Family Ties and Child Obesity in Italy

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Family

ties

and

child

obesity

in

Italy

$

Federico

Crudu

a,b,

*

,

Laura

Neri

a

,

Silvia

Tiezzi

a

a_Department_of_Economics_and_Statistics,_University_of_Siena,_Piazza_San_Francesco,_7/8,₅₃₁₀₀_Siena,_Italy b_CRENoS,_Cagliari,_Italy

ARTICLE INFO Articlehistory:

Received21January2020

Receivedinrevisedform25October2020 Accepted16November2020

Availableonline25November2020 Keywords:

Childobesity Conﬁdencesets Partialidentiﬁcation Peereffectswithinthefamily

ABSTRACT

ThispaperexaminestheimpactofoverweightfamilymembersonweightoutcomesofItalianchildren aged6–14years.Weuseanoriginaldatasetmatchingthe2012crosssectionsoftheItalianMultipurpose HouseholdSurveyandtheHouseholdBudgetSurvey. Sincetheidentificationofwithin-familypeer effectsisknowntobechallenging,weimplementouranalysisonapartiallyidentifiedmodelusing inferentialproceduresrecentlyintroducedintheliteratureandbasedonstandardBayesiancomputation methods.Wefindevidenceofastrong,positiveeffectofbothoverweightpeerchildreninthefamilyand ofoverweightadultsonchildrenweightoutcomes.Theimpactofoverweightpeerchildrenin the householdislargerthantheimpactofadults.Inparticular,theestimatedconfidencesetsassociatedto thepeerchildrenvariableispositivewithupperboundaroundoneorlarger,whiletheconfidencesetsfor theparameterassociatedtoobeseadultsoftenincludezeroandhaveupperboundthatrarelyislarger thanone.

1.Introduction

Inthelastdecadeschildrenoverweightandobesityprevalence have risen substantially in most countries. According toa new globalassessmentofchildmalnutritionbyUNICEF(2019)themost profound increase hasbeen in the 5–19 age group,where the globalrateofoverweightincreasedfrom10:3%in2000to18:4%in 2018.

Identifyingthedeterminantsofchildobesityis acompelling issuesinceobesityisnotonlyadirectthreatforchildren'shealth andacosttosociety,butalsohasdocumentedconsequencesfor adultlife,suchaseffectsonhealth(Llewellynetal.,2016),on self-esteem,bodyimageandconﬁdence,andonwages(Schwartzetal., 2011).

Itisrecognizedthatchildconsumptiondecisionsareaffectedby those of theirpeers (Dishionand Tipsord, 2011)and that peer effectsaremorepronouncedinchildrenthaninadolescents(Nie etal.,2015).While classroomorfriendspeereffects havebeen foundtoexplainchildhoodandadolescentsobesity(Asirvatham etal.,2014;Gwozdzetal.,2015;Nieetal.,2015),theroleof within-the-familypeers,e.g.interactionwithotheroverweightandobese peerchildreninthefamily,asadeterminantofchildoverweight andobesityhasnotyetbeeninvestigated.1

In fact, within-the-family social interaction could be an importantdeterminantofchild obesity,becausechildrenspend mostof theirtime in thefamily environment. A likely driving mechanism is imitation. Research in experimental psychology (Zmyjet al.,2012a,b;Zmyjand Seehagen,2013)postulatesthat prolonged individual experience with peers leads children to imitatepeersmorethanadults.Childrenimitatefamiliar behav-iourforsocialreasons,suchasidentiﬁcationwiththerolemodelor tocommunicatelikeness.Adultsarethenaturalmodelonwhich childrenrely inunfamiliarsituations whileageisanimportant indicator of likeness. With prolonged contact with peers (i.e. $_We _thank _three _anonymous _referees _and _the _Associate _Editor _Tinna

Ásgeirsdóttir, whose comments greatly improved thepaper. Comments from PamelaGiustinelli,GiovanniMellace,NicoleHair, andparticipantsatthe2018 WorkshoponInstitutions,IndividualBehaviorandEconomicOutcomes,Alghero, Italy;theIAAE2019Conference,Nicosia,Cyprus;theNordicHealthEconomicStudy Group(NHESG)2019,Reykjavik,Iceland;andthe2019AnnualHealthEconometrics Workshop,KnoxvilleTN,USAaregratefullyacknowledged.

*Correspondingauthorat:DepartmentofEconomicsandStatistics,Universityof Siena,PiazzaSanFrancesco,7/8,53100Siena,Italy

E-mailaddresses:federico.crudu@unisi.it(F.Crudu),laura.neri@unisi.it(L.Neri),

silvia.tiezzi@unisi.it(S.Tiezzi).

1 _An_exception_is_the_famous_study_by_Christakis_and_Fowler₍₂₀₀₇₎_focusing_on adults.Oneoftheirmainﬁndingswasthat,amongpairsofadultsiblings,ifone siblingbecameobese,thechancethattheotherwouldbecomeobeseincreasedby 40%andthatifonespousebecameobese,thelikelihoodthattheotherspouse wouldalsobecomeobeseincreasedby37%.

http://dx.doi.org/10.1016/j.ehb.2020.100951

ContentslistsavailableatScienceDirect

Economics

and

Human

Biology

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childreninthesameagegroup),childrenaremorelikelytoimitate behaviourfromthemthanfromadults,becausetheylearntotrust theirpeers andtorefer tothemfor learning alsoinunfamiliar situations. In this case imitation serves a cognitive function: prolongedcontactwithpeersleadschildrentobelievethatpeers are ascompetentasadults, i.e.a reliablemodel.Since children plausiblyspendextendedperiodsoftimewithfamilymembers, suchprolonged contact is reﬂectedin increasedlevelsof peers imitation. Ifimitation isthedriving mechanismthrough which within-the-familysocialinteractionaffectschildobesity,thenthe impactofpeerchildrenin thefamilyshouldbelargerthanthe impactofadults.

Thepurposeofthispaperistoinvestigatewhetherthepresence of other overweight/obese family members,i.e. children in the sameagegroupandadults,hasapositiveandsigniﬁcanteffecton theprobabilityofachildbeingoverweight/obese.Toaddressthis research question we use a unique cross-section of Italian householdscontainingdetailedinformationonfamilies’structure, composition,habits,andweightoutcomes.Weestimateabinary choicemodelwherethedependentvariableisabinaryindicator for each child being overweight or obese or not. The main explanatoryvariablesofinterestaretheshareofotheroverweight orobesechildreninthesameagegroupinthefamily,andtheshare ofoverweightorobeseadultsinthefamily.

To assess theimpactof childreninthesame agegroupand family(ourpeereffect),weuseanarrowpeer-groupdefinitionthat includesallchildrenaged6–14yearsbelongingtothesamefamily whethersiblingsornot.Whileassessingtheimpactofadultsdoes notposeparticularchallenges,within-the-familypeereffectsare particularly difficultto identify.Narrow definitions of thepeer group,suchasours,havebeenfoundtobemoreendogenousthan

broad ones, because of shared common traits, habits and

environments that maycause simultaneityeffects (Blacket al., 2017;Trogdonetal.,2008).Asharedenvironmentalsocomplicates theproblemofcontrollingforunobservedﬁxedeffects,because thelatentheterogeneitythat mayaffecttheweightoutcomeof each child is likely to affect theweight outcome of the other childreninthesamefamilyandagegroup.

