DASP MANUAL V2.2 - DASP Stata Packagedasp.ecn.ulaval.ca/modules/DASP_V2.2/DASP_MANUAL_V2.2.pdf ·...

USERMANUAL

DASPversion2.2

DASP:DistributiveAnalysisStataPackageBy

AbdelkrimAraar,Jean‐YvesDuclos

UniversitéLavalPEP,CIRPÉEandWorldBank

June2013

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TableofcontentsTableofcontents........................................................................................................................................................2ListofFigures..............................................................................................................................................................61 Introduction........................................................................................................................................................82 DASPandStataversions................................................................................................................................83 InstallingandupdatingtheDASPpackage............................................................................................93.1 InstallingDASPmodules.....................................................................................................................93.2 AddingtheDASPsubmenutoStata’smainmenu.................................................................10

4 DASPanddatafiles.......................................................................................................................................105 Mainvariablesfordistributiveanalysis..............................................................................................116 HowcanDASPcommandsbeinvoked?...............................................................................................117 HowcanhelpbeaccessedforagivenDASPmodule?...................................................................128 ApplicationsandfilesinDASP.................................................................................................................129 BasicNotation.................................................................................................................................................1410 DASPandpovertyindices.....................................................................................................................1410.1 FGTandEDE‐FGTpovertyindices(ifgt)...................................................................................1410.2 DifferencebetweenFGTindices(difgt).....................................................................................1510.3 Wattspovertyindex(iwatts).........................................................................................................1610.4 DifferencebetweenWattsindices(diwatts)...........................................................................1610.5 Sen‐Shorrocks‐Thonpovertyindex(isst)...............................................................................1710.6 DifferencebetweenSen‐Shorrocks‐Thonindices(disst).................................................1710.7 DASPandmultidimensionalpovertyindices..........................................................................1710.8 Multipleoverlappingdeprivationanalysis(MODA)indices............................................19

11 DASP,povertyandtargetingpolicies................................................................................................2011.1 Povertyandtargetingbypopulationgroups..........................................................................2011.2 Povertyandtargetingbyincomecomponents......................................................................21

12 Marginalpovertyimpactsandpovertyelasticities...................................................................2212.1 FGTelasticity’swithrespecttotheaverageincomegrowth(efgtgr)..........................2212.2 FGTelasticity’swithrespecttotheaverageincomegrowthwithdifferentapproaches(efgtgro)........................................................................................................................................2312.3 FGTelasticity’swithrespecttoGiniinequality(efgtineq)................................................2312.4 FGTelasticity’swithrespecttoGini‐inequalitywithdifferentapproaches(efgtine). 2412.5 FGTelasticitieswithrespecttowithin/betweengroupcomponentsofinequality(efgtg)......................................................................................................................................................................2512.6 FGT‐elasticitieswithrespecttowithin/betweenincomecomponentsofinequality(efgtc)......................................................................................................................................................................26

13 DASPandinequalityindices................................................................................................................2813.1 Giniandconcentrationindices(igini)........................................................................................2813.2 DifferencebetweenGini/concentrationindices(digini)...................................................2813.3 Generalisedentropyindex(ientropy)........................................................................................2913.4 Differencebetweengeneralizedentropyindices(diengtropy)......................................2913.5 Atkinsonindex(iatkinson).............................................................................................................3013.6 DifferencebetweenAtkinsonindices(diatkinson).............................................................30

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13.7 Coefficientofvariationindex(icvar)..........................................................................................3113.8 Differencebetweencoefficientofvariation(dicvar)...........................................................3113.9 Quantile/shareratioindicesofinequality(inineq)............................................................3113.10 DifferencebetweenQuantile/Shareindices(dinineq)..................................................3213.11 TheARAAR(2009)multidimensionalinequalityindex................................................32

14 DASPandpolarizationindices............................................................................................................3214.1 TheDERindex(ipolder)...................................................................................................................3214.2 DifferencebetweenDERpolarizationindices(dipolder)..................................................3314.3 TheFosterandWolfson(1992)polarizationindex(ipolfw)..........................................3414.4 DifferencebetweenFosterandWolfson(1992)polarizationindices(dipolfw)....3414.5 TheGeneralisedEsteban,GardinandRay(1999)polarisationindex(ipoger).......3414.6 TheInaki(2008)polarisationindex(ipoger).........................................................................36

