The Distribution of Consumption-Expenditure Budget Shares. Evidence from Italian Households

ContentslistsavailableatSciVerseScienceDirect

Structural

Change

and

Economic

Dynamics

j o ur n a l ho m e p age :w w w . e l s e v i e r . c o m / l o c a t e / s c e d

The

distribution

of

household

consumption-expenditure

budget

shares

Matteo

Barigozzi

a

_,

_Lucia

_Alessi

b

_,

_Marco

_Capasso

c

_,

_Giorgio

_Fagiolo

d,∗

a_{LondonSchoolofEconomics,London,UK}

b_{EuropeanCentralBank,FrankfurtamMain,Germany}

c_{UNU-MERITandSchoolofBusinessandEconomics,MaastrichtUniversity,TheNetherlands}

d_{Sant’AnnaSchoolofAdvancedStudies,LaboratoryofEconomicsandManagement,Pisa,Italy}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received11February2011

Receivedinrevisedform

26September2011 Accepted27September2011 Available online xxx JELclassification: D3 D12 C12 Keywords:

Householdconsumptionexpenditure

Budgetshares

Sumoflog-normaldistributions

a

b

s

t

r

a

c

t

Thispaperexploresthestatisticalpropertiesofhouseholdconsumption-expenditure

bud-getsharedistributions– definedastheshareofhouseholdtotalexpenditurespentfor

purchasing aspecificcategory ofcommodities – foralargesample ofItalian

house-holdsintheperiod1989–2004.Wefindthathouseholdbudgetsharedistributionsare

fairlystableovertimeforeachspecificcategory,butprofoundlyheterogeneousacross

commoditycategories.Wethenderiveaparametricdensitythatisableto

satisfacto-rily characterize(fromaunivariateperspective)householdbudgetshare distributions

and:(i)isconsistentwiththeobservedstatisticalpropertiesoftheunderlyinglevelsof

householdconsumption-expendituredistributions;(ii)canaccommodatetheobserved

across-categoryheterogeneityinhouseholdbudget-sharedistributions.Finally,we

taxono-mizecommoditycategoriesaccordingtotheestimatedparametersoftheproposeddensity.

Weshowthattheresultingclassificationisconsistentwiththetraditionaleconomicscheme

thatlabelscommoditiesasnecessary,luxuryorinferior.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Thestudy ofhouseholdbudgetallocation –i.e., how

the budgetof a household is allocated tobuy different

commodities–isoneofthemosttraditionaltopicsin

eco-nomics(PraisandHouthakker,1955).Householdbudget

shares contain useful information to shed light onthis

issue.Indeed,thehouseholdbudgetshareforagiven

com-modity category g is defined as the ratio between the

expenditureforthecommoditycategorygandtotal

house-holdresources,asmeasuredby,e.g.,totalexpenditureor

totalincome.

Inthelastdecades,thistopichasreceivedalotof

atten-tionbyappliedeconomists.Inparticular,manyeffortshave

beendevotedtodevelopstatisticaldemandfunctionsfor

homogeneousgroupsofcommodities,e.g.,byrelatingthe

∗ Correspondingauthor.

E-mailaddress:giorgio.fagiolo@sssup.it(G.Fagiolo).

expenditureofconsumersorhouseholdsforagiven

com-moditycategorytopricesandindividual-specificvariables

astotalexpenditureorincome,householdsize,

head-of-householdage,andsoon(Deaton,1992;Blundell,1988).A

paradigmaticexampleinthislineofresearchisthe

analy-sisofEngelcurves(Engel,1857;Deaton,1992;Blundell,

1988;Chai andMoneta,2010), whichdescribehow the

expenditureforagivencommoditycategoryvarieswith

household’stotalresources(Lewbel,2008).

Sucha researchprogram hasbeenmostly

character-izedbyatheory-drivenapproach(Attanasio,1999).Infact,

the parametric specifications that are employed in the

estimationofeach specificdemandfunctionarein

gen-eraltakentobe consistent(albeitina weakway)with

someunderlyingtheoryofhouseholdexpenditure

behav-ior, which very often is the standard model based on

utilitymaximization undertakenbyfullyrationalagents

(Banks et al., 1997; Blundellet al., 2007). Furthermore,

nomatter whetherparametric or non-parametric

tech-niquesareemployed,theestimationofdemandsystems

0954-349X/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved.

or Engel curves compresses household heterogeneity –

foranygivenincomeortotal expenditurelevel– tothe

knowledgeofthefirsttwo moments(atbest)of

house-holdexpenditurelevelorbudgetsharedistributionforthe

commoditycategoryunderstudy.

This of course is fully legitimate if the aim of the

researcher is to empirically validate a given

theoreti-cal model, or if there are good reasonsto believe that

thedistributionunderanalysiscanbefullycharacterized

byits first two moments. However, froma more

data-drivenperspective,constraininginthiswaytheexploration

of the statistical properties of the observed household

expenditurepatternsmaybeproblematicforanumberof

reasons.

First, heterogeneity of household

consumption-expenditure patterns is widely considered as a crucial

featurebecause,asPasinetti(1981)notices:“Atanygiven

levelofpercapitaincomeandatanygivenpricestructure,

the proportion of income spent by each consumer on

anyspecificcommoditymaybeverydifferentfromone

commodityto another”. This suggests that, in order to

fullycharacterizesuchheterogeneity,oneshouldperform

distributionalanalysesthatcarefullyinvestigatehowthe

shape–andnotonlythefirsttwomoments–ofhousehold

consumption expenditure and household budget share

distributions change over time and between different

commoditycategories.

Second,understandingheterogeneity maybe

impor-tant to build sound micro-founded, macroeconomic,

consumption models that go beyond the often

dis-putablerepresentative-agentassumption(Kirman,1992;

Hartley,1997;GallegatiandKirman,1999).Forexample,

Caselli and Ventura(2000) showthat models based on

therepresentative-agent assumption impose almost no

restrictionsonhouseholdconsumptionexpenditureand

budgetsharedistributions.Onthecontrary,ForniandLippi

(1997) demonstrate that heterogeneity is crucial when

aggregatingindividualbehaviorinmacromodels.

Further-more,Ibragimov(2005)providessupporttotheinsightthat

higher-than-twomomentscanhavearelevantimpacton

thedynamicsofmacromodels(ontheseandrelatedpoints,

seeHildenbrand,1994,amongothers).

Third,adoptingamoretheory-freeapproachfocusedon

distributionalanalysismayhelptodiscoverfreshstylized

factsrelatedtohowhouseholdsallocatetheir

consump-tionexpendituresacrossdifferentcommoditycategories.

In fact, theory-free approaches aimed at searching for

stylizedfactsarenot newineconomics and

economet-rics (see inter alia Kaldor, 1961; Hendry, 2000). More

recently,this perspective has been revived in the field

ofeconophysics,wherethestatisticalpropertiesofmany

interestingmicroandmacroeconomicvariables(e.g.,firm

sizeandgrowthrates,industryandcountrygrowthrates,

wealthandpersonalincome)havebeensuccessfully

char-acterizedbyusingparametrictechniques(Chatterjeeetal.,

2005;ClementiandGallegati,2005;Axtell,2001;Bottazzi and Secchi, 2006; Fagiolo et al., 2008). These studies

showthat,despitetheturbulencetypicallydetectedatthe

microeconomiclevel(e.g.,entryandexitoffirms;positive

andnegativepersistentshockstopersonalincome),there

existsanincredible highlevelofregularityintheshape

ofmicroeconomiccross-sectiondistributions,bothacross

yearsandcountries.

Notwithstandingsuchsuccessfulresults,similar

distri-butionalanalyseshave not beenextensivelyperformed,

sofar,onconsumption-relatedmicroeconomicvariables

suchashouseholdconsumptionexpendituresandbudget

shares,forwhichreliableanddetailedcross-sectiondata

arealsoavailable.Thisissomewhatsurprisingbecause– as

Attanasio(1999)notices–understandingconsumptionis

crucialtobothmicro-andmacro-economists,asitaccounts

forabouttwo-thirdsofGDPanditdecisivelydetermines

(andmeasures)socialwelfare.

Thereare onlythree exceptions–to thebestofour

knowledge–tothislackofdistributionalstudieson

house-hold consumption indicators. Chronologically, the first

instanceisthepioneeringworkofAitchison(1986)on

com-positionaldataanalysis.There,budget-sharedataareonly

toillustrateapplicationsofamultivariateapproachthat

aimsatparametricallymodelingdatadefinedby

construc-tiononthesimplex(seealsoMcLarenetal.,1995;Fryetal.,

1996,foranapplicationofcompositional-dataanalysisto

consumptionbudgetshareswithinthecontextofmodified

almost-idealdemandsystems).Nothingissaid,however,

ontheeconomicimplicationsofsuchanapproach,neither

themethodisappliedtootherdatabases.Weshallgoback

tothesepointsbelow.

More recently, Battistin et al.(2007) employ

expen-diture and incomedata fromU.K. and U.S.surveys and

showthattotalhouseholdconsumptionexpenditure

dis-tributionsarewell-approximatedbylog-normaldensities

(or,astheyputit,are“morelog-normalthanincome”).

In a complementary paper, Fagiolo et al. (2010) argue

that log-normalityisvalidonlyasa firstapproximation

forItaliantotalhouseholdconsumptionexpenditure

dis-tributions, while a refined analysis reveals asymmetric

departures from log-normality in the tails of the

dis-tributions. Both contributions focus on characterizing,

from a univariate perspective, the dynamics of

house-hold consumption-expenditure aggregate distributions

only. Theissue ofexploring thestatisticalproperties of

householdconsumptionexpenditureorbudgetshare

dis-tributionsdisaggregatedamongcommoditycategoriesis

notaddressed.