Inordertoprovidesomefurtherintuitiononthemechanicsof our problem and on the potential causal interpretation of themodel,letusconsiderasimpledirectedacyclicgraph(DAG)

(seeFig.1)torepresenttherelationshipamongthemainvariables ofourmodel.Ourmainproblemistostudytherelationshipofthe peereffectvariable(Peer)ontheobesityscore(Obesity)ofagiven child in thefamily.Wemay reasonablyconjecturethat Obesity woulddependonExogenousand Contextualvariablesaswellas otherUnobservedcharacteristics.Identifyingpeereffects maybe complicatedforanumberofreasons.First,inthecontextofagroup ofsiblingstheassignmenttoagivenfamilyisnonrandomandit wouldreasonablydependonthecharacteristicsoftheparents.2

Second,geneticandbehaviouralcharacteristicsmaybeimportant todeterminewhetheranindividualisobeseornot.Theformerset ofcharacteristicsmorethanthelattermaybedifﬁculttoobserve. However,theremayexistsomesuitableproxyvariablesthatmay workas mediatorsbetween the Unobserved variables and Peer, thesemaybephysicalcharacteristicssuchasadults’weightand height(orBMI)andhistoryofchronicdiseases.Ifthisisthecase,by controllingfortheExogenouseffectsandtheContextualeffectsin

Fig.1wemaybeabletoidentifythecausalrelationbetweenPeer andObesity.TheassumptionthatUnobserveddoesnotaffectPeer maybedifﬁculttomaintaininsomeapplications.Intheanalysisof peer effects in theclassroom context, for example, one would reasonablyassumethatsuchunobservedfactorsmayberelatedto familycharacteristicsandinparticulartoteacherquality.Inthis case,i.e.ifUnobservedaffectsPeer,identifyingthecausaleffectof PeeronObesitymaybeimpossible.

Duetothenarrowpeergroupandtothestructureofthedata, however, ouridentification problemremains hard tosolve.We resorttoabinarychoicemodelandtopartialidentificationresults for such models (see Section 4 for further details on the identification problemand, e.g., Blume et al., 2011; Brock and Durlauf,2001,2007).

Inferentialproceduresforpartiallyidentifiedmodelsareoften rathercomplicated. However,themethodweusein thispaper, introducedbyChenetal.(2018),iscomputationallyrathersimple andboilsdowntocalculatingconfidencesetsfortheparametersof interest by means of standard Bayesian computation methods. Consistentlywiththehypothesizeddrivingmechanism,wefind Fig.1.ThisDAGshowsthecausalrelationshipbetweenObesityandthepeereffectvariablePeer. Inthismodel,controlling forExogenousandContextualallowsone to identifythe causal effectof Peer on Obesity.

2

Theimplicitassumptionhereisthatfamilymembersareconsanguineous.

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evidenceofastrong,positiveandstatisticallysigniﬁcanteffectof overweight andobesepeerchildrenand asmallerpositive and generallystatisticallysigniﬁcant effectofoverweightand obese adultsinthefamilyonchildren'sobesity.

Ourcontributiontotheexistingliteratureisthreefold.First,to ourknowledgethisistheonlypaperstudyingthecausalroleof within-familypeereffectsonobesityasarelevanthealthoutcome. Ifpeersinthefamilyhaveimportantinﬂuencesonchildweight outcomes, policies affecting one child in the family may have beneﬁcialeffectsontheotherchildrenaswellasasocialmultiplier effect.

Second, asstressedby Blumeet al.(2011),theliterature on partial identification for social interaction models has evolved separately from that on the estimation of partially identified modelsviaboundsinitiatedbyTamer(2003)andusedinindustrial organization. This paper is an attempt to integrate these two bodiesofliteratureinaveryspecificcontext.

Finally,this istheonly studyonsocial interactionand child obesityinItaly.ObesityratesarelowinItalycomparedtomost OECDcountries,butthepictureisdifferentforchildren.According to the ﬁfth wave of the Italian Surveillance System Okkio alla Salute, in 2016 the prevalence rates of overweight (including obese) andobeseprimaryschoolchildrenwere30.6% and9.3%, respectively, withsouthern regionsdisplayinghigherratesthan northern regions (Lauria et al., 2019). The Surveillance System Okkio alla Salute (http://www.epicentro.iss.it/okkioallasalute/) monitors overweight and obesityofItalian children in primary schools(6–11yearsofage).TheSystem,promotedandﬁnancedby theItalianMinistryofHealth,wasstartedin2007andparticipates in the World Health Organization (WHO) European Childhood ObesitySurveillanceInitiative(COSI).Inaddition,familytiesare culturallystronginItalywhichmakessocialinteractionwithinthe familyaparticularlyinterestingissuetoexplore.

The remainder of the paper unfolds as follows. Section 2

summarizestheliterature.Section3describesthedata.Section4

discusses our identiﬁcation strategy. Section 5 presents the estimationmethodsandmainresults.Section6concludes.Finally, theAppendixcontainsadescriptionofstatisticalmatching,results for thefullsetsofparametersandtheresultsoftherobustness checks.

2.Childobesityandpeereffects

The mainrecognizedcauseof therisein child obesityis an imbalancebetweencalorieintakeandcalorieexpenditure.Thereis avastliteratureonthefactorsdrivingthisimbalance.Onestrand has addressed the relationship between maternal employment

and child obesity in many developed countries. Maternal

employment is usually associated with higher child weight

outcomes, because employed mothers may have less time to

pay attention to their children's diet (Cawley and Liu, 2012; Championetal.,2012;Fertigetal.,2009;Gainaetal.,2009;García et al.,2006;Greve, 2011; Gwozdzet al.,2013; Liu etal., 2009; Morrill, 2011, to cite only a few). Overall, these studies _ﬁnd empirical evidence ofa positive relationshipbetweenmaternal employmentandchildhoodobesity.However,thereisnoevidence of such positive relationshipin Italy.In Italy there is a female labourforceparticipationdivideanda childobesitydivide.The Southhasaverylowfemalelabourforceparticipationcomparedto theNorth,butchildobesityprevalenceismuchhigherintheSouth comparedtotheNorth(Brillietal.,2016).Arelatedfactoristhe increasing use of non-parental child care (informal care by a relative,carebyababy-sitterandcentre-basedcare)whichmay increase the likelihood of obesity (Herbst and Tekin, 2011; Hubbard,2008).Thegrowinguseofnon-parentalcaremayplay acrucialroleinshapingchildren'shabitsthroughthequalityofthe

foodofferedand thelevelofphysicalactivity.HerbstandTekin (2011) ﬁnd thatcentre-based care is associated withlargeand stable increases in BMI throughout its distribution, while the

impact of other non-parental arrangements appears to be

concentratedatthetailsofthedistribution.

Astrandofliterature,initiatedbyChristakisandFowler(2007), hasemergedinhealtheconomicsthataddressestheinfluenceof socialinteraction,particularlyofpeers,onhealthstatus.Intheir seminalpaper(ChristakisandFowler,2007)conductedastudyto determinewhetheradult obesitymight spread from person to person.Theirstartingpointwasthatpeopleembeddedinsocial networksarein_fluencedbythebehavioursofthosearoundthem suchthatweightgaininonepersonmightinfluenceweightgainin others.Thatstudyfocusedonsocialinteractionamongadults.A followupstudybythesameauthors(FowlerandChristakis,2008) producedevidenceofperson-to-personspreadofobesityalsoin adolescents.

Powell et al.(2015) haveidentifiedsocial contagion,i.e. the phenomenonwherebythenetworkinwhichpeopleareembedded influencestheirweightovertime,asoneofthesocialprocesses explainingtheriseofadultoverweightandobesity.Thegeneral finding is that weight-related behaviours of adolescents are affectedbypeercontacts(FowlerandChristakis,2008;Halliday andKwak,2009;MoraandGil,2013;Rennaetal.,2008;Trogdon etal.,2008).Thesestudiestakeadolescentsastherelevantage groupandtheclassroomorfriendsastherelevantnetwork.Much lessis knownaboutchildrenastherelevantagegroupand the familyastherelevantnetwork.3

Tothebestofourknowledgeonlyfourstudies,besidesours, analyzepeereffects amongchildrenastherelevantpopulation, andchildobesityastherelevantoutcome.Asirvathametal.(2014)

studypeereffectsinelementaryschoolsusingmeasuredobesity prevalenceforchildrencohortswithinschoolsandusingapanel datasetatgradelevelfromArkansaspublicschools.Theyfound that changes in theobesityprevalence at thehighest level are associatedwithchangesinobesityprevalenceatlowergradesand themagnitudeoftheeffectisgreaterinkindergartento fourth-gradeschoolsthaninkindergartentosixth-gradeschools.Nieetal. (2015)analyzepeereffectsonobesityinasampleof3–18yearsold childrenand adolescentsin China.Peereffects are foundtobe strongerinruralareas,amongfemalesandamongindividualsin theupperendoftheBMIdistribution.Gwozdzetal.(2015)analyze peereffectsonchildhoodobesityusingapanelofchildrenaged2– 9fromeightEuropeancountries.Theyshowthat,comparedtothe otherEuropeancountriesinthesample,peereffectsarelargerin Spain, Italy and Cyprus. These studies adopt a fairly broad deﬁnition of peereffects, either peers at thesame grade level withinaschoolorchildreninasimilaragegroupwithinaspeciﬁc community.Finally,Yajuanetal.(2016)estimatepeereffectson thirdgradestudents’BMIfromachildhoodobesityintervention programtargetedatelementary schoolsstudents inTexas.Peer effectswerefoundforstudentsaged8–11,withgenderdifferences inthepsychologicalandsocialbehaviouralmotivations.

