomesframeworkwithwithin-clusterspillovers,heterogeneoustreatmenteffects,andheterogeneousclusters.Weusethisframeworktoprovidesuf?cientconditionsforOLSestimandstorecovercausaldirectandspillovereffects.

Fifth,basedonourframework,wedesignedandconductedalarge-scale?eldexperimenttoestimatedirectandspillovereffectsofarandomizedcommunicationcampaignonpropertytaxcomplianceinAr-gentina.Ourexperimentsentpersonalizedletterstorandomlyselecteddwellingswithremindersabouttaxesdue,informationaboutthestatusoftheaccount,duedates,pastduedebt,andpaymentmethods.Whilethereisampleevidenceontheeffectoftaxremindersoncomplianceandcollection(

Antinyanand

Asatryan,

2024

),ourgoalwasto?ndevidenceonrelativelyelusivespillovereffectsfrominformationcampaignsontaxcollection.Wedesignedtheexperimentbasedonourmethodologicalresultstocapturespillovereffectsofourmailingsonneighborswholiveinthesamestreetblocksoftreatedindividualsbutwhodidnotreceivealetter.Ourresultsrevealhigherpaymentratesfortreatedindividuals,butalsofortheiruntreatedneighborsinthesamestreetblock,comparedtoaccountsinpurecontrolblockswherenoonereceivedtheletter.Spillovereffectsarelowerinmagnitudebutstillsubstantialandpreciselyestimatedinhigh-saturationstreetblocks,especiallywhenaccountingforexpected(pre-registered)heterogeneityinpastcompliance:paymentratesofuntreatedaccountsinhighsaturationblockswithabovemedianpastcomplianceincreasedby2.6percentagepoints,comparedtodirecteffectsofabout5.1percentagepoints.

Comparisonwithcurrentliterature.Ourpapercontributestoagrowingliteratureonexperimentaldesign(

Du?o,GlennersterandKremer,

2007;

BruhnandMcKenzie,

2009;

Bugni,CanayandShaikh,

2018,

2019;

Bai,

2022

)andinparticulartotheliteratureondesignandanalysisofexperimentsunder

4

spilloversorinterference(

HiranoandHahn,

2010;

Athey,EcklesandImbens,

2018;

Bairdetal.,

2018;

Basse,FellerandToulis,

2019;

Jiang,ImaiandMalani,

2023;

Puelzetal.,

2022;

Viviano,

2024;

Leung,

2022;

Liu,

2023

).Morespeci?cally,ourresultsgeneralizethoseof

HiranoandHahn

(2010

),

Hudgens

andHalloran

(2008

)and

Bairdetal.

(2018

)byallowingforclusterheterogeneity,heteroskedasticity,generaltreatmentassignmentmechanismsandwithin-groupcorrelationstructuresandalternativecriteriaforoptimaltreatmentassignment.

Inrelatedwork,

Athey,EcklesandImbens

(2018

),

Basse,FellerandToulis

(2019

)and

Puelzetal.

(2022

)deriverandomizationinferencetestsforageneralclassofnullhypothesesunderinterference.Acloselyrelatedstudyis

Jiang,ImaiandMalani

(2023

),whoanalyzetwo-stagecompletelyrandomizedexperimentsandproviderandomization-basedvarianceestimatorsandsamplesizeformulas.Ourre-sultscomplementthisliteraturebyconsideringdifferentestimands,differentassignmentmechanismsandbyconductingsuper-population-basedlarge-sample(insteadofdesign-based)inferenceinadoublear-rayasymptoticframework.Ourapproachallowsustodeterminetheroleofclusterheterogeneityintheasymptoticbehaviorofthetreatmenteffectestimators.

Ourpaperisalsorelatedtotheliteratureoninferenceinclusteredexperiments,whichareaparticularcaseofpartialpopulationexperimentswithonlytwosaturationsandnowithin-clustertreatmentvariation.

Bugnietal.

(2023

)studyinferenceinclusteredexperimentswithnon-ignorableclustersizesandderivevarianceestimatorsandvalidinferenceproceduresinasetupwithrandomclustersizes.WefurtherdiscusstherelationshipbetweenourresultsandthatpaperinSection

3.5.

Wealsocontributetoalargeempiricalliteratureonpropertytaxesandasmallbutgrowingempiricalliteratureonspillovereffectsintaxcompliance.Onpropertytaxes,recentcontributionsinclude

Brock-

meyeretal.