Thispaperisapreliminaryattempttofillthesegaps.

Todoso,weemploydatafromthe“SurveyofHousehold

IncomeandWealth”(SHIW)providedbytheBankofItaly

tostudyhouseholdconsumptionexpenditureandbudget

sharedistributionsforasequenceof8wavesbetween1989

and2004.Tofullyexploitthedatabase,wefocusonfour

commodity categories:nondurable goods,food,durable

goods,and insurancepremia. Notethat foodis actually

a subcategory of nondurable goods,but for its intrinsic

importanceweconsideritasaseparatecommodity

cat-egorythroughout thepaper.Sinceinsurancepremiaare

rarelystudiedintheconsumptionliteratureasacategory

onitsown,weexplicitlyconsidertheminthis studyto

understandwhethertheyexhibit differentdistributional

propertiesascomparedtomoretraditionalconsumption

categories.

We aim at empirically investigating the

distributions(andconsumptionexpendituredistributions)

of thesefourcommoditycategories and theirdynamics

withaparametricapproach,wherebyunconditional

distri-butionswemeanherenotconditionedtototalhousehold

resources,i.e.,incomeortotalexpenditures.More

specif-ically, we look for a unique, parsimonious, closed-form

univariate density family that: (i) is able to

satisfacto-rily fit observed unconditional household budget share

distributions, so as to accommodate the existing

het-erogeneityemergingacrosshouseholds,amongdifferent

commodity categories and over time; (ii) is consistent

withthestatisticalpropertiesofthe(observed)household

consumptionexpendituresdistributionsemployedto

com-putebudgetsharedistributions;(iii)featureseconomically

interpretable parameters that,once estimated,can help

onetobuildeconomicallymeaningfultaxonomiesof

com-moditycategories.

Webeginwithadescriptiveanalysisaimedat

empir-ically exploring thestability of householdbudget share

distributionsovertime.Estimatedsamplemomentsshow

thattheshapeofthehouseholdbudgetsharedistribution

ofeachgivencommoditycategorydoesnotdramatically

change over thetime interval considered.However, for

anygivenwave,thereemergesalotofacross-commodity

heterogeneityintheobservedshapesofhousehold

bud-getsharedistributions.Moreprecisely,weshowthatthe

underlyinghousehold-consumptionexpenditures

univari-ate distributions – for any given wave and commodity

category –are well-proxiedbylog-normal distributions

(withverydifferentparameters).

Yet,apreliminarymultivariateinvestigationindicates

that existing multivariate parametric density families

definedonthesimplex(i.e.,describingsharesthatsumto

one;seeAitchisonandEgozcue,2005,forareview)arenot

abletosatisfactorilymodelourdata.Therefore,weturnto

aunivariateapproachandwederiveanoriginalfamilyof

densities,definedovertheunitinterval,whichisconsistent

withthedetectedlog-normalityofhousehold

consump-tionexpendituresdistributions.Thepreciseformulationof

theclosed-formdensitycanbeshowntodependonthe

chosenapproximationfortherandomvariabledefinedas

thesumof(possiblycorrelated)log-normaldistributions.

Intheliteraturethereexisttwopossibleapproximations,

namelythelog-normalandtheinverse-Gamma,whichwe

bothfittoourdata.Tobenchmarkourresults,wealsofit

householdbudgetsharedistributionswithunivariateBeta

distributions,whichareinprincipleveryflexibledensities

definedovertheunitintervalbutlackanyconsistencywith

theshapeoftherandomvariableswhichhouseholdbudget

sharesstemfrom.

Wefindthat,inthecaseofItaly,forallthewavesunder

studyandforallthecommoditycategories,theproposed

density family – using either approximation –

outper-formstheBetainfittingobservedhouseholdbudgetshare

distributionsforthemajorityofcases.Indeed,according

tosimple measures ofgoodness-of-fit (e.g.,theaverage

absolutedeviation), theproposeddensity familyis able

tobetteraccommodatetheexistingshape-heterogeneity

that characterizes householdbudget share distributions

acrossdifferentcommoditycategories.Furthermore,the

estimated parameters of the proposeddensity allowto

reproduceaneconomicallymeaningfultaxonomyof

com-modity categories, which interestingly maps into the

traditionalclassificationofcommoditiesamongnecessary,

luxuryorinferiorgoods.

The paper is structured as follows. In Section 2 we

describethedatabasethatweemployintheanalysisand

wediscusssomemethodologicalissues.Section3presents

apreliminarydescriptiveanalysisofhousehold

consump-tionexpenditureandbudgetsharedistributions,whereas

Section 4 discusses multivariate analyses. In Section 5

we derivetheproposedfamily of univariate theoretical

densities.Section6presentsfittingresultsobtainedwith

thatdensity family,andcomparesthemwithBeta

vari-ates.Section7brieflyreportsonsomeinterpretationsof

ourexercisesintermsofcommoditycategorytaxonomies.

Finally,Section8concludes.

2. Dataandmethodology

Theempiricalanalysisbelowisbasedonthe“Survey

ofHouseholdIncomeandWealth”(SHIW)providedbythe

BankofItaly.TheSHIWisoneofthemainsourcesof

infor-mationonhouseholdincome and consumption inItaly.

Indeed,thequalityoftheSHIWisnowadaysverysimilarto

thatofsurveysinothercountrieslikeFrance,Germanyand

theU.K.SHIWdataareregularlypublishedintheBank’s

supplementstotheStatisticalBulletinandmadepublicly

availableonline1_{(seeBrandolini,1999;Battistinetal.,2003}

foradditionaldetails).

TheSHIWwasfirstlycarriedoutinthe1960swiththe

goalof gatheringdataonincome and savings ofItalian

households.Overtheyears,thesurveyhasbeenwidening

itsscope.Householdsarenowaskedtoprovide,in

addi-tiontoincomeandwealthinformation,alsodetailsontheir

consumptionbehaviorandeventheirpreferredpayment

methods.Sincethen,theSHIWwasconductedyearlyuntil

1987(exceptfor1985)andeverytwoyearsthereafter(the

surveyfor1997wasshiftedto1998).

Thepresentanalysisfocusesontheperiod1989–2004.

Wethereforehave8waves.Thesampleusedinthemost

recentsurveyscomprisesabout8000households(about

24,000 individuals distributed across about 300 Italian

municipalities).Thesampleisrepresentativeofthe

Ital-ianpopulationandisbasedonarotatingpaneltargetedat

4000units.Availableinformationincludesdataon

house-holddemographics(e.g.,ageofhouseholdhead,numberof

householdcomponentsandgeographicalarea),disposable

income,consumptionexpenditures,savings,andwealth.2

In this study, we employ yearly data on (nominal)

aggregate,householdconsumption expendituresand on

thefollowingdisaggregatedcommoditycategories:

non-durablegoods(N),durablegoods(D),andinsurancepremia

(I). Nondurable goods include also food (F), which we

1 _{http://www.bancaditalia.it/statistiche/indcamp/bilfait.}

2 _{Moreprecisely,thesampleunitisthehouseholddefinedinabroad}

senseasagroupofindividualsrelatedtoeachotherbylinksofblood,

marriageoraffection,livingtogetherandpooling(partof)theirincomes.

Thesurveywasrestrictedtohouseholdswithatleasttwomembers.Data

werenotaggregatedacrossfamilysizes.Formoredetails,seeBrandolini

consider as a separate (sub-)category of commodities.

Accordingtothedefinition of theBank ofItaly,

expen-dituresfornondurablegoodscorrespondtoallspending

onbothfoodandnon-fooditems,excludingexpensesfor

durablegoodsandinsurance,maintenance,mortgageand

rentpayments.Theexpendituresforfoodincludespending

onfoodproductsinshopsandsupermarkets,andspending

onmealseatenregularlyoutsidehome.Household

expen-dituresfordurablegoodscorrespondtoitemsbelongingto

thefollowingcategories:preciousobjects,meansof

trans-port, furniture, furnishings, household appliances, and

sundryarticles.Finally,thecommoditycategorylabeledas

“insurances”includesthefollowingformsofinsurance:life

insurance,privateorsupplementarypensions,annuities

andotherformsofinsurance-basedsaving,casualty

insur-ance,and healthinsurancepolicies.Notethat insurance

expenditure,strictusensu,mightbeconsideredasaform

ofsaving.However,weconsiderinsurancesasa

commod-itycategoryforbasicallytworeasons:(i)insuranceforms

mightalsobeseen asconsumption goods,insomuchas

theycoveractualexpenseswhicharebornebythe

house-hold(e.g.,forpharmaceuticalproducts);(ii)theinsurance

itselfmightbeconsidered–andindeedappearsin

theo-reticalmodels–asagood,whichanagentmightormight

notpurchase:theonlydifferencewithrespecttoa

tradi-tionalconsumptiongoodstandsinthefactthatthedegree

ofriskaversioninfluencestheamountofinsurance

pur-chased.ThemajordrawbackoftheSHIWdatabaseisthat

itdoesnotallowforfurtherdisaggregationof

consump-tioncategories intomore detailed groups. Furthermore,

thedisaggregationlevelofcategoriesasdurablegoodsand

insurancepremiaisnottotallycomparable,astheformer

certainlycomprisesmuchmoreitemsthanthelatter.As

alreadydiscussedabove,however,insurancepremiaare

notfrequently studiedin theconsumption literatureas

acategoryonitsown.Therefore,itseemsinterestingto

explicitlyaddresshereitsstudytobetterunderstandthe

extenttowhichitsstatisticalpropertiesdifferfromthose

ofmoretraditionalconsumptioncategories.