Noneoftheabovestudiesfocusesonthefamilyastherelevant peer group. The literature on the causal role of siblings on children's outcomes is recent and growing. This literature has focused on the effects of sibling health status on educational outcomes(Blacketal.,2017;Fletcheretal.,2012),ontheeffectof earlyhealth shocksonchild humancapital formation(Yiet al., 2015),ontheeffectsofteenmotherhoodontheirsiblings’short and medium term human capital development (Heissel, 2017, 2019),ontheeffectofsiblingsoneducationalchoicesandearly

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Nieetal.(2015)reportthatmostoftheempiricalliteratureonpeereffectsand obesityreferstoadolescentsoradultsandusesUSdata.

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careerearnings(Dustan,2018;JoensenandNielsen,2018;Nicoletti andRabe,2019;Qureshi,2018),andontheeffectsofhealthshocks toindividualsontheirfamilymembersconsumptionofpreventive care(FadlonandNielsen,2019).

Our study contributes to the latter strand of literature considering children as therelevant populationand obesityas therelevanthealthoutcome.Weconjecturethatthemechanism throughwhichthepeereffectplausiblyoperatesisviaimitationof goodandbadbehaviourssuchaseatinghabits.

3.Dataandmatching

Thechoiceofthefamilyastherelevantnetworktoanalyzepeer effects complicates the problem of controlling for unobserved ﬁxedeffects.Thus,theamountofavailableinformationisacrucial issue inourcase. Studiesofpeereffects and childhood obesity usually include information on economic characteristics of the household, such as income, in addition to personal and socio-demographic information, because low-income individuals are more likelytobeobese thanthose withhigh-income(Trogdon et al., 2008). Moreover, the relationship between income and weight is reported to vary by gender, race/ethnicity and age.4

LackingasingleItaliancrosssectioncontainingindividualweight outcomes, detailed family characteristics and socio-economic variables,weusedstatisticalmatching(SM)tomatchtwodatasets. Theﬁrstisthe2012crosssectionoftheMultipurposeSurveyon Households: Aspects of Daily Life (MSH) containing detailed informationonfamilycharacteristicsandtheweightoutcomeof

each member. The second is the 2012 cross section of The

HouseholdBudget Survey(HBS)covering detailsofcurrent and durableexpenditures.BothsurveysareconductedbytheItalian NationalStatisticalInstitute(ISTAT).

TheMSHfortheyear2012isalargenationallyrepresentative sample survey covering 19,330 households and 46,463 family members,includingchildrenaged6–14years.5_The_{questionnaire,}

administered by paper and pencil, contains three blocks of questions: a general questionnaire onindividualcharacteristics oftheﬁrstsixmembersofthehousehold;afamilyquestionnaire collecting information about household habits and lifestyles; a diaryofhealthandnutritionalinformationforeachmemberofthe

household. For children and adolescents aged 6–17 a binary indicator for whether the child is overweight or obese is also included.Identiﬁcationofachildasoverweightorobeseisbased onBMIthresholdvaluesforchildrenaged6_–17developedbyCole etal.(2000)andadoptedbytheInternationalObesityTaskForce (IOTF).The MSH doesnotcontain informationon expenditures that couldbeimportantcovariates inourempirical model.We obtainthisinformation fromthe2012crosssection oftheHBS which includes monthly consumption expenditures of 22,933 Italianhouseholds.ISTATusesaweeklydiarytocollect expendi-ture data on frequently purchased items and a face-to-face interview to collect data on large and durable expenditures. Current expenditures are classiﬁed into about 200 elementary goodsandservices.

The surveyalsoincludes detailed information on household structureandsocio-demographiccharacteristics(suchasregional location,householdsize,gender,age,educationandemployment conditionofeach householdmember).Forbothsurveys,annual samples are drawn independently according to a two-stage design.6Inadditiontohavingalargesetofvariablesincommon, the two surveys share many characteristics suchas the target population,samplingmethod,geographicframeanddata collec-tionprocedure.ThesecommoncharacteristicsallowustouseSM as an ideal method for combining information on households’ qualityof life and child weightoutcomes with information on households’consumptionexpenditures.7

Thesampleunderanalysisincludes3906observations.Theunit ofanalysisisdeﬁnedaschildagedbetween6and14years:the barplotinFig.2 panel(a)displayshowchildrenaredistributed acrosshouseholds.The3906childreninvolvedintheanalysisare distributedacross2954households.AsshowninFig.2panel(a), 2095childrenhavenosiblingsinthetargetagegroup:6–14years; 770householdshavetwochildreninthesameagegroup,foratotal number of children equal to 1540; 85 households have three childreninthesameagegroup,thusthetotalnumberofchildrenis 255; ﬁnally, 4 households have fourchildren in thetarget age group,foratotalamountofchildrenequalto16.

Foreachindividual,arichsetofcovariatesisavailable.Table1

showssummarystatisticsoftherelevantvariablesintheﬁnalSM Fig.2.Children'sdistributionandshareofobesechildren.

4_Food_Research_and_Action_Center:

http://frac.org/obesity-health/relationship-poverty-obesity. 5

AccordingtoboththeHBSandtheMSHafamilyorhouseholdisdeﬁnedasthe setofcohabitingpersonslinkedbymarriageorkinshipties,afﬁnity,adoption, guardianshiporaffection.

6 _Details_on_the_sampling_procedure_used_to_collect_data_in_both_surveys_can_be foundin: ISTAT (2012)IndagineMultiscopo sulle Famiglie,aspetti della Vita quotidiana,Anno2012,fortheMSHsurvey;andinISTAT(2012)File Standard-IndaginesuiConsumidelleFamiglie-Manualed’uso,anno2012,fortheHBSsurvey. Downloadableathttp://www.istat.it/it/archivio/4021.

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SMofthetwodatasetsisdetailedinAppendixA.

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dataset.Wedistinguishﬁvesetsofvariables:individual character-isticsofchildren(panelA),householdcharacteristics(panelB),in some cases related to the household's reference person (RP), behaviouralvariables(panelC),proxiesforgeneticcharacteristics (panel D), regional variables (panel E). More speciﬁcally, the individualcharacteristicsarethechildoverweight/obesity indica-tor(ourdependentvariable),genderandageforeachchildinthe household.Thereare1141overweight/obesechildrenoutof3906, thusoverweight/obesechildrenaccountfor29%ofchildrenaged 6–14years.Thechildren'smeanageis10andthepercentageof malechildrenis49:5%.

The peer effect variable is de_ﬁned as the share of other (overweightandobese)childreninthefamily(excludingthechild considered).Thisvariable,mgi,iscomputedastheratiobetween thenumberofobesechildreninfamilygexcludingthereference child i, nO

gi,and thetotal numberof children inthefamily,ng. Hence,

mgi¼ nO

gi ng

where,in ourdataset,1ng4,0nOgi3and0mgi 2 3.

Theminimumvalueofthevariablecorrespondstotwodifferent cases.Theﬁrstcaseoccurswhenchildihasnosiblingsandthe secondoccurswhenchildihasnoobesesiblings.Themaximum valueoccurswhentherearetwooutofthreeobesechildreninthe family.

