A New Generalized Variance Approach for Measuring Multidimensional Inequality and Poverty 2021 Ottó Hajdu.pdf

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SocialIndicatorsResearch
/s11205-021-02720-9
ORIGINALRESEARCH
ANewGeneralizedVarianceApproachfor Measuring
MultidimensionalInequalityand Poverty
Ottó Hajdu1
Accepted:15May2021
©TheAuthor(s)2021
Abstract
Thepapersuggestsanewgeneralizedvarianceconceptformeasuringmultidimensional
inequalityofastratifiedsociety,basedonmultivariatestatisticalmethods,wherethemem-
bersofsocietyformacloudintheobliquespaceofdimensionsofinequality,suchas
income,-

cloudintoacompositecompactmetriccharacterizingboththeshapeandtheinnerstruc-

asaunidimensionalweightedcombinationofthedimensions,ournewcompositeindex
.
First,weintroducetheInequalityCovarianceMatrix(ICM)assignedtothecloud,withele-
,weproposetheGener-
alizedVariance(GV)ofICMtomeasurethecompositeGeneralizedVarianceInequality
(GVI),toevaluatethestratum-specificstructureoftheoverallinequality,we
suggestanewtwo-,wedividethetotalGVIintobetween-
groupsandwithin-,inthesecondstagethecontributionsofthestrata
tothewithin-groupsinequalityand,thecontributionsofthedimensionstothebetween-
,
decomposableintostratumeffectsand,,
includingthelog-dimensionsintheanalysis,GVIyieldsanEntropyCovarianceMatrix
,theGVIofcensoredpoverty

yetsolvedinthetraditionalliteraturesofar.
KeywordsGeneralizedentropy ·Generalizedvariance ·Multidimensionalinequality ·
Poverty ·Theil-indices ·Wilks ·Lambda
*OttóHajdu
******@
1Departmentof ComparativeEconomics,Instituteof BusinessEconomics,EötvösLoránd
University,Egyetemtér1-3,1053 Budapest,Hungary
Vol.:(0123456789)13:.
O. Hajdu
1Introduction
Measuringmultidimensionalinequalityofastratifiedsociety,first,needsdefiningacom-
positemultivariateindex,whichisdecomposableintosubgroupeffectsgeneratedbythe
(households)formamultidimensionalpopu-
-economicinequality
dimensionsareforinstanceincome,consumption,expenditure,property,,
giventhestratification,theoverallpopulationcloudembodiestheoverallinequalitypre-
sentinthesociety,
paperdistinguishesgroupsfromcloudsand,dimensionsfromvariables.
Thefundamentalaimofthisarticleistoelaborateamethodologicalframeworkfor
analysingthestructureofinequalitiesinastratifiedsystemofskewedclouds,withspe-
cialregardtothebetween-strataandwithin-stratadecompositionoftheoverallinequality.
Oncetheinequalitymeasurementisdefined,themeasurementofpovertyisalso,basedon
thetheoryofcensoreddistributions.

Consideringthekeyfieldsofmeasuringeconomicinequalityandpoverty,thefundamental
contributionsofthetraditionalliteraturesofar,withoutwishingtobeexhaustive,areas
follows.
First,intermsofdimensionality,dimensionsarelatentfactorsforwhichmanifestindi-

one-dimensionalapproachontheoneside,oramulti-dimensionalapproachontheother
,theone-dimensionalapproachessentiallymultivariateiftheinequality-pov-
ertyindicatorsusedarecondensedintoasingledimensionwithappropriateweightsasa
,puremultidimensionalapproachisbothmultidimensional
andmultivariateinthesensethatitdefinesseveralfactors(dimensions)anddefinesthem
bycorrespondingmanifestindicators.
Thenextcentraltaskistodefineaconcise,compositemeasure—inequality-poverty
index—thatmeetspre-establishedreasonablerequirements,so--
omsdefinehowaspecificindexrequiredtorespondtostructuralchangeshappenedinthe
%decompositionofthecompos-
iteindexintosubgrouprelative(percentage)contributions,providedastratifiedpopula-
,
theimportanceoffactors,,theorderofimportanceof
,the
importanceofindicatorsshouldberankedwithinagivenfactor.
Below,wehighlightthestudiesthathavefundamentallyinfluencedtherelevantlitera-
-

follows.
(1987)and,fortheunit-
consistentdecomposabilityseeZheng(2005).
 al.(1984)andFoster
andShneyerov(1999).
13:.
ANewGeneralizedVarianceApproachfor Measuring…
,infact,thefundamental
Giniindex,theSen-Shorrocks-Thonindex(Shorrocks,1995)and,theAtkinsonindex-
family(Atkinson,1987).