(2020

)studyofMexicoCity,

Bergeron,TourekandWeigel

(2024

)and

Weigel

(2020

)fortheDemocraticRepublicofCongo,and

Krause

(2020

)forHaiti,amongothers.Thelattertwoareran-domizedcontrolledtrials,andinbothcases,theauthorsaddressthepresenceofspillovers,butinex-postanalysisratherthanintheexperimentaldesigns.Theeffectofsocialinteractionsintaxcomplianceinter-ventionshasremainedarelativelyelusiveissueinthebroaderexperimentalcomplianceliterature.Somenotableexceptionsare

Pomeranz

(2015

),whodetectsenforcementspilloversuptheVATchaininChilean?rms,

Drago,MengelandTraxler

(2020

)whostudyenforcementspilloversofTVlicensinginspectionsonuntreatedhouseholdsinAustria,and

Boningetal.

(2020

)whoanalyzedirectandnetworkeffectsfromin-personvisitsbyrevenueof?cersonvisitedandnon-visited?rmsintheUnitedStates(seethereviewin

PomeranzandVila-Belda,

2019

,formorestudiescoveringspillovereffects).InArgentina,arecentstudyby

Carrillo,CastroandScartascini

(2021

)?ndsneighborhoodspillovereffectsfromaprogramthatrandomlyawarded400taxpayerswiththerepairofasidewalk.Whereasthesepapers?ndspillovereffectsintaxcompliance,theiroriginalexperimentswerenotdesignedtocapturetheseeffects.Webuildonthesepioneeringworkswithaninterventiondesignedwiththepurposeofcapturingspillovers.

Thepaperisorganizedasfollows.Section

2

illustratesthepracticalimportanceofclusterheterogeneitywhenconductingpowercalculations.InSection

3

,wesetupourframeworkandderivethemainresults.In

5

Section

4

,weimplementourmethodsinalarge-scalerandomizedcommunicationcampaign,wedescribetheadministrativedatausedintheanalysis,theempiricalstrategy,andevidenceofdirectandspillovereffects.Section

5

providessomepracticalrecommendationsfordesigningandanalyzingpartialpopulationexperiments.Section

6

concludes.

2WhyisClusterHeterogeneityImportant?

Weconsiderapopulationwhereunitsaregroupedintomutuallyexclusiveandindependentclusters.Com-monexamplesofthistypeofclusteringarestudentsinschools(

MiguelandKremer,

2004;

Beuermann

etal.,

2015

),familymembersinhouseholds(

Barrera-Osorioetal.,

2011;

FoosanddeRooij,

2017

),jobseekersinlocallabormarkets(

Crponetal.,

2013

),employeesin?rmsororganizations(

Du?oandSaez,

2003

),orhouseholdsinneighborhoods,villagesorothergeographicadministrativeunits(

Angelucciand

DeGiorgi,

2009;

IchinoandSch…undeln,

2012;

HaushoferandShapiro,

2016;

GinandMansuri,

2018

).Inourapplication,alocalpropertytaxreminderinformationcampaign,thepopulationofinterestconsistsoftaxpayersinresidentialblocks.Withinthispopulation,westudyanexperimentaldesignwheretreatmentassignmentscanvarybothbetweenandwithinclusters.

Figure

1

showsthedistributionofclustersizesinsixpartialpopulationexperiments,includingouranalysissampleand?vepublishedpapers(

Crponetal.,

2013;

GinandMansuri,

2018;

Haushoferand

Shapiro,

2016;

IchinoandSch…undeln,

2012;

Imai,JiangandMalani,

2021

).The?gurerevealssubstantialvariationinclustersizes.Whenclustersizesareheterogeneous,itislikelythatthedistributionofoutcomeswillvaryacrossclustersaswell.Forinstance,onemayexpectthemeanandthevarianceoftheoutcometobedifferentinlargeclusterscomparedtosmallclusters.Werefertothevariationinoutcomedistributionsacrossclustersasdistributionalheterogeneity.