The SHIW database includes a variable recording

householdtotal(aggregate)expenditure.Thisquantityis

reportedby households in the SHIWindependently on

theirexpenditurefordisaggregatedcommoditycategories.

Therefore,itdoesnotnecessarilycorrespondtothesumof

expendituresofthethreemacroconsumptioncategories

consideredhere(N,DandI),thesummakingupon

aver-age80%oftotal expenditures.Inwhat follows,weshall

employtotal householdexpenditureasaproxyfortotal

householdresources(moreonthatbelow).Household

bud-getsharesareaccordinglycomputedasratiosofnominal

yearlyquantities.Moreformally,ourdatastructure

con-sistsofthedistributionofyearlyhouseholdbudgetshares

definedas: Bh,t_i =C h,t i Ch,t, (1) wheret∈T={1989,1991,1993,1995,1998,2000,2002,

2004} are survey waves, i∈{N, F, D, I} are the four

commoditycategories,C_ih,t isthe(nominal)consumption

expenditureofhouseholdh=1,...,Ht forthe

commod-itycategoryi,andCh,t_{isthe(nominal)totalconsumption}

expenditureofhouseholdh,asreportedintheSHIW.All

household consumption expenditure observations have

been preliminary weighted using appropriate sample

weightsprovidedbytheBankofItaly.Weightingensures

thatsocio-demographicmarginaldistributionsareinline

withthecorrespondingdistributionsfoundinItalian

Sta-tisticalOffice(ISTAT)populationstatisticsandlaborforce

surveys.3_{Outliers–definedasobservationsgreaterthan10}

standarddeviationsfromthemean–havebeenremoved.4

Sinceineachwavethereweresomecasesofunrealistic

(e.g.,zeroor negative) aggregateconsumption

expendi-turefigures,wedroppedsuchobservationsandwekept

onlystrictly positiveones.Wealsodropped households

forwhich yearlyexpendituresforatleastone

commod-itywaslargerorequaltototalexpenditure(asreportedin

theSHIW).Sinceweruleoutborrowing,Bh,t_i ∈(0,1).

Finally,weexcludedzeroobservationsfromthe

anal-ysis,forthefollowingreasons:(i)itisnotclearwhether

zeroentriesfornondurablegoodsmeananull

consump-tion expenditure or rather theyare due to mistakes in

datacollection;(ii)thedecisionwhethertobuyadurable

goodor aninsuranceornot (dependingforexampleon

whetherhouseholdincomeexceedssomethreshold)is

dif-ferentfromandprecedesthedecisiononthebudgetshare

possiblyallocated tothis good;wefocusonthesecond

stepofthedecisionalprocess;(iii)thedegreeofbunching

atzeroonetypicallyfindsinthedistributionsofdurable

goodsandinsuranceexpendituresisinfluencedbyfactors

whicharelikelytovaryoverthebusinesscycle;(iv)

sam-plemomentscomputedincludingzeroeswouldbepoorly

informative,giventherelativelylargeproportionofzero

observations.Therefore,weendedupwithachanging(but

stillverylarge)numberofhouseholdsineachwaveHt(see

Table3).

Inwhatfollows,weshallalsomakeuseofan

alterna-tiverepresentationofourdatathatallowsonetolookat

budget-share vectorsaspointsin thesimplex. For each

householdhandwavet,considerthevector

Bh,t_=(Bh,t

N ,Bh,tD ,Bh,tI ,BOh,t), (2)

whereBh,tO isthebudgetsharesforallothercommodity

categoriesdefinedas:

Bh,tO =1−(Bh,tN +Bh,tD +Bh,tI ), (3)

ObviouslythisimpliesthatBh,tliesinthesimplex,whichin

ourcasehasa3-dimensionalEuclideanspacestructure.In

otherwords,weshalldefinethehouseholdconsumption

expenditurecategory“Others”(O)asCOh,t=Ch,t−(CNh,t+

CDh,t+CIh,t)andcomputebudgetsharesasin(1)fori∈{N,

D,I,O}.

3_{Aninterestingandopenissueconcernstheimpactthatweighting,}

throughpopulationtrends,mayhaveonthedistributionalpropertiesof

budgetshares.Additionaldataonphenomenasuchaspopulation

age-ing,increasingshareofforeignpopulation,loweringfertilityrateand

intra-countrymigrationflowsmayshedsomelightontheimplications

ofdemographictrendsforourresults.

4_{Thechoiceof10standarddeviationswasdoneinordertomaximize}

thenumberofobservationstoberetainedinthesample.Ourexercises

showthatchoosingasmallernumberofstandarddeviationsdoesnot

0 0.5 1 1.5 2 2.5 x 105 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10 5

Total Consumption Quantiles

Income Quantiles

Fig.1.Quantile–quantileplotcomparingincomeandtotalconsumption

expendituredistributions.Wave,2004;Y-axis,incomequantiles;X-axis,

consumptionquantiles.

Twoimportantpointsdeservetobediscussed.First,we

usetotalexpendituresinsteadofincometoproxy

house-holdtotalresourcesand computebudgetshares.Thisis

primarilydoneinordertoseparatetheproblemof

allo-catingtotalconsumptiontovariouscommoditiesfromthe

decisionofhowmuchtosaveoutofcurrentincome.Notice

that this is commonpractice in the relevant literature.

Indeed,duetotherelativelyhigherreliabilityof

expendi-turedata(ascomparedtoincomeones),mostofempirical

studiestypicallyusehouseholdconsumptionexpenditures

eveniftheoreticalmodelsareoriginallydevelopedinterms

oftotalincome(see,e.g.,Banksetal.,1997).Sinceincome

isavailableintheSHIWdatabase,wereplicatedour

exer-cises by defining household budget shares in terms of

household-incomeratioswithoutanyappreciable

differ-ences in theresultsas faras descriptive analyseswere

concerned.Quantile–quantileplotscomparinghousehold

incomeandtotalconsumptionexpendituredistributions,

reported in Fig.1 for wave 2004,show that as income

increasestheproportionofincomedevotedtototal

con-sumptionalsoincreases.

Second,asalreadymentioned,thisstudyisnot

explic-itlyconcernedwiththeestimationofEngelcurves,either

withparametricornon-parametricapproaches(Engeland

Kneip,1996;ChaiandMoneta,2008).Conversely,wetreat

householdbudgetsharesasagnosticvariables thathave

aneconomicmeaning‘perse’.Moreover,notethatEngel

curvesdescribetherelationshipbetweenconditional

aver-agesofhouseholdconsumption expenditures(orbudget

shares) for a particular commoditycategory and levels

ofincomeortotalconsumptionexpenditure,where

aver-agesarecomputedconditionaltolevelsofincomeortotal

consumptionexpenditure,andpossiblyotherexplanatory

variables. In this paper, we begin instead to study the

statistical propertiesof unconditionalhouseholdbudget

sharedistributions,thatis–foranycommoditycategory

andwave–wepooltogetherhouseholdsirrespectiveof

theirincomeor total consumptionexpenditure,and we

consequentlystudytheshapeoftheensuingdistributions

andtheirdynamics.Inotherwords,wedonotcompress

the overall across-household heterogeneity existing for

each commodity category and wave, as done in

Engel-curve studies. This is because the goal of the paper is

simplytocharacterizethedistributionalshapeof

uncon-ditional household budget share distributions and not

how they change with household total budget. As we

brieflyrecallintheconcludingsection,thismight

envis-ageapossibleextensionofthepresentwork.Onemight

indeed condition household budget share distributions,

foreach commoditycategoryand wave, tototal

house-holdresourcesandinvestigatehowconditionalhousehold

budgetsharedistributionschangeasincomeortotal

con-sumptionexpenditureincreases.

3. Apreliminarystatisticalanalysis

In this section,we beginwith a preliminary,mainly

descriptive,statistical analysisofItalian household

con-sumption expenditure and budget share distributions,

mainlyfocused oninvestigating whethersuch

distribu-tions–andtheircorrelationstructure–exhibitstructural

changesovertime.