15 DASPanddecompositions....................................................................................................................3915.1 FGTPoverty:decompositionbypopulationsubgroups(dfgtg).....................................3915.2 FGTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue(dfgts)4015.3 AlkireandFoster(2007)MDindexofpoverty:decompositionbypopulationsubgroups(dmdafg).........................................................................................................................................4215.4 AlkireandFoster(2007:decompositionbydimensionsusingtheShapleyvalue(dmdafs).................................................................................................................................................................4215.5 FGTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue(dfgts)4315.6 DecompositionofthevariationinFGTindicesintogrowthandredistributioncomponents(dfgtgr).........................................................................................................................................4515.7 DecompositionofchangeinFGTpovertybypovertyandpopulationgroupcomponents–sectoraldecomposition‐(dfgtg2d)..............................................................................4615.8 DecompositionofFGTpovertybytransientandchronicpovertycomponents(dtcpov)..................................................................................................................................................................4915.9 Inequality:decompositionbyincomesources(diginis)....................................................5115.10 Regressionbaseddecompositionofinequalitybyincomesources.........................5215.11 Giniindex:decompositionbypopulationsubgroups(diginig)..................................5815.12 Generalizedentropyindicesofinequality:decompositionbypopulationsubgroups(dentropyg)...................................................................................................................................5915.13 Polarization:decompositionoftheDERindexbypopulationgroups(dpolag).5915.14 Polarization:decompositionoftheDERindexbyincomesources(dpolas)........60

16 DASPandcurves.......................................................................................................................................6016.1 FGTCURVES(cfgt)..............................................................................................................................6016.2 FGTCURVEwithconfidenceinterval(cfgts)...........................................................................6216.3 DifferencebetweenFGTCURVESwithconfidenceinterval(cfgts2d).........................6216.4 LorenzandconcentrationCURVES(clorenz).........................................................................6216.5 Lorenz/concentrationcurveswithconfidenceintervals(clorenzs)............................6316.6 DifferencesbetweenLorenz/concentrationcurveswithconfidenceinterval(clorenzs2d).........................................................................................................................................................6416.7 Povertycurves(cpoverty)...............................................................................................................6416.8 Consumptiondominancecurves(cdomc)................................................................................6516.9 Difference/Ratiobetweenconsumptiondominancecurves(cdomc2d)....................66

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16.10 DASPandtheprogressivitycurves.........................................................................................6616.10.1 Checkingtheprogressivityoftaxesortransfers........................................................6616.10.2 Checkingtheprogressivityoftransfervstax..............................................................67

17 Dominance..................................................................................................................................................6717.1 Povertydominance(dompov).......................................................................................................6717.2 Inequalitydominance(domineq).................................................................................................6817.3 DASPandbi‐dimensionalpovertydominance(dombdpov)............................................68

18 Distributivetools......................................................................................................................................6918.1 Quantilecurves(c_quantile)...........................................................................................................6918.2 Incomeshareandcumulativeincomesharebygroupquantiles(quinsh)................6918.3 Densitycurves(cdensity)................................................................................................................6918.4 Non‐parametricregressioncurves(cnpe)...............................................................................7118.4.1 Nadaraya‐Watsonapproach.................................................................................................7118.4.2 Locallinearapproach...............................................................................................................71

18.5 DASPandjointdensityfunctions.................................................................................................7118.6 DASPandjointdistributionfunctions........................................................................................72

19 DASPandpro‐poorgrowth..................................................................................................................7219.1 DASPandpro‐poorindices.............................................................................................................7219.2 DASPandpro‐poorcurves..............................................................................................................7319.2.1 Primalpro‐poorcurves...........................................................................................................7319.2.2 Dualpro‐poorcurves...............................................................................................................74

20 DASPandBenefitIncidenceAnalysis..............................................................................................7520.1 Benefitincidenceanalysis...............................................................................................................75