Fig.2(panelb)showstheconditionaldistributionoftheshareof otherobese/overweightchildreninthefamilygiven thenumberof childreninthetargetagegroupwithinthefamily.Ofcourse,ifthe reference child has no siblings, the share is zero. As tothe children having siblings inthe age group6–14,wecanobservethe followingpicture:the numberofchildreninfamilieswithtwochildreninthetargetagegroup is1540,71_%ofthemhasasiblingwithnormalweight(share¼0) and21%hasanobese/overweightsibling(share¼1=2¼0:5);the numberofchildreninfamilieswiththreechildreninthetargetage groupis255,52%theof themhassiblingswithnormalweight (share¼0), 33% of them has one obese/overweight sibling (share¼1=3¼0:33), and the remaining 13% has two obese/ overweight siblings(share¼2₌₃¼0_:67).Finally,thenumber of childreninfamilieswithfourchildreninthetargetagegroupis16, 12:5%ofthemhasnormalweightsiblings(share¼0),62:5%has1 obese/overweight sibling(share¼1=4¼0:25) and25%of them hastwoobese/overweightsiblings(share¼2=4¼0:5).

FurthercharacteristicsshowninTable1includetheshareof overweightandobeseadultfamilymembers(42%),householdsize (4onaverage)and adummy forchildrenbornfroma previous marriage,theemploymentstatus oftheRP(threedummiesfor whetherthehouseholdRPisemployed,astudentorhousewife, retiredorinotheremploymentpositions(e.g.military,unableto work,detained)),dummiesforthelevelofeducationofthemother (fivedummiesforwhetherthemotherholdsaMaster'sdegree,a Bachelor'sdegree,hasattendedHighSchool,JuniorHigh,oronly PrimarySchool)andadummyforwhetherthehouseholdlivesina centralornorthernItalianregion.Inaddition,weincludemonthly current expenditure, whose average value is 2131 Euros. This variable is important as it captures contextual effects and we conjecturethatitssupport(237–16,998Euros)issufficientlylarge toensurethat anonlinear relationshipwiththeshare ofobese children in the family (the endogenous effect) exists. We also include a setof variables capturing behaviours of siblingsin a wideragegroup(6–18years)comparedtothetargetagegroup, becauseoldersiblingscouldinfluencethebehavioursofyounger ones.Suchvariablesincludetheshareofsiblings(excludingthe child underconsideration)agedbetween6and18watchingTV

everyday,havinglunchathome,practicingphysicalactivitiesona regular basis and walkingtoschool, dummiesfor whether the parentsconsumesodadrinksorsmoke.Wealsoincludethechild's daily average fruit portions and the adults daily average fruit portions.As proxiesfor thegeneticvariables weusethemean heightandweightoftheadultmembersofthefamilyandtwo dummyvariablesforwhethertheRPorherspousesufferfroma chronicdiseaseordiabetes.Finally,weusetwoadditionalvariables at the regional level: the 2012 consumer price index (CPI) (2010=100)andthepercentageofobeseadultsbyregionin2012. 4.Identi_ﬁcation

Ouraimistoassesswhetherthepresenceofotheroverweight/ obesefamilymembers, i.e.children inthesameagegroupand adults,hasapositiveandsignificanteffectontheprobabilityofa childbeingoverweight/obese.Ifimitationbehaviouristhedriving mechanismwealsoexpectthat theimpactofoverweight/obese peerchildreninthefamilyislargerthantheimpactofoverweight/ obeseadults.Weuseanarrowpeer-groupdefinitionthatincludes allchildrenaged6–14yearsbelongingtothesamefamily(whether siblingsornot).Narrowdefinitionsofpeergroupshavebeenfound tobemoreendogenous than broadones.In particularTrogdon etal.(2008)reportthatbroadermeasuresofsocialnetworks(e.g. grade-levelpeergroups)aremoreexogenousthannarrowones (e.g. children in the same family) as they are likely to be determined by different causal mechanisms. While grade-level peereffectsmaybedrivenbyBMIrelatedsocialnormsandbody imageconcerns,family-levelpeereffectsmayalsooperatethrough additional channels such as the influence of diets, habits and physicalactivities.ChristakisandFowler(2007)showedthatthe influenceoftheweightoffriends,familymembersandneighbours decreaseswithincreasingdegreesofseparationfromtheperson underinvestigation.Despitethelargeempiricalliteratureonsocial interactioninavarietyofcontexts,identifyingsucheffectsremains aformidablechallenge.

LetusconsiderthenotationandthedeﬁnitionsinBrockand Durlauf (2007). We assume that individual binary weight out-comesaredeterminedbyﬁvesetsoffactors:

(i)observable individual-speciﬁccharacteristicsknownalso as theexogenouseffects,measuredbyanr-vectorXi;

(ii)unobservable individual characteristics summarized by a scalar

ei

;

(iii)observablegroupcharacteristics,measuredbyans-vectorYg;

these are known as contextual effects and may directly inﬂuenceindividualdecisions:forexample,peers’ character-isticssuchasparents’income,educationoroccupationmay inﬂuencechildren'sweight;

(iv)unobservable (totheeconometrician)groupcharacteristics, measuredbyascalar

a

gthatmayaffectindividualoutcomes;

theseare knownascorrelated effects:forexample,genetic characteristics mayaffect theweightof all childrenin the samefamily;

(v)theaverageoutcome inthepeergroupexcludingthechild underconsideration,mgi.Itisameasureoftheshareofobese children in the family that could affect each individual outcome(seeSection3forthedeﬁnitionofmgi).

Thus,ourmodelofsocialinteractioncanbedescribedasinEqn 1below

vi

¼kþc0Xiþd0YgþJmgiþ

ei

þ

a

g ð1Þ

where

vi

isabinaryindicatorthattakesvalueoneif,accordingtoa BMIscore,individualiisoverweight/obeseandzerootherwise.One

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importantadvantageofusingabinarychoicemodelisthat,undera large support on Yg, the data reveal a non-linear relationship

between mgi and Yg. This implies that theso-called reﬂection problem(Manski,1993)doesnotariseinthebinarychoicecase.8

Identiﬁcationinbinarychoicemodelsofsocialinteractionhas beenthoroughlyexploredinBrockandDurlauf(2007)(seealso

Blumeetal.,2011,forasurvey).Intheirbaselineresult,with

a

g¼0

andrandomassignment,themodelparametersareidentifiedupto scaleandthereflectioneffectthattypicallycharacterizes linear-in-meansmodelsis notpresent.The mainargumentreliesonthe support of thecontextual effects Yg to be sufficiently largeto

establisha nonlinearrelationshipwithmgi.Inadditiontothat, theyderiveidentiﬁcationresultsalsoforthecaseofnonrandom assignmentprovidedthat

a

g¼0.

With respect to our context, the assumption of random

assignmentisverydifﬁculttojustify.Sinceindividualswithina groupareconsanguineouswithhighprobability,theirassignment toagivengroupg dependsoncommongenetictraits. Unfortu-nately,ourdatasetdoesnotcontainanyexplicitinformationon thegeneticfactorsthatdetermineobesityinthegroup.However,

we may proxy it with mean adult height and weight in the

householdandbyaddingtwodummyvariablesforwhetherthe householdheadorherspousesuffers froma chronicdiseaseor diabetes. We canconjecturethat ourproxies capture sufficient information from the fixed effect to guarantee identification. However,wecannotbesurethatallrelevantassumptionsaremet. Whenpoint identification isnot possible,Brockand Durlauf (2007) describe a number of situations where at least partial identificationcanbeachieved.Inparticular,theyprovethatunder nonrandomassignment,providedthat

a

g¼0andthesupporton

Ygissufﬁcientlylargetoruleoutthereﬂectionproblem,J>0andJ

is largeenoughtoproducemultipleequilibria.Thismeansthat groupgmaycoordinateonanequilibriumexpectedaveragechoice level otherthanthelargestof thepossibleequilibriaassociated withitwhileanothergroupg0maycoordinateonanequilibrium other than the lowest possible expected average choice level amongthoseitcouldhaveattained.Oneofthesituationswhere thismayhappenisthecaseofassortativematching,wherehigher groupqualityisrelatedtohigherindividualquality.Inourcontext, thismayrefertothecasewhereindividualswithinaspeciﬁcgroup share common genetic traits or the same eating habits. It is, though,importanttostressthatthemultipleequilibriumresultsin

Brock and Durlauf (2007) hold for large groups. In particular,

Krauth (2006) suggests that multiplicityof equilibria for small groupsmayhappenforlowerthresholdvaluesofJ.