3.1. Householdexpendituredistributions

Let us start with household consumption

expendi-turedistributions.Fig.2 showskerneldensityestimates

of thelogs of household consumption expenditure

dis-tributions for waves 1989, 1993, 1998, and 2002 (for

thesake ofexposition).5 _{Apreliminaryvisualinspection}

of thefourpanels indicatesthat, withtheexception of

insurancepremia,theshapeofanygivenhousehold

con-sumptionexpendituredistributiondoesnotdramatically

changeovertime. Thisevidenceseemstobeconfirmed

byTable1,wherewereportestimatedsamplemoments

forloggedhouseholdconsumptionexpenditure

distribu-tions, and by Fig. 3, which shows theirevolution over

time. We will go backtothis questionin more details

in the next section. For the moment, notice that

sam-ple means show a positive trend in time because we

are considering nominalquantities, but we expect also

real values to display a rightward average shiftdue to

economic growth. However,insurance expenditure

dis-tributions display a more pronounced trend, which is

probablyduetotheobservedstructuralincreasein

expen-ditureforinsurancepremiafromthelate90salsoinreal

terms. Furthermore, Table 2 shows sample correlations

and p-values for the null hypothesis of no correlation

5 _{Allkerneldensityplotsinthepaperhavebeenobtainedusingdensity}

estimatescomputedwiththefunctionkdensfromtheStatasoftware

pack-age(http://ideas.repec.org/c/boc/bocode/s456410.html).Densitieshave

beenestimatedon100pointsforthelogconsumptionlevels,and50points

forthebudgetshares.AGaussiankernelhasbeenused.Thebandwidth

hasbeenselectedaccordingtothe“ruleofthumb”suggestedbySilverman

(1986).Althoughseveralrefinementsofthe“ruleofthumb”havebeen

presentedintheliterature(see,e.g.,SheatherandJones,1991),weopted

fortheoriginalmethodby(Silverman,1986)becauseitwastheonlyone

Fig.2.Kernel-densityestimatesofloggedhouseholdconsumptionexpendituredistributions:evolutionovertime.Solidline,year1989;dashed-dotted

line,year1993;dottedline,year1998;dashedline,year2002.

between household consumption expenditure

distribu-tionsfordifferentcommoditiesin2004,wherecorrelation

coefficientsarecomputed(hereandinwhatfollows)only

for households with non-zero expenditure for all

com-modity categories.As expected, thecorrelations among

householdconsumptionexpendituredistributionsareall

stronglypositiveandsignificant.

Theparabolicshapeofloggedhouseholdconsumption

expenditurekernelsinFig.2hintstothepossibilitythatthe

(disaggregated)distributionsmightbewell-proxiedby

log-normaldensities.Noticethattheexistingliteratureshows

thataggregate householdconsumption expenditure

dis-tributionsaretypicallylog-normallydistributed(Battistin

etal.,2007).InFagioloetal.(2010)weshowthat,asafirst

approximation,similarevidenceistruealsofortheItalian

totalhouseholdconsumptionexpenditure.Itisthen

worth-whiletocheckifalsocommodity-disaggregatedhousehold

consumptionexpenditures behavethe sameway.Fig.4

indicatesthata log-normal densityprovidesreasonable

fitsalsoforourhouseholdconsumptionexpenditure

dis-tributionsdisaggregatedacrossourcommoditycategories.

Table1confirmsthisfinding,asthelogsofdisaggregated

householdconsumptionexpendituredistributionsexhibit

skewnessandkurtosisvaluesveryclosetowhatwouldbe

expectediftheoriginaldistributionswerelog-normal(i.e.,

0and3respectively).

As discussed in Battistin et al. (2007), consumption

and income data generally suffer from under reporting

(especiallyinthetails)andoutliers,andItaliandataare

not an exception (Brandolini, 1999). In order to

min-imize the effect of gross errors and outliers, we have

employed robust statistics toestimate themoments of

householdconsumptionexpendituredistributions(Huber,

1981). More specifically, we have estimated the third

momentwithquartileskewness(GroeneveldandMeeden,

1984)andkurtosisusingMoors’soctile-basedrobust

esti-mator(Moors,1988).Robust-estimatoranalysissupports

log-normalityofhouseholdconsumptionexpenditure

dis-tributions.Infact,accordingtostandardbootstraptests,

robust skewnessand kurtosisof loggedhousehold

con-sumptionexpendituredistributionsareoftenclosetotheir

expectedvaluesinnormalsamples(0and1.233,

respec-tively).Sincelog-normalityseemstobeagoodproxyalso

for COh,t distributions, it is temptingtocheck for

multi-variatelog-normalityofthe4-dimensionalvectorCh,t=

(CNh,t,CDh,t,CIh,t,COh,t). Unfortunately, a standard

Energy-testformultivariatenormality(SzekelyandRizzo,2005)

stronglyrejectsthenullhypothesiswithp-valuecloseto

zero(weshallgobacktothispointinSection4).

3.2. Householdbudget-sharedistributions

We turn now toa descriptive analysisof household

budgetsharedistributions.Fig.5showstheplotsof

Table1

Momentsofloggedhouseholdconsumptionexpendituredistributionsvs.waves.Avg,averagevaluesoverthewholeperiod;TC,totalconsumption;N,

nondurables;D,durables;I,insurances;F,food.ThefigureslabeledasN+D+Ionlyrefertohouseholdswithnon-zeroexpenditureforeachcommodity

category.

Stats Waves Avg

1989 1991 1993 1995 1998 2000 2002 2004 N Nobs. 7409 7209 6223 6258 5588 6277 6361 6281 6451 Mean 8.551 8.518 8.715 8.876 8.863 8.949 8.942 9.073 8.811 Std.dev. 1.014 1.064 0.962 0.918 0.939 0.939 0.982 0.909 0.966 Skewness 0.201 0.111 0.281 0.108 0.007 0.102 0.174 0.143 0.141 Kurtosis 2.844 2.893 2.927 2.703 2.710 2.727 2.647 2.810 2.783 D Nobs. 2534 2352 2082 1856 2091 1920 1833 1961 2079 Mean 6.554 6.713 6.529 6.900 6.902 7.078 6.881 6.879 6.805 Std.dev. 1.626 1.656 1.615 1.593 1.560 1.588 1.635 1.568 1.605 Skewness −0.041 0.033 0.028 0.017 0.123 0.047 0.158 0.296 0.083 Kurtosis 2.698 2.678 2.561 2.509 2.503 2.454 2.560 2.683 2.581 I Nobs. 1780 1928 2257 2961 2652 2575 2175 2164 2312 Mean 5.501 5.604 5.737 5.853 6.198 6.359 6.358 6.551 6.020 Std.dev. 1.500 1.599 1.557 1.567 1.476 1.398 1.408 1.408 1.489 Skewness −0.069 −0.216 −0.269 −0.284 −0.504 −0.166 −0.228 −0.279 −0.252 Kurtosis 2.555 2.788 2.689 2.649 3.218 2.641 2.965 3.592 2.887 F Nobs. 7409 7228 6235 6261 5596 6281 6366 6281 6457 Mean 7.738 7.808 8.014 8.108 8.089 8.119 8.133 8.241 8.031 Std.dev. 0.978 1.059 0.969 0.913 0.930 0.947 0.975 0.919 0.961 Skewness 0.214 0.109 0.253 0.099 0.017 0.108 0.138 0.116 0.132 Kurtosis 2.753 2.844 3.029 2.744 2.770 2.805 2.635 2.853 2.804 N+D+I Nobs. 896 904 1099 1225 1310 1162 930 1016 1068 Mean 9.148 9.156 9.269 9.484 9.435 9.607 9.640 9.716 9.432 Std.dev. 0.932 1.030 0.904 0.830 0.909 0.877 0.945 0.861 0.911 Skewness 0.326 0.078 0.335 0.186 0.011 0.174 0.232 0.109 0.182 Kurtosis 2.848 3.131 2.818 2.534 2.718 2.420 2.558 2.721 2.718 TC Nobs. 7416 7237 6245 6274 5598 6282 6370 6285 6463 Mean 8.907 8.905 9.093 9.256 9.300 9.369 9.377 9.510 9.215 Std.dev. 1.028 1.095 0.978 0.925 0.934 0.939 0.975 0.908 0.973 Skewness 0.230 0.150 0.240 0.120 0.077 0.132 0.262 0.236 0.181 Kurtosis 2.828 2.865 2.855 2.690 2.659 2.704 2.659 2.866 2.766

firstvisualinspectiondoesnotseemtodetectstrongtime

dependence in theshape of budget-share distributions.

Conversely,as expected,theirshapes differsignificantly

acrosscommoditycategories.Householdbudgetshare

dis-tributionsfornondurablegoodsandfoodarerelativelybell

shapedandalargemassofobservationsisshiftedtowards

the rightof the unit interval. Kernels of durable goods

andinsurancepremiaareinsteadmuchmoreright-skewed

andmonotonicallydecreasing.Notealsothat

insurance-premiakernelsexhibitarelevantirregularityintheright

tail,duetoasmallsample-sizeproblem.Thestrong

across-commodityheterogeneitythatclearlyemergesintheshape

ofhouseholdbudgetsharedistributionssuggeststhatin

ordertofindaunique,parsimonious,parametricstatistical

modelabletosatisfactorilyfitthedata,onewouldrequire

averyflexibledensityfamily.

Estimatedsamplemomentsofhouseholdbudgetshare

distributions are reported in Table 3. On average, 68%

oftotalhouseholdexpendituresisrelatedtonondurable

goods, while food accounts for 33% of the total. Much

lessisspentondurablegoodsandinsurancepremia,as

theyrespectively represent– onaverage– 13% and 5%

oftotalhouseholdconsumptionexpenditures.No

appre-ciabledifferencesarefoundbyreplacingthedenominator

Table2

Correlationsamonghouseholdconsumptionexpendituredistributionsandp-values(inbrackets)forthenullhypothesisofnocorrelation.Wave2004.TC,

totalconsumption;N,nondurables;D,durables;I,insurances;F,food.