21 Disaggregatingthegroupeddata......................................................................................................8022 Appendices..................................................................................................................................................8422.1 AppendixA:illustrativehouseholdsurveys............................................................................8422.1.1 The1994BurkinaFasosurveyofhouseholdexpenditures(bkf94I.dta).........8422.1.2 The1998BurkinaFasosurveyofhouseholdexpenditures(bkf98I.dta).........8522.1.3 CanadianSurveyofConsumerFinance(asubsampleof1000observations–can6.dta)...........................................................................................................................................................8522.1.4 PeruLSMSsurvey1994(Asampleof3623householdobservations‐PEREDE94I.dta).............................................................................................................................................8522.1.5 PeruLSMSsurvey1994(Asampleof3623householdobservations–PERU_A_I.dta).................................................................................................................................................8622.1.6 The1995ColombiaDHSsurvey(columbiaI.dta)........................................................8622.1.7 The1996DominicanRepublicDHSsurvey(Dominican_republic1996I.dta).86

22.2 AppendixB:labellingvariablesandvalues.............................................................................8722.3 AppendixC:settingthesamplingdesign..................................................................................88

23 Examplesandexercises.........................................................................................................................9023.1 EstimationofFGTpovertyindices...............................................................................................9023.2 EstimatingdifferencesbetweenFGTindices..........................................................................9623.3 Estimatingmultidimensionalpovertyindices.....................................................................10023.4 EstimatingFGTcurves...................................................................................................................10323.5 EstimatingFGTcurvesanddifferencesbetweenFGTcurveswithconfidenceintervals..............................................................................................................................................................11123.6 Testingpovertydominanceandestimatingcriticalvalues...........................................115

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23.7 DecomposingFGTindices.............................................................................................................11623.8 EstimatingLorenzandconcentrationcurves......................................................................11923.9 EstimatingGiniandconcentrationcurves............................................................................12523.10 Usingbasicdistributivetools.................................................................................................12923.11 Plottingthejointdensityandjointdistributionfunction..........................................13523.12 Testingthebi‐dimensionalpovertydominance............................................................13823.13 Testingforpro‐poornessofgrowthinMexico...............................................................14123.14 BenefitincidenceanalysisofpublicspendingoneducationinPeru(1994)....147

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ListofFiguresFigure1:Ouputofnetdescribedasp..................................................................................................................9Figure2:DASPsubmenu......................................................................................................................................10Figure3:UsingDASPwithacommandwindow.......................................................................................11Figure4:AccessinghelponDASP....................................................................................................................12Figure5:EstimatingFGTpovertywithonedistribution......................................................................13Figure6:EstimatingFGTpovertywithtwodistributions....................................................................13Figure7:Povertyandthetargetingbypopulationgroups..................................................................21Figure8:DecompositionoftheFGTindexbygroups.............................................................................39Figure9:DecompositionofFGTbyincomecomponents......................................................................44Figure10:SectoraldecompositionofFGT..................................................................................................48Figure11:Decompositionofpovertyintotransientandchroniccomponents...........................50Figure12:DecompositionoftheGiniindexbyincomesources(Shapleyapproach)..............52Figure13:FGTcurves...........................................................................................................................................61Figure14:Lorenzandconcentrationcurves..............................................................................................63Figure15:Consumptiondominancecurves...............................................................................................66Figure16:ungroupdialogbox..........................................................................................................................83Figure17:Surveydatasettings........................................................................................................................88Figure18:Settingsamplingweights..............................................................................................................89Figure19:EstimatingFGTindices...................................................................................................................92Figure20:EstimatingFGTindiceswithrelativepovertylines...........................................................93Figure21:FGTindicesdifferentiatedbygender......................................................................................94Figure22:EstimatingdifferencesbetweenFGTindices.......................................................................97Figure23:EstimatingdifferencesinFGTindices.....................................................................................98Figure24:FGTdifferencesacrossyearsbygenderandzone.............................................................99Figure25:Estimatingmultidimensionalpovertyindices(A)..........................................................101Figure26:Estimatingmultidimensionalpovertyindices(B)..........................................................102Figure27:DrawingFGTcurves.....................................................................................................................104Figure28:EditingFGTcurves........................................................................................................................104Figure29:GraphofFGTcurves.....................................................................................................................105Figure30:FGTcurvesbyzone.......................................................................................................................106Figure31:GraphofFGTcurvesbyzone....................................................................................................107Figure32:DifferencesofFGTcurves..........................................................................................................108Figure33:Listingcoordinates.......................................................................................................................109Figure34:DifferencesbetweenFGTcurves............................................................................................110Figure35:DifferencesbetweenFGTcurves............................................................................................111Figure36:DrawingFGTcurveswithconfidenceinterval.................................................................112Figure37:FGTcurveswithconfidenceinterval....................................................................................113Figure38:DrawingthedifferencebetweenFGTcurveswithconfidenceinterval.................114Figure39:DifferencebetweenFGTcurveswithconfidenceinterval ( 0) ...........................114Figure40:DifferencebetweenFGTcurveswithconfidenceinterval ( 1) ...........................115Figure41:Testingforpovertydominance...............................................................................................116Figure42:DecomposingFGTindicesbygroups....................................................................................117Figure43:Lorenzandconcentrationcurves...........................................................................................120