Intuitively,withheterogeneousclusters,thevarianceofanestimatorofinterest,suchasadifference

inmeansbetweenunitsintreatedanduntreatedclusters(wede?netheestimatorsofinterestpreciselyinthenextsection),canbedecomposedintofourparts:

V[]≈varianceunderuncorrelatedobservations(1)

+clusteringwithequally-sizedclusters(2)

+clustersizeheterogeneity(3)

+clusterdistributionalheterogeneity(4)

The?rsttermisthevariancethatwouldbeobtainedifobservationswereuncorrelatedwithinclusters.Thesecondtermisanadjustmentfactorthataccountsforthewithin-clustercorrelation,oftenknownasthe“designeffect”orthe“Moultonfactor”(after

Moulton,

1986

)thatdependsontheaverageclustersize.Theterminthethirdlinerepresentstheadditionalvariationduetotheheterogeneityinclustersizes,

6

whichintuitivelyaccountsforthevarianceofclustersizes(

Moulton,

1986

,alsoderivesthisadjustmentforarandomeffectsmodel).Finally,thelastcomponentaccountsforthebetween-clusterheterogeneityinoutcomedistributions.Whiletheneedtoaccountforwithin-clustercorrelations(lines(1)and(2))iswell-understoodfordesigningandanalyzingclusteredexperiments,theadjustmenttermsthataccountforclusterheterogeneityaretypicallyassumedawaybytheliteratureonexperimentaldesign(e.g.

Bloom,

2005;

Du?o,GlennersterandKremer,

2007;

HiranoandHahn,

2010;

Bairdetal.,

2018

).

Tonumericallyillustratetheimportanceofappropriatelyaccountingforclusterheterogeneityinthisdesign,weconsiderthesimplesettingofaclusterRCT(whichisaparticularcaseofapartialpopulationexperiment)where“afew”clustersare“large”.Speci?cally,weconsiderasampleof200clusters,indexedbyg=1,...,200,eachhavingsizeng.The?rst10clusterscontain100units,ng=100,andtheremaining190clusterscontain25unitseach,ng=25(thesevaluesarechosentomatchthemedianvaluesintheliteratureinFigure

1

).Weassumethetreatmenthasnoeffect,andtheoutcomeofuniti=1,...,nginclustergisgivenbyarandomeffectsmodel:Yig=αg+νg+ωig,νg,1/2),ωigN(0,1/2)withνgindependentofωigandwhereαgisa(non-random)interceptwithαg=0ifng=25andαg=1ifng=100.ThismodelimpliesthattheaverageoutcomeisE[Yig]=1inlargeclustersandE[Yig]=0insmallclusters.Inaddition,V[Yig]=1andthewithin-clustercorrelationbetweenoutcomesiscor(Yig,Yjg)=0.5.

Figure

2

plotsthreepowerfunctionsforthedifferenceinmeansbetweentreatedanduntreatedclustersthataresearchermayconsiderwhendesigningthisexperiment.Theshort-dashedcurverepresentsthepowerfunctionthatisobtainedwhenignoringbothsourcesofheterogeneity,thatis,consideringonlythetermsinlines(1)and(2)ofthevarianceformula.Usingthisformula,theMDEat80%power,giventhissamplesize,is0.29standarddeviations.However,whenaccountingforthevariationinclustersizes,thecorrespondingpowerfunctionisrepresentedbythelong-dashedcurve.Accordingtothiscurve,thepowertodetectaneffectof0.29isnot80%but69%,sotheexperimentisunderpowered.Furthermore,thetruepowerfunctionthataccountsforbothsourcesofheterogeneity(sizesandoutcomedistributions)isrepresentedbythesolidcurve.Thiscurveshowsthatthetruepowertodetectaneffectof0.29inthissettingwithheterogeneousclustersis48%,signi?cantlybelowthedesiredpowerof80%.Thisnumericalexerciseshowshowignoringheterogeneitymayresultinseverelyunderpoweredexperiments.WeprovidefurtherexamplesoftheimportanceofaccountingforheterogeneityinSection

4.

3AnalysisofPartialPopulationExperiments

3.1Setup

Weconsiderasampleofobservations(units)thataredividedintomutuallyindependentclustersg=

1,...,G,whereeachclustergcontainsngobservationsi=1,...,ngandthetotalsamplesizeisn=

7

Σg.Weviewclustersizesasnon-random(see

Bugnietal.,

2023;

SasakiandWang,

2022

,foranalternativesamplingapproachwhereclustersizesarerandom).Inapartialpopulationexperiment,clustersarerandomlydividedintocategoriesorsaturationsdenotedbyTg∈{0,1,2,...,M},wherebyconventionTg=0denotesapurecontrolcluster(i.e.aclusterwherenounitistreated).LetP[Tg=t]=qt∈(0,1)denotetheprobabilitythatclustergisassignedtosaturationt.Withineachcluster,abinarytreatmentDigisassignedtounitswithprobabilityP[Dig=1|Tg=t]whereP[Dig=0|Tg=0]=1

.2

WeletDg=(D1g,D2g,...,Dngg)9bethevectorofunit-leveltreatmentassignmentsinclusterg,D=

(D,...,D)9andT=(T1,...,TG)9.Figure

A.3

providesanexampleofapartialpopulationdesign

withfoursaturations.NoticethatbothstandardRCTswithindependentobservationsandclusterRCTsareparticularcasesofpartialpopulationexperiments,aswefurtherillustrateinSection

3.5.