N D I F D 0.40 – – – (0.00) – – – I 0.49 0.39 - – (0.00) (0.00) – – F 0.87 0.29 0.44 – (0.00) (0.00) (0.00) – TC 0.92 0.58 0.51 0.80 (0.00) (0.00) (0.00) (0.00)

1989 1991 1993 1995 1998 2000 2002 2004 6 6.5 7 7.5 8 8.5 9 9.5 Sample Mean

Nondurables Durables Insurances Food Total

1989 1991 1993 1995 1998 2000 2002 2004 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Sample Std Dev

Nondurables Durables Insurances Food Total

1989 1991 1993 1995 1998 2000 2002 2004 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 Sample Skewness

Nondurables Durables Insurances Food Total

1989 1991 1993 1995 1998 2000 2002 2004 2.6 2.8 3 3.2 3.4 3.6 Sample Kurtosis

Nondurables Durables Insurances Food Total

Fig.3. Samplemomentsofloggedhouseholdconsumptionexpendituredistributionsandtheirevolutionovertime.

ofhouseholdbudgetshareswiththesumofnondurables,

durablesandinsuranceexpenditure(i.e.,byreplacingthe

variabletotalexpenditureprovidedbytheBank ofItaly

withthesumofthecommoditycategoriesofexpenditure

employedinthisstudy).

Fig.6plotsthetimeevolutionofthefirstfoursample

momentsofhouseholdbudgetsharedistributions.In

gen-eral,momentsarerelativelystableovertime.Theexception

isrepresentedagainbyinsurancepremia,whichdisplay

highlyincreasingmoments from1989 on.Inparticular,

skewnessandkurtosisexhibitbigjumpsin1995,andthen

movetohigherlevels.Instead,standarddeviationsteadily

increasesfrom1989on.Noticealsothatskewnesssignsdo

notchangeovertime:theyarealwaysnegativefor

non-durablegoodsandalwayspositiveforallothercategories

(seeTable3).

Themainmessage comingfromtheforegoingvisual

analysisisthathouseholdconsumptionandbudgetshare

distributionsdidnotdramaticallychangetheirstructural

propertiesovertime.However,allpreviouskernelplots

werenotcomplementedbytheirconfidenceintervals.This

prevents one to correctly compare kernel density

esti-matesandtoassesswhetherkernelsforsuccessiveyears

arereallydifferent.Tobetterexplorethispoint,weplot

kernel-estimates95%confidencebandsforboth

consump-tionlevelsandbudgetsharesinthefirst(1989)andfinal

(2002)wave.6_{Ifconfidenceintervalsdonotoverlap,wecan}

concludethatthereissomestatisticalevidencefor

non-constantkerneldensitiesovertime,atleastasfarasthe

changebetween1989(firstwave)and2002(lastwave)is

concerned.

Fig.7showssomeexamplesofconfidencebandsplots

for both household consumption log-levels and budget

shares.Due tospace constraints,we onlyplot durables

andnon-durables,butourmainresultsholdalsoforthe

othertwocategories.Thisexerciseshowsthatindeed

con-fidence intervalsoverlap only for someintervalsof the

domain,i.e.,thedensityestimatesarenotstatistically

con-stantovertimeovertheentireranges.Nevertheless,the

densityestimatesseemtopreserveovertimesome

prop-ertiesrelatedtotheshapeofthedistribution,properties

thatareidiosyncratictothetypeofcommodityconsidered.

6_{Confidenceintervalshavebeenestimatedwiththeoptionciofthe}

Statafunctionkdens,with100bootstrapreplications.Somepartsofthe

lowerboundoftheconfidenceintervaldonotshowupintheplotsbecause

5 6 7 8 9 10 11 12 100

101 102 103

Log Nondurables Expenditure

density (log−scale) Data Normal fit 2 4 6 8 10 12 100 101 102 103

Log Durables Expenditure

density (log−scale) Data Normal fit 2 4 6 8 10 12 100 101 102 103

Log Insurances Expenditure

density (log−scale) Data Normal fit 4 5 6 7 8 9 10 11 12 100 101 102 103

Log Food Expenditure

density (log−scale)

Data Normal fit

Fig.4. Anexampleofnormalfitstologgedhouseholdconsumptionexpendituredistributions.Wave,2000.

Inotherwords,mostofthecharacteristicsoftheempirical

distributions(relativepositionofthemode,etc.)seemto

dependmoreonthecommoditycategoryunderanalysis

andnotontheparticularcross-sectionalwave.Therefore,

wecanconcludethatourcross-wavecomparisonsbring

evidenceinfavoroftheneedforfindingatheoretical

dis-tributionthatcanaccommodatedifferentbudgetshares,

which areheterogeneousacrossgoodsmuch morethan

overtime.

Therelativeconstancyoftheshapeofhousehold

expen-diture and budget-share distributions is a strongresult

alsoinlightoftheintroductionoftheEuroin2001and

theincreaseinaveragenominal(andreal)individual

con-sumptionlevelsexperiencedintheobservedperiodand

alreadynoticedabove.Ourresultsthat,despiteacommon

trendduetoeconomicgrowththatpushedupaverage

con-sumptionlevels,therewasabalancedturbulencewithin

eachdistributions,withsomehouseholdsmovingtowards

theuppertailandothersmovingtowardsthelowertail,

withouthowever changing verymuch the shape ofthe

distribution.Inotherwords,theoverallimpactof

macroe-conomicdynamicsontheshapeofthemicrodistribution

wasrelativelyweak.

We now turn to study the correlation structure of

householdbudgetsharedistributions.Table4reportsthe

correlationmatrixfor2004,togetherwiththep-valuesfor

thenullhypothesisofnocorrelation.Fig.8plotsinstead

thetimeevolutionofthecorrelationsbetweenthe

distri-butionsofnondurablegoods,durablegoodsandinsurance

premia.Note that all correlations are fairly stable over

timeandexhibitsignsconsistentwiththeeconomic

intu-ition.Indeed,nondurablesarenegativelycorrelatedwith

durables–theaveragecorrelationbeing−0.54%–which

in turn are negatively correlated with food (here the

averagecorrelationcoefficientis−0.3%).Negative

correla-tionsindicatethatwhenhouseholdsincreasetheirrelative

expenditurefordurablegoods,theytendtoreducetheir

relativeexpenditurefornondurablegoods,includingfood.

Noticealsothatthecorrelationbetweeninsuranceandall

othercategoriesisstatisticallynonsignificant.

Aremarkisinorder.Correlationanalysisisknownto

besubjecttomanydifficultiesinthecaseofcompositional

data(Aitchison,1986,chapter3.3),mainlyduetothe

pres-enceof the linear constraint imposing that the sumof

budgetsharesmustbeone.Notice,however,thatinour

Fig.5.Kernel-densityestimatesofhouseholdbudgetsharedistributions:evolutionovertime.Solidline,year1989;dashed-dottedline,year1993;dotted

line,year1998;dashedline,year2002.

B’sisnotobtainedasthesumofhouseholdconsumption

expenditureforN,D,andI(seeabove).

We have also performed a robust-moment analysis,

usingmedianandmeanabsolutedeviationasrobust

esti-matorsforlocationand scaleparameters,in additionto

therobust-skewnessestimatoralreadyintroducedabove.

Results confirm, overall, our previous findings: robust

momentsfor (logged) householdconsumption

expendi-tureandbudgetsharedistributionsarestableovertime,

withthesameexceptionsfoundbefore.

4. Amultivariateparametricapproach

The most direct strategy to determine a parametric

modelthat is abletosatisfactorily fit budget-share

dis-tributionswouldbetoemploytherepresentation in(2)

and(3),andfollowamultivariateapproach.SinceBh,t_,h=

1,...,Hliesinthesimplexforeachwave,itis

straight-forward toapply standard techniques of compositional

dataanalysis. Unfortunately, this lineof attack facesin

ourcaseanumberofdifficulties.First,asnotedin

Mateu-Figuerasetal.(2007a),“thereisadearthofsuitablemodels

withwhichtoadequatelymodelcompositionaldatasets”

(p.217),and someofthem(i.e.,theDirichletclass)are

straightforwardlyrejectedbythedata.Indeed,theDirichlet

class(Connor,1969)requiresonetoassumethatoriginal

householdconsumptionexpendituredataareindependent

andalltheoff-diagonalelementsofthecorrelationmatrix

arenegative(seeAitchison,1986,Chapter3).Second,and

mostimportanthere,almostallviablealternativesrelyon

mathematicaltransformationsoftheoriginalbudget-share

vector Bthat, albeit veryuseful forstatistical purposes,

arenotabletopreservetheoriginaleconomicmeaningof

budget-sharesasfractionsofhouseholdtotalexpenditure

devotedtoaparticularclassofcommodities.Indeed,the

basicideaofcompositionalanalysisistotransform

com-positionaldata(i.e.,sharevectorslyingonthesimplex)in

suchawaytoobtainvectorsofquantitiesdefinedonreal

spaces.

Thisinthecasealsoofthemostsimpleofsuch

trans-formations,namelytheadditivelogisticratio(ALR),which

inourcasewouldmapthevectorB=(BN,BD,BI,BO)lying

inthesimplex,intothevector:

ALR(B)=log(B_−V/BV)∈R3, (4)

whereVisoneofthe4originalcomponents{N,D,I,O}and

Table3

Momentsofhouseholdbudgetsharedistributionsvs.waves.Avg,averagevaluesoverthewholeperiod;N,nondurables;D,durables;I,insurances;F,food.

ThefigureslabeledasN+D+Ionlyrefertohouseholdswithnon-zeroexpenditureforeachcommoditycategory.