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Figure44:Lorenzcurves..................................................................................................................................121Figure45:Drawingconcentrationcurves................................................................................................122Figure46:Lorenzandconcentrationcurves...........................................................................................123Figure47:DrawingLorenzcurves...............................................................................................................124Figure48:Lorenzcurves..................................................................................................................................124Figure49:EstimatingGiniandconcentrationindices.........................................................................126Figure50:Estimatingconcentrationindices...........................................................................................127Figure51:EstimatingdifferencesinGiniandconcentrationindices...........................................128Figure52:Drawingdensities..........................................................................................................................129Figure53:Densitycurves.................................................................................................................................130Figure54:Drawingquantilecurves............................................................................................................131Figure55:Quantilecurves...............................................................................................................................131Figure56:Drawingnon‐parametricregressioncurves......................................................................132Figure57:Non‐parametricregressioncurves........................................................................................133Figure58:Drawingderivativesofnon‐parametricregressioncurves........................................134Figure59:Derivativesofnon‐parametricregressioncurves...........................................................134Figure60:Plottingjointdensityfunction.................................................................................................135Figure61:Plottingjointdistributionfunction........................................................................................137Figure62:Testingforbi‐dimensionalpovertydominance...............................................................139Figure63:Testingthepro‐poorgrowth(primalapproach).............................................................142Figure64:Testingthepro‐poorgrowth(dualapproach)‐A............................................................143Figure65:Testingthepro‐poorgrowth(dualapproach)–B..........................................................145Figure66:Benefitincidenceanalysis..........................................................................................................148Figure67:BenefitIncidenceAnalysis(unitcostapproach).............................................................150

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1 IntroductionTheStatasoftwarehasbecomeaverypopulartooltotransformandprocessdata.Itcomeswithalargenumberofbasicdatamanagementmodulesthatarehighlyefficientfortransformationoflargedatasets.TheflexibilityofStataalsoenablesprogrammerstoprovidespecialized“.ado”routinestoaddtothepowerofthesoftware.ThisisindeedhowDASPinteractswithStata.DASP,whichstandsforDistributiveAnalysisStataPackage,ismainlydesignedtoassistresearchersandpolicyanalystsinterestedinconductingdistributiveanalysiswithStata.Inparticular,DASPisbuiltto:

Estimatethemostpopularstatistics(indices,curves)usedfortheanalysisofpoverty,inequality,socialwelfare,andequity;

Estimatethedifferencesinsuchstatistics; Estimatestandarderrorsandconfidenceintervalsbytakingfullaccountofsurvey

design; Supportdistributiveanalysisonmorethanonedatabase; Performthemostpopularpovertyanddecompositionprocedures; Checkfortheethicalrobustnessofdistributivecomparisons; Unifysyntaxandparameteruseacrossvariousestimationproceduresfordistributive

analysis.ForeachDASPmodule,threetypesoffilesareprovided:

*.ado: Thisfilecontainstheprogramofthemodule*.hlp: Thisfilecontainshelpmaterialforthegivenmodule*.dlg:

Thisfileallowstheusertoperformtheestimationusingthemodule’sdialogbox

The *.dlg files in particularmakes theDASP package very user friendly and easy to learn.Whenthese dialog boxes are used, the associated program syntax is also generated and showed in thereviewwindow.Theusercansavethecontentsofthiswindowina*.dofiletobesubsequentlyusedinanothersession.