bySen(1976)andCowellandKuga(1981a,b).
(2009).
,concerningcomparisons,
relativewelfareanddominanceconditionsareavailableinDucloset al.(2006)while,
comparingmultidimensionalindicesofinequalityisgiveninLugo(2005).
,seeMussard
et al.(2003),Bourguignon(1979),Aristondoet al.(2010)andDagum(1997).
,seeTsui(1999),
Mussardet al.(2003).
-
spectiveseeLugoandMaasoumi(2008).
-called
datamatrixseeforinstanceAlkireandFoster(2011).
 Contributions
Thefundamentalaimofthispaperistointroduceanewconceptsuitabletoanalysethe
structureofinequalityconsideringastratifiedsocietywithoutliers,formingasystemof
skewedcloudsintheobliqueinequalityspacedefinedbycorrelated,asymmetricaldimen-
,ourconceptworksasastep-by-
stepprocedureratherthanasingle,composite,-
alizedVarianceInequality(GVI)procedureisbasedmainlyonfundamentalstatisticaltools
ofmultivariatediscriminantanalysis.
TheGVIapproachisatwo-,thetotalpopulationinequalityis
measuredanddividedintoawithin-clouds+between-cloudsstructure(%+20%).
Then,secondly,weallocatethe80%within-cloudsinequalitytostrataandthe20%
between-cloudsinequalitytodimensions.
Tomeasurethecompositeoveralldegreeofthepopulationinequality,wedefinethe
InequalityCovarianceMatrix(ICM)whichconsistsofallthepairwisecovariancesamong
-
,itsdeterminantmeasures
,provided
astratifiedsociety,theoverallpopulationcovariancematrixisadditivelydecomposed
accordingtothestratification.
Basedonthesetwoproperties,asabasicresult,thepaperproposestheGVgeneralized
variancemetrictomeasurethecomposite,multivariatedegreeofinequality(dispersion)
,inthecaseofastratifiedpopulation,wehavean
overallGVmeasureforthepopulationontheonesideand,separatedsingleGVsforthe
sub-,thefinaltaskistoallocatethewithin-clouds
inequalityamongstrataand,thebetween-,

stratacanberankedaccordingtotheirimportanceinexplaininginequalitybasedontwo
typesofstatisticaltests:testingtheequalityofthestratum-specificgeneralizedvariances
and,testingtheequalityofthestratum-specificcentroids.
13:.
O. Hajdu
Further,havingthelog-dimensionsincludedamongdimensionsleadstotheinterpretation
ofthecovariancebetweentheso-
(shownbythispaper)thesumoftheTheil-RedundancyIndex(TRI)andtheMeanLogarithmic
Deviation(MLD)index,,
theInequalityCovarianceMatrixcontainsthecovarianceandthevariancesofdimension
“income”anditslogarithmthereforetermedEntropyCovarianceMatrixdenotedbyECM.
2HighlightedPropertiesand Contributionsof the GVIProcedure
Providedastratifiedpopulationofmultidimensionalclouds,GVIgivesthebetween-
clouds+within-,thestratum-
specificcontributionstotheaveragewithin-cloudsinequalityand,thedimension-specific
contributionstothebetween--
dimensionalbecauseitmeasuresclouds,ratherthanarbitrary,univariate,weightedsumsof

stratificationisreportedbythestandardWilks’-specific
contributionsofthestratatotheaveragewithin-cloudsinequality(computedviathestandard
Box-Mstatistic)itispossibletotesttheequalityofstratum-
thedimensionsandthestrataarerankedaccordingtotheirimportanceinexplaininginequal-
:the
correlationsamongthedimensionsand,thedistortionsduetodistributionalasymmetrywith

inference.
Forthesakeofclearcomputationsandinterpretations,thepaperprovidescomputational
resultsbasedonartificialdataand,besides,empiricalresultsbasedonHungarianHouseholds’
expendituredatasurveyedbytheHungarianCentralStatisticalOfficeintheyearof2003.
ThecomputationsandplotsarecarriedoutbythemeansoftheSPSSandStatistica,andR
packages.
 2introducesthenewGeneralizedVari-
anceInequalityGVIconcept,yieldingtheGVIinequalityandGVPpovertyindicesbasedon
theideaoftheInequalityCloudsand,theInequalityCovarianceMatrixICMdefinedinthe
 3givestheMultivariateDiscriminantAnalysisoftheGeneralizedVariance
Inequality,withspecialregardtothebetween-groupsandwithin-groupssub-groupdecompo-
sition,
- 4
introducestheGeneralizedVarianceEntropyconceptGVE,basedontheEntropyCovariance

 5suggests—
defines—thewithin-groupsCensorizedHead-CountRatiopovertyapplicationbasedonthe
 ,Sect. 7
concludes.
13:.
ANewGeneralizedVarianceApproachfor Measuring…
:Ownelaboration(basedonSharma,1996)
3TheGeneralizedVarianceInequalityConcept:GVI
Followingthefundamentalprinciplethatinequalityisessentiallyaspecialaspectofdis-
persion(Tsui,1999),thispapersuggeststhegeneralizedvariance(GV)multivariatemetric
tomeasurethedegreeofinequality,-
ancemeansthedeterminantofthecovariancematrixCofthemultivariatedataapplied.
Dependingonthefeatu