Stats Waves Avg

1989 1991 1993 1995 1998 2000 2002 2004 N Nobs. 7409 7208 6223 6258 5588 6277 6361 6281 6451 Mean 0.717 0.702 0.703 0.699 0.667 0.675 0.668 0.666 0.687 Std.dev. 0.141 0.154 0.151 0.142 0.149 0.143 0.147 0.146 0.147 Skewness −0.873 −0.736 −0.797 −0.686 −0.698 −0.726 −0.652 −0.610 −0.722 Kurtosis 3.674 3.199 3.583 3.425 3.317 3.508 3.500 3.339 3.443 D Nobs. 2534 2352 2082 1856 2091 1920 1833 1961 2079 Mean 0.130 0.148 0.118 0.127 0.137 0.136 0.118 0.105 0.127 Std.dev. 0.138 0.150 0.143 0.144 0.151 0.149 0.144 0.137 0.144 Skewness 1.640 1.591 1.967 1.752 1.755 1.648 1.994 2.233 1.822 Kurtosis 5.767 5.663 7.182 6.139 6.146 5.610 7.072 8.207 6.473 I Nobs. 1780 1928 2257 2961 2652 2575 2175 2164 2312 Mean 0.039 0.043 0.048 0.049 0.062 0.062 0.060 0.066 0.054 Std.dev. 0.040 0.042 0.051 0.057 0.063 0.066 0.067 0.079 0.058 Skewness 2.116 1.799 2.609 3.954 2.544 3.270 3.700 4.281 3.034 Kurtosis 9.798 7.403 14.980 39.336 14.909 23.049 27.494 33.127 21.262 F Nobs. 7409 7228 6235 6261 5596 6281 6366 6281 6457 Mean 0.335 0.364 0.369 0.343 0.323 0.310 0.314 0.305 0.333 Std.dev. 0.122 0.139 0.138 0.127 0.124 0.117 0.124 0.117 0.126 Skewness 0.405 0.402 0.285 0.389 0.557 0.457 0.530 0.586 0.452 Kurtosis 3.051 2.938 2.863 3.018 3.563 3.373 3.218 3.423 3.181 N+D+I Nobs. 896 904 1099 1225 1310 1162 930 1016 1069 Mean 0.805 0.790 0.794 0.809 0.815 0.817 0.803 0.798 0.804 Std.dev. 0.153 0.159 0.175 0.150 0.151 0.158 0.142 0.172 0.157 Skewness −0.423 −0.129 −0.083 0.125 0.220 0.258 0.129 0.741 0.105 Kurtosis 4.790 5.211 5.079 5.173 4.884 6.287 4.888 6.382 5.337

NotethatthechoiceofVamong{N,D,I,O}doesnot

affect,froma statisticalpoint ofview,the

goodness-of-fit oftheparametricmodel chosenfor representingthe

data (see Aitchison, 1986, Chapter 7), but introduces a

stringenttrade-offasfaraseconomicinterpretationis

con-cerned.Ontheonehand,afirstobviouschoicewouldbe

tosetV=Oandstudythelog-ratiovectorlog(BN/BO,BD/BO,

BI/BO)=log(CN/CO,CD/CO,CI/CO).Thiswouldkeepthefocus

onthethreemostimportantcommoditycategories, but

theinterpretabilityoftheresultsintermsofthe“O”

cat-egorydefinedex-postwouldbestronglyreduced.Onthe

otherhand,lettingV∈{N, D,I}wouldimprove the

eco-nomicinterpretation,aslog-ratioswouldrepresentexcess

of expenditureforone commoditycategoryin terms of

anothermeaningful categoryonthe(−∞,+∞)support

(with0standingforequalexpenditure).However,setting

V∈{N,D,I}wouldimplytoloseoneimportantdimension

oftheanalysis–i.e.,thestudyoftheshapeofone

impor-tantexpenditurecategoryforwhichwehavereliabledata

–whichwouldbeinsteadpreservedwiththechoiceV=O.

Forthisreason,wehavechosenheretopresentallresults

withV=O.Inbothcases,theoriginalsetupintermsof

well-definedeconomicbudget-sharevariablesislost.

Despitethislackofinterpretabilityofresults,wehave

attempted to fit our data with the two most-widely

employed multivariate distributions that employ ALR

transformations(AitchisonandEgozcue,2005),namelythe

additive-logistic multivariate normal (ALN) distribution

(Aitchison,1986,Chapter6)andtheadditive-logistic

multi-variateskew-normal(ALSN)distribution(Mateu-Figueras

etal.,2007a;Mateu-FiguerasandPawlowsky-Glahn,2007).

Formally,BisALN-distributedifALR(B)ismultivariate

nor-mallydistributed.Noticethatthisnecessarilyappliesif(CN,

CD,CI,CO)ismultivariatelog-normallydistributed,which

doesnotseemtobethecaseforourdataaccordingtoa

multivariatenormalityEnergy-test(seeSection3.1).

Simi-larly,BisALSN-distributedifALR(B)isamultivariateskew

normal(seeAzzaliniandDallaValle,1996;Azzaliniand

Capitanio,1999,foraformaldefinition).Inotherwords,the

multivariateskew-normaldistributionisanextensionof

themultivariatenormalallowingforamoderate(positive

ornegative)skewness.

Table5summarizesourresultsforwave2004(but

find-ingsareverysimilarinallthewaves).Asthefirstthreelines

ofthetableshow,normalityofBisrejectedbothjointly

–accordingtotheEnergytest–andseparatelyforeach

marginalaccordingtobothJarque–Bera(BeraandJarque,

1980,1981)and Lilliefors (Lilliefors, 1967).Thisimplies

thatALNdoesnot seemtoprovide agood multivariate

descriptionofourALN-transformeddata.Notice,however,

Table4

Correlationsamonghouseholdbudgetsharedistributionsandp-values

(inbrackets)forthenullhypothesisofnocorrelation.Wave2004.N,

nondurables;D,durables;I,insurances;F,food.

N D I D −0.55 – – (0.00) – – I 0.02 0.05 – (0.44) (0.08) – F 0.48 −0.31 0.04 (0.00) (0.00) (0.17)

1989 1991 1993 1995 1998 2000 2002 2004 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Sample Mean

Nondurables Durables Insurances Food

1989 1991 1993 1995 1998 2000 2002 2004 0.04 0.06 0.08 0.1 0.12 0.14 Sample Std Dev

Nondurables Durables Insurances Food

1989 1991 1993 1995 1998 2000 2002 2004 −1 0 1 2 3 4 Sample Skewness

Nondurables Durables Insurances Food

1989 1991 1993 1995 1998 2000 2002 2004 0.5 1 1.5 2 2.5 3 3.5

Log − Sample Kurtosis

Nondurables Durables Insurances Food

Fig.6.Samplemomentsofhouseholdbudgetsharedistributionsandtheirevolutionovertime.

thatinadditiontothethirdstandardizedmoment(i.e.,3rd

momentaboutthemeandividedbythestandarddeviation

),alsothe1coefficient(definedas1=0.5(4−)m3/3,

wherem3isthethirdmoment),theD’Agostinoskewness

test(D’Agostino,1970),andtherobust-skewnessestimator

ofGroeneveldandMeeden(1984)allsuggestsome

(mod-erate) positive skewness.Asmentioned, a skew-normal

distributioncanonlyaccommodatemoderateskewness,

i.e.,valuesof1intheinterval[−0.995,0.995](Azzaliniand

DallaValle,1996).Itisthusnaturaltocheckwhetherthe

Table5

Multivariateanalysisforthelog-ratiovectorlog(BN/BO,BD/BO,BI/BO),whereN,nondurables;D,durables;I,insurances;F,food;Wave,2004.Sample

Skewness:3rdmomentaboutthemeandividedbythestandarddeviation.1indexdefinedas0.5(4−)m3/3,wherem3isthe3rdmoment.Thed-test

(Mateu-Figuerasetal.,2007b)formultivariateskewnormalityisdistributedasa2

3.

Statistic Standardizedlog-ratios

log(BN/BO) log(BD/BO) log(BI/BO)

Multivariatenormality Energytest 4.9518(0.0000) Univariatenormality Lilliefors 0.0529(0.0010) 0.0725(0.0010) 0.0305(0.0514) Jarque-Bera 627.2247(0.0000) 56.0317(0.0000) 65.5013(0.0000) Skewness Sampleskewness 0.8702 0.5843 0.0408 1index 0.3728 0.2504 0.0175 D’Agostinotest 6.0741(0.0000) 4.3550(0.0000) 0.3267(0.0741) Robustskewness 0.0574(0.1049) 0.2138(0.0000) −0.0689(0.0697) Multivariateskew-normality d-Test 35.2373(0.0000) Univariateskew-normality QuadraticAD 16.2662(0.0010) 10.1291(0.0175) 15.8801(0.0012)

4 6 8 10 12 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Log Non−Durables 1989 2002 0 2 4 6 8 10 12 14 0 0.05 0.1 0.15 0.2 0.25 Log Durables Density Density 1989 2002 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 Non−Durables BS Density 1989 2002 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 Durables BS Density 1989 2002

Fig.7.95%Confidencebandsforkernelestimates.Comparisonbetweenfirst(1989)andfinal(2002)wave.Toptwopanels:householdexpenditure distributions.Bottomtwopanels:householdbudget-sharedistributions.

jointBdistributioncanbesatisfactorilyproxiedbya

mul-tivariateALSNdistribution.Thisamountstotestif,jointly,

ALR(B)ismultivariateskew-normal.Weemployherethe

dtestdiscussedin Mateu-Figuerasetal. (2007b),which

underthenullofmultivariateskew-normalityshouldbe

distributed here as a 2

3. AsTable 5 shows, the null is

rejected,asalsohappensforthenullhypothesesof

skew-normality of the three marginal log-ratio distributions,

1989 1991 1993 1995 1998 2000 2002 2004 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Sample Correlations N vs D N vs I N vs F D vs I D vs F I vs F

accordingtostandard quadratic Anderson–Darlingtests (seeD’AgostinoandStephens,1986).Ofcourse,rejection

byAnderson–DarlingEDF-basedtestsmaysimplybedueto

asizeeffect,asoursamplesarealwaysverylarge(see,e.g.,

BentlerandBonett,1980).Tocontrolforsuchaproblem,we

haverunGoFtestsonbinneddataandonrandomlydrawn

smaller-sizedsub-sampleswithoutnoticinganymajor

dif-ferencesinourresults.Weshallgobacktothispointin

Section6.