TheobservedoutcomeofinterestforunitiinclustergisdenotedbyYigandweletYg=(Y1g,...,Yngg)9bethevectorofobservedoutcomesinclusterg.Inpartialpopulationexperiments,theestimandsofin-terestaretypicallycomparisonsofaverageoutcomesbetweentreatedoruntreatedunitsintreatedclusterstopurecontrolunits,E[Yig|Dig=d,Tg=t]-E[Yig|Tg=0],pooledacrossclusters.Inthe?rstpartofthepaper,wetaketheseestimandsasgivensincetheyarethemostcommonlyanalyzedestimandsintheempiricalliterature.InSection

3.6

,wesetupapotentialoutcomesframeworktorigorouslyjustifythecausalinterpretationoftheseestimands.Letμg(d,t)=E[Yig|Dig=d,Tg=t]betheconditionalexpectationoftheoutcomeinclusterggivenassignment(d,t).Weconsiderthefollowingsamplemeansestimators:

where1=1(Tg=t),N=Σi1(Dig=d)andY-gd=ΣiYig1(Dig=d)/N,de?nedwhenever

N>0.TheseestimatorsarecommonlycomputedbyrunninganOLSregressionoftheoutcomeon

afullsetofindicators(1(Dig=d,Tg=t))(d,t),withoutanintercept.Thus,inwhatfollows,werefertotheseestimatorsasOLSestimators.Ourparameterofinterestisthevectorofcluster-size-weightedaverageofcluster-speci?cdifferencesinmeans:

Wenotethatourframeworkcaneasilyaccommodateotherparameterswithdifferentweightingschemes,

suchasthesimpleaverageacrossclustersΣg(d,t)-μg(0,0))/G.

2Inpractice,somedesiredsaturationsmaynotcoincidewiththeobservedproportionoftreatedunitsforsomeclustersizes.Forinstance,ifP[Dig=1|Tg=t]=0.5butngisodd,theobservedproportionoftreatedcannotbeexactly0.5.Appendix

A.4

proposesanassignmentmechanismthatensuresthattheexpectedproportionoftreatedcoincideswithP[Dig=1|Tg=t].

8

3.2AsymptoticBehaviorofOLSEstimators

WenowstudytheasymptoticdistributionoftheOLSestimatorsde?nedinEquation(

5

)andfunctionsthereof.Weconsideradouble-arrayasymptoticsettingwheretheclustersizesareallowed,butnotre-quired,togrowwiththesamplesize.Thistypeofapproximationismoreappropriatethantheboundedclustersizeapproachwhengroupscanbelargeandheterogeneousinsize,butwenotethatthesettingswithboundedclustersizesand/orequally-sizedclustersarenestedasparticularcasesofouranalysis

.3

Weconsiderthefollowingsamplingscheme.

Assumption1(Sampling)

(i)(Yg,,Dg,,Tgg.

(ii)Foreachgandforalli=1,...,ng,E[Yi|Dig=d,Tg=t]=μ(d,t)forall(d,t)andforalllsuchthatE[|Yig|l|Dig=d,Tg=t]<∞.

(iii)Foreachgandforalli=1,...,ng,P[Dig=d|Tg=t]=pg(d|t)andP[Dig=d,Djg=d,|Tg=

t]=pg(d,d,|t)foralld,d,andt.

Part(i)statesthatclustersaremutuallyindependent,astandardassumptionintheclusteringliterature.

Noticethatwedonotrequireclusterstobeidenticallydistributed,sooutcomedistributionscanbehet-erogeneousacrossclusters.Part(ii)statesthataverageconditionaloutcomesarethesameforallunits

inthesamecluster.Inwhatfollowswede?neμ(d,t)=μg(d,t)forl=1toreducenotation.Part

(iii)statesthattheunit-leveltreatmentprobabilitiesarethesamewithinacluster.Notethatwithin-clusterassignm