We adopt Brock and Durlauf (2007) approach to (partial) identiﬁcation.Intheempiricalexerciseweneedtoaccommodate thelargesupportassumptiononYg.Morespeciﬁcally,weconsider

twocases.Inthefirstcasethevariablewithlargesupportisthelog of expenditures. The secondspecification includes alsoaverage adultweightandaverageadultheight.Itisinterestingtonotice, though,thatthereseemtobenocleartheoreticalguidelineson how many variables withlargesupportwould be necessaryto avoidthereflectioneffect(seee.g.Blumeetal.,2011,p.907).

Dealing with unobservable heterogeneity in the context of socialinteractionmodelsisgenerallyaverychallengingtasksince, assuggestedinBlumeetal.(2011),

ei

and

a

gareundertheorized.9

Nonetheless,therearestillanumber ofapproachesthatcanbe exploited when sufﬁcient information is available. The most problematicissueinoursettingishowtodealwiththegroupﬁxed effects

a

g. The simplest solution here is just to deﬁne

a

g¼

d0YgþJmgi(BlumeandDurlauf,2006).Thisis,weapproximate

a

g withobservables and changethe number of variables in Yg to

assessthestabilityoftheestimates.Ourmodelwillincludealarge numberofgroupcharacteristicsthatmayreasonablydetermine obesity and that are either related to genetic factors or to behaviouralfactors.10

Directestimationof

a

gviagroupdummieswouldbeimpossible

inourcontextduetothelargenumberoffamilies(2954)compared tothenumberofindividuals(3906).Wecouldhoweveridentifya restrictednumberofgroupsbyclusteringfamilieswithcommon

characteristics. The resulting number of groups would be

considerably smaller than the total number of families.

Allocating the families to speciﬁc groups may be done via an appropriate clustering algorithm. We givemore details onthis

approach and on the corresponding results in Appendix B

(TablesB19andB20).

BrockandDurlauf(2007)proposearathercleverwaytodeal with

a

g.Theysuggestspecifying

a

g asalinearfunctionofYgand

constructing an auxiliary variable Wi¼F1_e ðPð

vi

¼1jXi;Yg;

a

gÞÞ

where Fe is the distribution of

ei

. This would correspond to Wi¼kþc0Xiþd0YgþJmgiþ

a

g.Theconstructionof thesample analogue for Wi would rely on the existence of suitable

information.11_In_our_case,_once_again,_the_limited_availability_of

data does not allow us to consider this alternative. A further interesting possibility is due to Graham (2008), where

a

g is

interpretedasarandomeffect.Hence,Cov½

a

g;

ei

¼0,fori2g.This

approachisjustiﬁed,atleastinGraham'sclassroomproblem,by therandomassignmentofteacherstoclassrooms.

5.Estimationandinference

ItisinterestingtonoticethattheresultsinBrockandDurlauf (2007)differfromtheclassicalapproachestopartialidenti ﬁca-tion. The latter case involves the identiﬁcationof bounds and theirsubsequentestimationbymeansofappropriatestatistical procedures.Instead, Brockand Durlauf's (2007) theory-depen-dentapproachstudieshowintroducingunobserved heterogene-itywouldaffectthepropertiesofthemodel.Furthermore,theydo notestablishprobabilitybounds(Blumeetal.,2011).12_Hence,_we

assumethat,givena certainparameter space

Q

,thereexistsa subsetof

Q

,say

Q

I,suchthatF0¼Fufor

u

2

Q

IwhereF0isthe

truedistributionofthedataandF_uisourparametricmodel.We referto

Q

Iastheidentiﬁedsetforwhichanappropriateestimator

has to be found. In what follows, we focus our attention on conﬁdencesetsforindividualparameters.Inthisregard,weﬁnd thefollowingdecompositionuseful:

u

¼ð

_m

0_;

_h

0_Þ0_,_where

_m

_is_the

parametervector we areinterested in and

h

can beseen as a nuisanceparameter.Wedenotetheidentiﬁedsetforthesubvector

m

asMI.

We tackle the estimation problem by using a method

introducedin Chenet al. (2018).The conﬁdencesets produced usingthis approach aresimple tocalculate,workwellinﬁnite

8

Areﬂectionproblemariseswhenthedependentvariable(weightoutcomeof childi) and the explanatory variable of interest (peer variable mgi) are simultaneously determined, causing an endogeneity issue.

9

Instrumentalvariables maybea viable optionto deal withﬁxed effects. However, social interaction models do not generally suggest a theoretical justiﬁcationtoexcludevariablesfromthemodelitself.Thisfeatureisknownas openendedness(Blumeetal.,2011).

10

Arecentstrandofliteraturestressesthatanysimilarityinweightduetoshared householdenvironmentsisundetectableandignorable(CawleyandMeyerhoefer, 2012;Kinge,2016;Wardleetal.,2008)

11

Fortheproblemofsocialinteractionsintheclassroomexample,Brockand Durlauf(2007)suggestusingtestscorestorecoverasampleanalogueforWi.

12

See,e.g.,Manski(2003)andMolinari(2020)foracomprehensivetreatmentof partialidentiﬁcation.

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samples and asymptotically achieve frequentist coverage. The estimated confidence sets can be compared to the confidence intervals produced by standard estimation methods for binary choice models under the assumption of point identification. Intuitively,onemayarguethat(lackofpoint)identificationmay not be anissue if confidence sets and confidence intervalsare similar.

Inthissectionwedescribehowwebuildvalidconﬁdencesets usingProcedure1andProcedure3inChenetal.(2018).Theyare

both simple to compute but the former tends to produce

conservativeconﬁdencesetswhilethelattercanonlybeapplied to scalar subvectors of the parameter vector of interest. The associated numerical results are collected in Tables 2–9. AppendixBcontainstherobustnesscheckresults.

5.1.Conﬁdencesets

The methods proposed in Chen et al. (2018) exploit some classical ideas of Bayesian computation. The estimation of the conﬁdencesetsis infactbased onsamplingfromtheposterior distributionoftheparameters.Hereweprovideabriefdescription ofthethreeproceduresintroducedintheirpaper.Consideringthe discussioninSection4onthetreatmentoftheﬁxedeffect

a

g,the

modelthatweestimateis

vi

¼Z0i

u

þui

whereZi¼ð1;X0i;Y0g;mgiÞand

u

¼ðk;c0;d 0

;JÞ0 _is_a _{p-dimensional}

vector where p¼2þrþs.13 Let us consider a parametric loglikelihoodfunctionthatdependsonaparametervector

u

that takesvaluesinaset

Q

andthedataZi

LNð

u

Þ¼ 1 N XN i¼1 log fð

u

;ZiÞ:

Let us denote the identiﬁed set as

Q

I¼f

u

2

Q

:F0¼Fug,

whereF_uisourparametricmodelandF0isthetruedistributionof

thedata.Theposteriordistribution,say

P

N,of

u

giventhedataZis

d

P

Nð

u

;ZÞ¼

exp ðNLNð

u

ÞÞd

P

ð

u

Þ

R

Q exp ðNLNð

u

ÞÞd

P

ð

u

Þ

where

P

ð

_u

Þisapriordistribution.The100

a

%conﬁdenceset,say b

Q

afor

QI

iscomputedinathreestepprocedure: Table1

Summarystatistics.a

Mean S.d. Min. Max. Obs.