5. Towardsaunivariateparametricmodelfor budgetsharedistributions

Theforegoinganalysissuggeststhat–fromapurely

sta-tisticalperspective–ourdatacannotbeeasilyproxiedby

themostlyemployedparametricmultivariatedistributions

definedonthesimplex.Moreimportantly,such

represen-tationsoftenrequiremathematicaltransformationsofthe

datathatwouldstronglyreducethepossibilityof

econom-ically interpreting theresults, let asidethe difficultyof

treatingsub-categoriesasfood.Inwhatfollows,wewill

thereforeturntoaunivariateanalysisaimingat

determin-ingaparsimonious,parametric,modelabletosatisfactorily

fithouseholdbudgetshare(marginal)distributions,

with-out using mathematical transformations of theoriginal

variablesthat mayundermine theeconomic

interpreta-tionsoftheresults.

We lookfora family of univariate densities,defined

ontheunit interval,holdingatleastthefollowingthree

desirable features. First, the family of densities fitting

householdbudget sharesshouldbeconsistent withthe

statistical properties of the underlying household

con-sumptionexpenditureunivariatedistributionsemployed

tocomputebudgetshares.Second,it shouldbeflexible

enoughtoaccommodate–foreachwave–theobserved

across-commodityheterogeneityin theshape of

house-holdbudgetsharedistributions.Third,theparametersof

thedensityshouldembodysomeeconomicmeaningand

allowonetotaxonomizecommoditycategoriesaccording

totheir(highandlow)level.

LetC1,...,CKbetheexpenditurelevelsofagiven

house-holdinarepresentativetimeperiod,whereKisthenumber

ofcommoditycategoriesconsidered.Thehousehold

bud-getshareofcommoditycategoryiisdefinedas

Bi=C_Ci =₁₊₍



1 j/=iCj)/Ci = 1 1+



_j/=iZj(i)= 1 1+Si (5)

whereSi is thesumoftheK−1 randomvariables Zj(i),

eachbeingequaltotheratiobetweenCjandCi,withj=1,

...,i−1,i+1,...,K.Obviously,Bi∈(0,1)asrequired.From

Eq.(5),itfollowsthatthecumulativedistributionfunction

(cdf)ofBireads: FBi(x)=Prob{Bi<x}=Prob



1+Si>1_x



=1−FS_i



₁ x−1



, (6)

wherex∈(0,1)andFSiisthecdfofSi.Therefore,the

prob-abilitydensityfunction(pdf)ofBiisgivenby:

fBi(x)dx= 1 x2fSi



₁ x −1



dx, (7)

wherefSi isthepdfofSi.Thismeansthatcharacterizing

thedistributionofBirequiresstudyingthedistributionof

Si=



j/=iCj/Ci=



j/=iZj(i).Given theempiricalevidence

above,therearegoodreasonstoassumethatexpenditure

levelsCiarealllog-normallydistributed,atleastasafirst

approximation.ThisimpliesthattheratiosZj(i)arealso

log-normallydistributed,as:

Prob{Zj(i)<z}=Prob{log(Cj)−log(Ci)<log(z)}

=Prob{Dj(i)<log(z)}. (8)

Sincelog(Cj)andlog(Ci)arenormallydistributed(and

possiblycorrelated),theirdifferenceDj(i)willalsobe

nor-mal.Henceexp(Dj(i))willbelog-normallydistributed.

Asaresult,theshape ofhouseholdbudgetshare

dis-tributionBifullydependsontheshapeofthesumofthe

K₋1log-normalvariatesZj(i)s.NoticethatingeneralZj(i)

willnotbeuncorrelated.Indeed,theCismaybecorrelated

becauseofhouseholdpreferences.Thisseemstobethecase

fromourempiricalevidence,aswehavealreadynoticed

statistically significant correlations between household

consumptionexpendituredistributions(seeTable2).The

significant correlation between household consumption

expendituredistributionsthusimpliesthatZj(i)–aswell

ashouseholdbudgetsharedistributions–willnotbe

inde-pendent.

Accordingtotheliterature,theredoesnotexistaclosed

form for thepdf of a sumof log-normal (correlated or

uncorrelated)randomvariablesandonlyapproximations

areavailable,seeBeaulieuetal.(1995)forthecaseof

inde-pendentsummandsandMehtaetal.(2006)forthecaseof

correlatedsummands.Thebaselineresultisthatthe

dis-tributionofSicanbewellapproximatedbyalog-normal

distribution,whoseparametersdependinanon-trivialway

ontheparametersofthelog-normalstobesummedup

andtheircovariancematrix.Manymethodsareavailable

tofindapproximationstotheparametersoftheresulting

log-normaldistribution,see,e.g.,Fenton(1960),Schwartz

and Yeh(1982)andSafak andSafak (1994).We arenot

interestedhereinthisissuebecausewecandirectly

esti-matetheparametersoftheresultingdistributionforBivia

maximumlikelihood.

The log-normal proxy to the sum of log-normals is

not, however, theonly approximation available. Indeed

MilevskiandPosner(1998)showthatwhenK→∞thenSi

convergesindistributiontoananinverse-Gamma(Inv)

density,which performswellinapproximatingthesum

alsoforverysmallK.Moreformally,theInvrandom

vari-ableisdefinedastheinverseofa randomvariable,i.e.,

ifX∼ (,p−1)thenX−1∼Inv(,p).Thereforetheremay

besomegainsinconsideringanInv proxytoSiinstead

ofalog-normalone.Ofcourse,theextenttowhicheither

approximationistobepreferredisanempiricalissue.For

thisreason,weshallconsiderbothproxiesinourempirical

0 0.5 1 1.5 2.5 x Density m = 0 m = −1 m = 1 s = 1 0 0.5 1 2 4 x Density s = 1.3 s = 1.8 s = 2 s = 2.5 m = 0 0 0.5 1 3 6 x Density s = 1.3 s = 1.8 s = 2 s = 2.5 m = −1 0 0.5 1 3 6 x Density s = 1.3 s = 1.8 s = 2 s = 2.5 m = 1

Fig.9.TheLN-Bapproximationtohouseholdbudgetsharedistributions.DifferentshapesoffBiasparametersmandschange.

In thecase Si hasa log-normalpdf withparameters

(m,s),then: fSi(x;m,s)= 1 xs√2exp



−(log(x)−m) 2 2s2



. (9) Using(7),weget: fB_i(x;m,s)= 1 x(1−x)s√2 exp



−(log(1−x)−log(x)−log(m))

2

2s2



(10)

Inwhatfollowsweshallrefertodensity(10)asthe

LN-Bdensity.NotethattheLN-Bisalreadyapdfgiventhat

itsintegral over [0,1]isone.In Fig.9 weshowa

vari-etyofshapesderivedfrom(10)forselectedvaluesofthe

parametersmands.Ifm>0(m<0)thedistributionis

right-skewed(left-skewed),ifm=0itissymmetric.If0<s≤1.5

thedistributionisbell-shaped,if1.5<s_≤2.5itisbimodal,

whileifs>2.5itisU-shaped.Thisseemstoconfirmthat

despiteitsparsimony,thedensity(10)issufficiently

flexi-bletoaccommodatedifferentshapesforhouseholdbudget

sharedistributions.

Ontheotherhand,ifweassumeanInv

approxima-tionforthedistributionofasumoflog-normals,thenthe

distributionofSidependsontwoparameters(,p)andits

pdfreads: fSi(x;,p)= p (p)x−p−1exp



− x



(11)

Onceagain,using(7)weobtainthepdfofBi(henceforth,

Inv-B),whichreads:

fBi(x;,p)= p x2_(p)



₁ x−1



−p−1 exp

− x 1−x

. (12)

Fig.10showstheshapeofthedensity(12)forselected

val-uesofandp.Weimmediatelyseethat(12)isalwaysan

asymmetricdistribution,asfBi(1;,p)=0foranyvalues

oftheparameters,whileifp>1fBi(0;,p)=0butifp≤1

fB_i(0;,p)>0.Theinterpretationofthetwoparametersis

lessstraightforwardthaninthepreviouscase.Noticethat

forsmallvaluesofpthefunctionismonotonically

decreas-ing,whileaspincreasesarightward-shiftingmaximum

emerges.Whenp<(p>)themaximumisattainedfor

x<0.5(x>0.5),whileifp=themaximumisaroundx=0.5:

thisisthemostsymmetriccasewecanmodelwiththis

0 0.5 1 3 6 x Density p = 1 θ = 0.5 p = 2 θ = 1 p = 0.5 θ = 0.1 p = 3 θ = 1 p > θ 0 0.5 1 3 6 x Density p = 1 θ = 2 p = 2 θ = 3 p = 0.5 θ = 1 p = 1 θ = 3 p < θ 0 0.5 1 3 6 x Density p = 1 θ = 1 p = 2 θ = 2 p = 0.5 θ = 0.5 p = 3 θ = 3 p = θ

Fig.10. TheInv-Bapproximationtohouseholdbudgetsharedistributions.DifferentshapesoffBiasparameterspandchange.

than(10),weshallretainitinourfittingexercisesforthe

sakeofcomparison.