A.Individualcharacteristics

Childobesity 0.292 0.455 0 1 3906

Age 9.987 2.595 6 14 3906

Gender(male) 0.495 0.500 0 1 3906

B.Householdcharacteristics

Shareofotheroverweight/obesechildren 0.072 0.174 0 0.667 3906

Shareofoverweight/obeseadults 0.420 0.351 0 1 3906

Householdsize 4.120 1.018 2 11 3906

Childrenbornofpreviousmarriage 0.007 0.086 0 1 3906

Monthlyexpenditure(Euro) 2131 1346 237 16,998 3906

EmployedRP 0.813 0.390 0 1 3906

StudentorhousewifeRP 0.053 0.225 0 1 3906

Retiredorotheremp.statusRP 0.023 0.150 0 1 3906

Mother'seducation(Master) 0.126 0.331 0 1 3906

Mother'seducation(Bachelor) 0.029 0.169 0 1 3906

Mother'seducation(HighSchool) 0.348 0.476 0 1 3906

Mother'seducation(JuniorHigh) 0.412 0.492 0 1 3906

Mother'seducation(PrimarySchool) 0.068 0.252 0 1 3906

Centralornorthernregion 0.578 0.494 0 1 3906

C.Behaviouralcharacteristics

Siblings(regularly)practisingsport 0.176 0.244 0 0.75 3906

Siblingslunchathome 0.186 0.246 0 0.75 3906

Siblingswalkingtoschool 0.089 0.193 0 0.75 3906

SiblingsTVwatchingeveryday 0.810 0.317 0 1 3906

Parentssodadrinks 0.155 0.362 0 1 3906

Parentssmoking 0.372 0.483 0 1 3906

Childrenaveragefruitportions 1.125 0.754 0 5.5 3906

Adultsaveragefruitportions 1.147 0.778 0 5.5 3906

D.Proxiesforgeneticcharacteristics

Meanadultweight(kg) 70.676 9.159 35.5 117.5 3906

Meanadultheight(cm) 168.959 5.694 110 193 3906

Chronicdisease 0.251 0.439 0 1 3906

Diabetes 0.034 0.181 0 1 3906

E.Othercharacteristics

CPI(2010=100) 106.022 0.614 104.6 108.1 3906

%obeseadultsbyregion 24.954 5.476 17.7 36.1 3906

a_This_table_includes_summary_statistics_on_individual_{characteristics}_of_children_(panel_A),_household_{characteristics}_(panel_B),_in_some_cases_related_to_the_household's referenceperson(RP),behaviouralvariables(panelC),proxiesforgeneticcharacteristics(panelD),regionalvariables(panelE).

13

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Table 2

Conﬁdence intervals and conﬁdence sets for the logit model.a

Dependent variable: child obesity

Logit CCT1 CCT3 Logit CCT1 CCT3 Logit CCT1 CCT3

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Share of other obese children ½0:913; 1:695 ½0:568; 2:046 ½0:925; 1:689 ½0:706; 1:526 ½0:301; 1:991 ½0:711; 1:513 ½0:679; 1:530 ½0:328; 1:956 ½0:690; 1:529 constant ½1:744; 1:173 ½3:060; 2:446 ½1:744; 1:130 ½3:623; 22:087 ½13:770; 19:093 ½3:292; 19:093 ½3:576; 22:700 ½11:224; 19:916 ½3:329; 18:600 gender ½0:158; 0:448 ½0:028; 0:579 ½0:158; 0:443 ½0:157; 0:453 ½0:011; 0:578 ½0:160; 0:452 ½0:150; 0:448 ½0:061; 0:545 ½0:154; 0:447 age ½0:121; 0:383 ½0:310; 0:438 ½0:113; 0:375 ½0:085; 0:428 ½0:415; 0:697 ½0:078; 0:428 ½0:099; 0:420 ½0:218; 0:567 ½0:092; 0:416 age2 _{½0:025; 0:001} _{½0:036; 0:012} _{½0:024; 0:000} _{½0:028; 0:002} _{½0:043; 0:015} _{½0:028; 0:002} _{½0:028; 0:001} _{½0:035; 0:004} _{½0:027; 0:002}

Share of obese adults ½0:626; 1:042 ½0:441; 1:228 ½0:628; 1:042 ½0:537; 0:962 ½0:322; 1:224 ½0:541; 0:960 ½0:031; 0:651 ½0:170; 0:797 ½0:023; 0:651 log expendituresðEuroÞ ½0:284; 0:054 ½0:387; 0:048 ½0:283; 0:056 ½0:136; 0:116 ½0:294; 0:269 ½0:135; 0:116 ½0:117; 0:139 ½0:210; 0:184 ½0:115; 0:136

Households characteristics No No No Yes Yes Yes Yes Yes Yes

Regional characteristics No No No Yes Yes Yes Yes Yes Yes

Genetic proxies No No No No No No Yes Yes Yes

Behavioural variables No No No No No No Yes Yes Yes

N, p 3737, 7 3737, 7 3737, 7 3737, 19 3737, 19 3737, 19 3737, 30 3737, 30 3737, 30

a

This table provides 95% confidence intervals using the standard logit model as if it were identified and 95% confidence sets using the approach described inChen et al. (2018). CCT1 and CCT3 denote Procedure 1 and Procedure 3 respectively. The models are estimated using all the families.

Table 3

Dependent variable: child obesityð6 age 11Þ

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Household characteristics No No No Yes Yes Yes Yes Yes Yes

N, p 2503, 7 2503, 7 2503, 7 2503, 19 2503, 19 2503, 19 2503, 30 2503, 30 2503, 30

a_{This table provides 95% confidence intervals using the standard logit model as if it were identified and 95% confidence sets using the approach described in}

Chen et al. (2018). CCT1 and CCT3 denote Procedure 1 and Procedure 3 respectively. The models are estimated using all the families with children with 6 age 11.

F. Crudu, L. Neri and S. Tiezzi Economics and Human Biology 40 (202 1) 1 0 095 1 8

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Table 4

(1) (2) (3) (4) (5) (6) (7) (8) (9)

N, p 1234, 7 1234, 7 1234, 7 1234, 19 1234, 19 1234, 19 1234, 30 1234, 30 1234, 30

a_{This table provides 95% confidence intervals using the standard logit model as if it were identified and 95% confidence sets using the approach described in}_{Chen et al. (2018)}_{. CCT1 and CCT3 denote Procedure 1 and Procedure 3}

respectively. The models are estimated using all the families with children with 12 age 14.

Table 5

Conﬁdence intervals and conﬁdence sets for the probit model.a

Probit CCT1 CCT3 Probit CCT1 CCT3 Probit CCT1 CCT3

(1) (2) (3) (4) (5) (6) (7) (8) (9)

N, p 3737, 7 3737, 7 3737, 7 3737, 19 3737, 19 3737, 19 3737, 30 3737, 30 3737, 30

a

This table provides 95% confidence intervals using the standard probit model as if it were identified and 95% confidence sets using the approach described inChen et al. (2018). CCT1 and CCT3 denote Procedure 1 and Procedure 3 respectively. The models are estimated using all the families.

Crudu, L. Neri and S. Tiezzi Economics and Human Biology 40 (202 1) 1 0 095 1 9

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Procedure1(Wholeparametervector) 14 drawBsamplesf

_u

ð1Þ;...;

u

ðBÞ_g

fromtheposteriordistribution

P

NviaaMonteCarloMarkovchain(MCMC)sampler14;

15 calculatetheð1

_a

ÞquantileoffLNð

u

ð1ÞÞ;...;LNð

u

ðBÞÞg,say

zN

_;_a;

16

deﬁne b

Q

aas b

Q

a¼f

u

2

Q

:LNð

u

Þ

zN;

_ag.