6. Measuringthegoodnessoffit

Intheprevioussection,wehavederivedtwoalternative,

parsimonious,approximationsofunivariatebudget-share

distributions, which appear – at least in principle –

flexibleenoughtoaccommodatetheobservedshape

het-erogeneityandareconsistentwiththeempiricallydetected

log-normalityofhouseholdconsumptionexpenditure

dis-tributions.

Tocheck how well theforegoing approximationsfit

thedata,wefirstlyestimatetheparametersof(10)and

(12) via maximum likelihood. Results are reported in

Table 6, together with asymptotic standard deviations

forthe parameters ofLN-B and Inv-B.We shall

com-mentparameterestimatesinSection7,wheretheywill

beemployedtoclassifythecommoditycategoriesunder

study. In the rest of this section, we focus instead on

goodness-of-fitconsiderations.

To evaluate the performance of the two proposed

proxies in fitting household budget share distributions

ascomparedtoalternativedistributions,wechooseasa

benchmarktheunivariateBetadensity(Evansetal.,2000),

whosepdfreads:

b(x;˛,ˇ)= x˛−1(1−x)ˇ−1

BE(˛,ˇ) , (13)

where x∈[0,1]and BEistheBetafunction.Notice that

theBetaalsodependsononlytwoparametersand

typi-callyisflexibleenoughtoaccommodatemanyalternative

shapes. However,it lacksanyconsistencyrequirements

withrespecttotheunderlyingshapeofhousehold

con-sumption expenditure distributions, because in general

itcannotbederivedasthedensityofhouseholdbudget

sharesstemmingfromlog-normallydistributed

expendi-turelevels.

We firstly employ a goodness-of-fit tests based on

empiricaldistributionfunctionstatistics.Morespecifically,

werunthreewidelyusedtests:Kuiper(KUI;Kuiper,1962),

Cramér-vonMises(CvM;PearsonandStephens,1962)and

quadraticAnderson–Darling(AD2;AndersonandDarling,

1954).We areinterested inunderstandingwhetherthe

theoreticaldistributionswesuggestareabletoproxythe

empiricaldatabetterthanplausiblealternatives,because

of their consistencywith thefindings above on

Table6

Estimatedparametersandasymptoticstandarddeviations(inparentheses)ofLN-BandInv-Bvs.waves.N,nondurables;D,durables;I,insurances;

F,food. LN-B Waves 1989 1991 1993 1995 1998 2000 2002 2004 N m −1.04(0.009) −0.98(0.010) −0.98(0.010) −0.95(0.010) −0.77(0.010) −0.81(0.009) −0.78(0.009) −0.77(0.009) s 0.76(0.006) 0.83(0.007) 0.83(0.007) 0.77(0.007) 0.74(0.007) 0.72(0.006) 0.75(0.007) 0.74(0.007) D m_s 2.47_1.33(0.019)(0.026) 2.28_1.33(0.019)(0.027) 2.69_1.42(0.031)_(0.022) 2.57_1.40(0.023)(0.033) _1.372.43_(0.021)(0.030) _1.372.45_(0.022)(0.031) _1.352.64_(0.022)(0.032) _1.342.80(0.030)_(0.021) I m_s 3.74_1.18(0.020)(0.028) 3.66_1.24(0.020)(0.028) 3.61_1.31(0.028)_(0.020) 3.61_1.31(0.024)_{(0.017) 1.25}3.25_(0.017)(0.024) _1.183.22_(0.016)(0.023) _1.173.26_(0.018)(0.025) _1.203.17(0.026)_(0.018) F m 0.74(0.007) 0.61(0.008) 0.59(0.008) 0.71(0.008) 0.80(0.008) 0.86(0.008) 0.85(0.008) 0.89(0.008) s 0.59(0.005) 0.66(0.005) 0.66(0.006) 0.61(0.005) 0.62(0.006) 0.60(0.005) 0.62(0.006) 0.60(0.005) Inv-B Waves 1989 1991 1993 1995 1998 2000 2002 2004 N p_ 1.79_0.47(0.008)(0.027) 1.45_0.37(0.007)(0.022) 1.38_0.34(0.022)_(0.007) 1.52_0.41(0.025)_{(0.008) 0.67}1.93_(0.013)(0.034) _0.661.96_(0.012)(0.032) _0.581.74_(0.011)(0.028) _0.621.81(0.030)_(0.012) D p_ 0.66_3.09(0.106)(0.016) 0.68_2.70(0.095)(0.017) 0.57_2.78(0.015)_(0.108) 0.63_3.08(0.017)_{(0.124) 2.81}0.64_(0.106)(0.017) _2.830.64_(0.112)(0.017) _3.120.61_(0.127)(0.017) _3.260.58(0.015)_(0.130) I p_ 0.99(0.029) 0.95(0.027) 0.86(0.022) 0.78(0.017) 0.92(0.022) 0.93(0.023) 0.91(0.024) 0.71(0.018) 23.14(0.877) 20.14(0.738) 15.95(0.549) 13.37(0.409) 12.69(0.398) 12.50(0.398) 12.46(0.433) 7.30(0.265) F p_ 3.07_5.41(0.092)(0.048) 2.38_3.51(0.061)(0.037) 2.52_3.68(0.043)_(0.069) 2.86_4.82(0.048)_{(0.089) 5.00}2.73_(0.098)(0.049) _6.023.02_(0.111)(0.051) _5.342.77_(0.098)(0.046) _6.093.00(0.051)_(0.112)

asoursamplesarealwaysverylarge,thenullassumption

that empirical data are drawn from any given

theoret-ical distribution would bring the traditional EDF-based

goodness-of-fittests,whichevaluatethedeviationofthe

empiricalCDFfromthetheoreticaloneateachsingle

obser-vation,torejectthenullhypothesisatanyprobabilitylevel,

despiteonlyminordiscrepancies(seeBentlerandBonett,

1980).Therefore, weshall evaluateinwhat followsthe

goodness of fit of our competing densities afterhaving

groupedloggedobservationsamongequallyspacedbins

andcomputedteststatisticsoversuchbins.Inthisway,

theeffectofadiscrepancybetweentheoreticaland

empiri-calprobabilitydensitywillstronglyaffectthestatisticonly

ifthesamediscrepancyis recurringfrequentlywithina

particularinterval (bin),whiletheexistenceofsporadic

outlierswillonlyhaveaminoreffectontheteststatistic.

Thisprocedurealsoalleviatesthedifficultiescomingfrom

thegenerationofpseudo-randomnumbersdistributedas

in(10)or(12).Thisisnotatrivialtaskandtheissueisone

ofthemainpointsofourresearchagenda.

Table7reportsthegoodness-of-fitstatisticsonlyforthe

AD2test,asingeneralallthreetestsagreeonwhetherthe

benchmarkisoutperformedorunderperformedbyeither

specification of the proposeddensity. Accordingto test

statistics,theBetafitisneverthebestone.Withrespect

totheLN-BandtheInv-Bdensities,thelatterseemsto

deliverabetterfitinthecaseofnondurableanddurable

goods(andaccordingtoKUIandCvMtests,alsointhecase

offood).However,thelevelsofthestatisticsdonottake

intoaccountthedifferentinfluencesexertedbythe

sam-plesizeontheresultsofthebinningprocedurefordifferent

distributions.Moreover,adensitymayperformrelatively

betterthananotheroneeventhoughbothprovideavery

baddescriptionoftheempiricalsample.Inordertoperform

amorestatisticallysoundcomparison,wehavetherefore

proxiedviasimulationthedistributionsofthetest

statis-ticswhenusingourprocedureofgroupingtheobservations

intobins,andwehavecomputedtherelevantp-valuesof

theempiricalvaluesobtainedbefore,i.e.,theprobability

masstotherightoftheobservedstatistics.

More specifically,to proxy thedistribution of a test

statisticfor a given theoretical density and a

commod-itycategoryofhouseholdbudgetsharesofsizen,weuse

thefollowingprocedure:(i)generateviaa

bootstrap-with-replacementmethodarandomsampleofobservationsof

thesamesizenastheobservedsample,thengroupthe

observationsintoLbins; (ii)ontherandomlyextracted

sample,re-estimatetheparametersofthetheoretical

fre-quency by maximum likelihood and compute the test

statistic;(iii)repeatthisprocedurealargenumberoftimes

mtogettheproxyforthedistributionoftheteststatistic.

Ofcoursetheforegoingstepsshouldberepeatedforany

givenempiricalsample,i.e.,foranywave and

commod-itycategoryconsidered,andforanyofthethreedensities

studied.Inwhatfollows,wehavesetm=1000andwehave

consideredL=100bins.

Thepicturegivenbythep-valuesisgenerallyconsistent

withthatgivenbythestatistics,howeverinadozencases

thep-values fortheBetadensityareatleastasgood as

thosefortheLN-BortheInv-Bdensities,notwithstanding

lowervaluesofthestatistics.Still,intermsofp-values,the

LN-BortheInv-BdensitiesoutperformtheBetadensity

inthe78%,59%and78%ofthecasesaccordingtoKUI,CvM

andAD2,respectively.

Together with standard goodness-of-fit tests, we

employalsothe averageabsolutedeviationasa simple

measure of goodness-of-fit. The average absolute