Itispossibletoadaptprocedure1toconstructconﬁdencesets forthesubsetvector

m

.Theso-calledprojectionconﬁdencesetfor MI isdeﬁned as bMproj_a ¼f

m

:ð

m

0;

h

0Þ02 b

Q

a;forsome

h

g. The

pro-jection conﬁdencesetisknowntobeconservativeinparticular whenthedimensionofthesubvector

m

issmallerincomparison with the dimension of

u

. Let us now deﬁne the set Hm¼ f

_h

:ð

_m

0_;

_h

0_Þ0₂

_Q

_g_and_the_pro_ﬁle_likelihood_for_M

I

PLNðMIÞ¼ inf m2MI sup h2HmLNð

m

;

h

Þ:

Let

D

ð

u

b

Þ;b¼1;...;Bbeanequivalenceset,i.e.asetof

u

2

Q

that produce the same likelihood values and let

Mð

_u

b_Þ_¼_f

_m

:ð

_m

0_;

_h

0_Þ0₂

_D

_ð

_u

b

Þ;forsome

h

g.Then,theproﬁle likeli-hoodforMð

u

b

Þis

PLNðMð

u

bÞÞ¼ inf _m_2Mð_ub_Þ sup _h_2H_mLNð

m

;

h

Þ:

Wecannowdescribethesecondprocedureforasubvector

m

of

u

:

Procedure2(Subvector)

(a) drawBsamplesf

_u

ð1Þ;...;

u

ðBÞ_g

fromtheposteriordistribution

P

NviaaMCMCsampler;

(b) calculate the ð1

a

Þ quantile of

fPLNðMð

u

ð1ÞÞÞ;...;PLNðMð

u

ðBÞÞÞg,say

zN;

_a;

(c) deﬁne Mba as

b

Ma¼f

m

2M: sup _h2HmLNð

m

;

h

Þ

zN;

_ag.

Wenowdescribeasimpleprocedureforscalarsubvectors.Let usdeﬁnethelikelihoodratio

LRNð

u

Þ¼2N LNðb

u

ÞLNð

u

Þ

foramaximizer b

u

.Procedure3canbeimplementedintwosimple steps

Procedure3(Scalarsubvector) (a) calculateamaximizer b

u

;

(b) deﬁne Mb_a as b M_a¼f

_m

2M: inf _h2HmLRNð

m

;

h

Þqx 2 1 ag, where qx21 a isthe

a

quantileofthe

x

2

1distribution.

As suggested in Chen et al. (2018), the conﬁdence sets are compared totheconﬁdenceintervalsprovidedby thestandard probitandlogitmodels.

5.2.Results

Tables 2–9 in this section contain95% confidenceintervals obtained using the standard logit and probit models as if identification were possible and 95% confidence sets obtained usingProcedure1andProcedure3oftheapproachdescribedin Section5.1anddenotedasCCT1andCCT3respectively.Tables2–7

showestimatesconsideringfamilieswithatleastonechild(i.e.all thefamilies).Inadditiontothat,weconductouranalysisinsubsets ofthedatabasedonage.Twosubsetsareconsidered6age11 (Tables3and6)and12age14(Tables4and7).Moreover,each modelisestimatedusingfoursetsofcovariates.15Tables8and9, ontheotherhand,displaytheestimatesofaparsimoniousmodel thatincludesthenumberofchildrenasaregressoraswellasthe averageheightandweightofadultsinthefamily.Alsoforthese modelsweconsidertheagesubsetsdescribedabove.AppendixB showssimilarmodelsforfamiliesthatincludeeithermorethan onechild or onlyonechild. These resultsserve thepurposeof checkingthestabilityofthecon_ﬁdencesets.

Ourdependentvariableisa binaryvariableforachild being overweight/obese. Theexplanatory variables of interestare the shareofotheroverweightandobesechildreninthehousehold(the peereffect)andtheshareofoverweightandobeseadultsinthe household. The other covariates introduced in the models are described in Section 3. Speciﬁcally, these are the individual characteristicsofthechild(gender,ageandage2₎_and_household

characteristics, like the share of other overweight and obese children in the household (the key covariate), the share of overweightandobeseadultsinthehouseholdandthehousehold consumptionexpendituresinlogs–thematchedvariable.Astothe effectofthekeycovariate,weobservethatthepresenceofother obesechildreninthefamilyhasapositiveeffectontheprobability thatachildbeobese.Thisresultisrobusttoallthespecificationsof themodelweconsidered.Theresultsobtainedwiththestandard binarychoicemodel,eitherlogitorprobit,areverysimilartothe confidencesetscomputedviaCCT3.Thismaysuggestthatifthere is no point identification, this has only a mild effect on the confidenceintervals. TheconfidencesetsobtainedviaCCT1 are generallylargerthan thosebuiltwithCCT3: this isinlinewith whatissuggestedinChenetal.(2018).Furthermore,wefindthat theeffectofobeseadultsinthefamilyisgenerallysmallerthan thatofpeerchildren.Wealsofindthatbyincludingthegenetic proxiesandthebehaviouralvariablestheconfidencesetstendto movetotheleftandinsomecasestheyincludezero.Ifwelookat theresultsfor thetwo agesubsetswenotice somedifferences. However,theymaybecausedbythedifferenceinsamplesize.The modelspecificationinTables8and9showsasizableshifttowards the right of the confidence sets associated to the peer effect variable.Thisresultisobservedforallagesubsets.Theconfidence sets tend to get larger when we consider the subset of older children;however,alsothiseffectmaybecausedbythereduced samplesize.Theeffectoftheshareofobeseadultsseemstobe more ambiguous asit is smallerin comparison withthe other model specifications and, for the subset of older children, it includes zero.Thisresult mayalso dependon theinclusion of averageadultweightandheight,astheyarerelatedtotheshareof obeseadultsinthefamily.Astothevariablegenderofthechild,its effectis significant in almostall themodel specifications,with confidence sets defined in a positive subset of the real line, meaningthattheprobabilityofbeingoverweight/obeseislarger

14

Chenetal.(2018)suggestusingasequentialMonteCarlosamplerasMCMCmay benumericallyunstable.Wedonotexperiencesuchproblemsinourapplication.

15 _Covariates_include_household,_behavioural_and_regional_{characteristics}_as_well asproxiesforgeneticcharacteristics.Householdcharacteristicsincludehousehold size,whetherthehouseholdlivesinaNorthernorCentralItalianregion,the employmentstatusof theRP, thelevelofeducation ofthemother.Regional characteristicsincludesageneralCPIattheregionallevelandtheregionalshareof obeseadultsin2012.Geneticproxiesincludethemeanheightandweightofthe adultmembersofthefamilyandtwodummyvariablesforwhethertheRPorher spousesufferfromachronicdiseaseordiabetes.Behaviouralvariablesincludesthe shareofsiblings(excludingthechildunderconsideration)agedbetween6and18 watchingTVeveryday,havinglunchathome,practicingphysicalactivitiesona regularbasisandwalkingtoschool,dummiesforwhethertheparentsconsume sodadrinksorsmoke.Thefullsetofconﬁdenceintervalsandconﬁdencesetscanbe foundinTablesB1–B3andTablesB10–B12.

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Table 6

(1) (2) (3) (4) (5) (6) (7) (8) (9)

N, p 2503, 7 2503, 7 2503, 7 2503, 19 2503, 19 2503, 19 2503, 30 2503, 30 2503, 30

a _{This table provides 95% confidence intervals using the standard probit model as if it were identified and 95% confidence sets using the approach described in}_{Chen et al. (2018)}_{. CCT1 and CCT3 denote Procedure 1 and Procedure 3}

respectively. The models are estimated using all the families with children with 6 age 11.

Table 7

(1) (2) (3) (4) (5) (6) (7) (8) (9)

N, p 1234, 7 1234, 7 1234, 7 1234, 19 1234, 19 1234, 19 1234, 30 1234, 30 1234, 30

a

This table provides 95% confidence intervals using the standard logit model as if it were identified and 95% confidence sets using the approach described inChen et al. (2018). CCT1 and CCT3 denote Procedure 1 and Procedure 3 respectively. The models are estimated using all the families with children with 12 age 14.

Crudu, L. Neri and S. Tiezzi Economics and Human Biology 40 (202 1) 1 0 095 1 11

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Table 8

6 age 14 6 age 11 12 age 14

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Share of obese adults ½0:020; 0:687 ½0:323; 1:073 ½0:034; 0:672 ½0:130; 0:917 ½0:268; 1:338 ½0:138; 0:917 ½0:691; 0:594 ½1:394; 1:326 ½0:673; 0:576 log expendituresðEuroÞ ½0:255; 0:021 ½0:368; 0:104 ½0:254; 0:025 ½0:273; 0:002 ½0:433; 0:162 ½0:272; 0:001 ½0:381; 0:071 ½0:559; 0:273 ½0:374; 0:063 number of children ½0:228; 0:024 ½0:342; 0:092 ½0:227; 0:028 ½0:216; 0:019 ½0:329; 0:154 ½0:212; 0:018 ½0:420; 0:011 ½0:693; 0:229 ½0:414; 0:008 average adult weight ½0:013; 0:042 ½0:005; 0:059 ½0:013; 0:041 ½0:003; 0:032 ½0:024; 0:052 ½0:002; 0:031 ½0:030; 0:085 ½0:001; 0:118 ½0:031; 0:085 average adult height ½0:051; 0:016 ½0:070; 0:004 ½0:051; 0:016 ½0:050; 0:008 ½0:074; 0:012 ½0:049; 0:008 ½0:078; 0:011 ½0:118; 0:027 ½0:077; 0:012

N, p 3737, 10 3737, 10 3737, 10 2503, 10 2503, 10 2503, 10 1234, 10 1234, 10 1234, 10

a

This table provides 95% confidence intervals using the standard logit model as if it were identified and 95% confidence sets using the approach described inChen et al. (2018). CCT1 and CCT3 denote Procedure 1 and Procedure 3 respectively. The models include the number of children in the family as a regressor.

Table 9

6 age 14 6 age 11 12 age 14

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Share of obese adults ½0:012; 0:411 ½0:215; 0:642 ½0:020; 0:407 ½0:074; 0:552 ½0:206; 0:825 ½0:084; 0:546 ½0:379; 0:363 ½0:802; 0:777 ½0:376; 0:351 log expendituresðEuroÞ ½0:153; 0:013 ½0:232; 0:067 ½0:151; 0:014 ½0:166; 0:001 ½0:262; 0:099 ½0:165; 0:003 ½0:223; 0:037 ½0:376; 0:183 ½0:221; 0:035 number of children ½0:135; 0:015 ½0:206; 0:055 ½0:133; 0:016 ½0:130; 0:011 ½0:213; 0:091 ½0:129; 0:010 ½0:230; 0:008 ½0:373; 0:145 ½0:226; 0:004 average adult weight ½0:008; 0:025 ½0:002; 0:036 ½0:008; 0:025 ½0:002; 0:020 ½0:014; 0:032 ½0:001; 0:020 ½0:017; 0:049 ½0:000; 0:067 ½0:017; 0:049 average adult height ½0:031; 0:010 ½0:043; 0:003 ½0:031; 0:010 ½0:030; 0:005 ½0:045; 0:009 ½0:030; 0:005 ½0:045; 0:007 ½0:068; 0:016 ½0:045; 0:007

N, p 3737, 10 3737, 10 3737, 10 2503, 10 2503, 10 2503, 10 1234, 10 1234, 10 1234, 10

a _{This table provides 95% confidence intervals using the standard probit model as if it were identified and 95% confidence sets using the approach described in}

Chen et al. (2018). CCT1 and CCT3 denote Procedure 1 and Procedure 3 respectively. The models include the number of children in the family as a regressor.

F. Crudu, L. Neri and S. Tiezzi Economics and Human Biology 40 (202 1) 1 0 095 1 12

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for malesthanfor females.The childageis insignificant inthe quadratic polynomial specification in almost all the model specifications. As to the effect of household consumption expenditures,introducedinthemodelasalogarithmic transfor-mation,itseffectissignificant,bothforthelogitandprobitmodels, either for the standard binary choice model or for the one estimatedviaCCT3,onthesetincludingallchildrenaged6–14 yearsandforthesubsetofchildrenaged6–11years,onlywhenthe modelsdonotincludeotherhouseholdbehaviouralvariables.In suchcases,theconfidencesetsaredefinedinanegativesubsetof the realline(see columns(1) and (3) inTables2, 3, 5, and 6) meaningthattheprobabilityofbeingoverweight/obesedecreases withhouseholdconsumptionexpenditures.Itthusseemsthatthe impact ofconsumption expenditures, viewed asa proxyof the economic status ofthe household,ismediated bytheincluded householdandbehaviouralcovariates.Onepossibleinterpretation ofthisresultisthatthosefamiliesinbettereconomicconditions can offer better opportunities for a healthy diet and physical activity.Ontheotherhand,resourceconstraintsleadtoalackof opportunitiesandtoalackofinformationontheingredientsofa healthychildren'sdietandonhealthybehaviours.

6.Conclusion

This paper contributes to the literature on child obesityby assessingtheeffectofpeersonchildren'sweightoutcomesinthe contextofanarrowpeergroup.Weassessedwhetherthepresence of overweight andobese family members– otherchildren and adults – affects children's weightoutcomes.To thebest ofour knowledgenostudyhasyetanalyzedtheimpactoftheobesity statusofothermembersofafamilyonchildobesity.Wechoseto carryoutouranalysisnotpresumingpointidentificationforour models. With respect to that aspect, we contribute to the integration, albeit in a rather specificcontext, of theliterature onpartialidentificationforsocialinteractionmodelsandthaton partiallyidentifiedmodelsinindustrialorganization(Blumeetal., 2011;Tamer,2003).

WeusedadatasetonItalianchildrenresultingfromstatistical

matching of the 2012 cross sections of two surveys, the

Multipurpose Household Survey and the Household Budget

Survey, both supplied by ISTAT. To provide valid inference for ourpartiallyidentiﬁedmodelsweusethemethodproposedby

Chenetal.(2018).Wefoundevidenceofastrong,positiveimpact of overweight and obese peer children in the family and of overweightandobeseadultsonchildweightoutcomes. Interest-ingly,inallempiricalmodelswefindthattheimpactofoverweight andobesepeerchildreninthehouseholdislargerthantheimpact ofadults.Wealsofindthat,whengeneticproxiesandbehavioural variablesareadded,theimpactofthepresenceofoverweightand obeseadultsisdriventozerowhiletheimpactofoverweightand obesepeerchildrenremains positive.Ourresultsareconsistent withstudiesonchildimitationbehaviourandtherolemodelage (Zmyjetal.,2012a,b;ZmyjandSeehagen,2013),whereprolonged contactwithpeersledchildrentoimitatepeersmorethanadults. Despitegrowingratesofchildobesity,empiricalevidenceon thefactorsaffectingItalianchildweightoutcomesremainspoor. Furtherexplorationofcausalpathwayslinkingsocialinteraction withinthefamilyandchildobesityisthereforedesirable.Weshow thatthepresenceoverweight/obeseparentsand/orpeersiblingsis animportantfactoraffectingchildobesityinItaly.Inparticular,we showthatthepresenceofotheroverweightandobesechildrenin the family is themost important factoraffecting child obesity. Indeed,whentherichestmodelspecificationisused,i.e.whenwe includeproxiesforgeneticvariables(columns7,8,9inTables2,4,

5,and7),theshareofobeseadultsisnolongersigniﬁcant,while thepeereffectvariableisstillsigniﬁcant.Thisresultsuggeststhat

within-the-family obesity is driven more by peer siblings interactionthanbyinteractionsbetweentheparentandthechild. In general, it seems that family characteristics and behaviours affectchildrenhabitsandtheirprobabilityofbeingobese.Inthis

context, family-based programmes, based on collaborative

approaches,mayhelppreventingchildobesity.Particularattention shouldbepaidtohouseholdswithmorethanoneoverweightchild where acollaborative approachcouldhave muchmore impact. Moreover,sincesiblingsrelationshipsarethelongestlastingones, theuseofatruefamily-basedapproachintakingactionagainst childhoodobesitywillincreasethelikelihoodthatchangesinchild healthbehaviourswillbesustainable(see,e.g.,BergeandEverts, 2011).

Authorcontributions

Federico Crudu: Methodology; formal analysis; writing – originaldraft;writing– review&editing;software.LauraNeri: Conceptualization; data curation; formal analysis; writing – originaldraft;writing–review&editing.SilviaTiezzi: Conceptu-alization;datacuration;writing–originaldraft;writing–review& editing.

AppendixA.Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,athttps://doi.org/10.1016/j.ehb.2020.100951.

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