2 DASPandStataversionsDASPrequires

o Stataversion10.0orhighero adofilesmustbeupdated

Toupdatetheexecutablefile(from10.0to10.2)andtheadofiles,see:http://www.stata.com/support/updates/

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3 InstallingandupdatingtheDASPpackageIngeneral,the*.adofilesaresavedinthefollowingmaindirectories:

Priority Directory Sources1 UPDATES: OfficialupdatesofStata *.adofiles2 BASE: *.adofilesthatcomewiththeinstalledStata software3 SITE: *.adofilesdownloadedfromthenet4 PLUS: ..5 PERSONAL: Personal*.adofiles

3.1 InstallingDASPmodulesa. Unzipthefiledasp.zipinthedirectoryc:b. Makesurethatyouhavec:/dasp/dasp.pkgorc:/dasp/stata.tocc. IntheStatacommandwindows,typethesyntax

netfromc:/dasp

Figure1:Ouputofnetdescribedasp

d. Typethesyntaxnetinstalldasp_p1.pkg,forcereplacenetinstalldasp_p2.pkg,forcereplacenetinstalldasp_p3.pkg,forcereplace

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3.2 AddingtheDASPsubmenutoStata’smainmenuWithStata9,submenuscanbeaddedtothemenuitemUser.

Figure2:DASPsubmenu

ToaddtheDASPsubmenus,thefileprofile.do(whichisprovidedwiththeDASPpackage)mustbecopiedintothePERSONALdirectory.Ifthefileprofile.doalreadyexists,addthecontentsoftheDASP–providedprofile.dofileintothatexistingfileandsaveit.Tocheckifthefileprofile.doalreadyexists,typethecommand:findfileprofile.do.

4 DASPanddatafilesDASPmakes itpossible tousesimultaneouslymore thanonedata file.Theusershould,however,“initialize”eachdatafilebeforeusingitwithDASP.Thisinitializationisdoneby:

1. Labelingvariablesandvaluesforcategoricalvariables;2. Initializingthesamplingdesignwiththecommandsvyset;3. Savingtheinitializeddatafile.

UsersarerecommendedtoconsultappendicesA,BandC,

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5 MainvariablesfordistributiveanalysisVARIABLEOFINTEREST.Thisisthevariablethatusuallycaptureslivingstandards.Itcanrepresent,forinstance, incomeper capita, expenditures per adult equivalent, calorie intake, normalizedheight‐for‐agescoresforchildren,orhouseholdwealth.SIZEVARIABLE.Thisreferstothe"ethical"orphysicalsizeoftheobservation.Forthecomputationofmany statistics, we will indeed wish to take into account how many relevant individuals (orstatisticalunits)arefoundinagivenobservation.GROUPVARIABLE.(ThisshouldbeusedincombinationwithGROUPNUMBER.)Itisoftenusefultofocusone’s analysis on some population subgroup. We might, for example, wish to estimate povertywithinacountry’sruralareaorwithinfemale‐headedfamilies.OnewaytodothisistoforceDASPto focus on a population subgroupdefined as those forwhom someGROUP VARIABLE (say, area ofresidence)equalsagivenGROUPNUMBER(say2,forruralarea).SAMPLINGWEIGHT.Samplingweightsaretheinverseofthesamplingprobability.Thisvariableshouldbesetupontheinitializationofthedataset.

6 HowcanDASPcommandsbeinvoked?Statacommandscanbeentereddirectlyintoacommandwindow:Figure3:UsingDASPwithacommandwindow

Analternativeistousedialogboxes.Forthis,thecommanddbshouldbetypedandfollowedbythenameoftherelevantDASPmodule.

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Example:dbifgt

7 HowcanhelpbeaccessedforagivenDASPmodule?TypethecommandhelpfollowedbythenameoftherelevantDASPmodule.Example:helpifgtFigure4:AccessinghelponDASP

8 ApplicationsandfilesinDASPTwomaintypesofapplicationsareprovidedinDASP.Forthefirstone,theestimationproceduresrequireonlyonedatafile.Insuchcases,thedatafileinmemoryistheonethatisused(or“loaded”);itisfromthatfilethattherelevantvariablesmustbespecifiedbytheusertoperformtherequiredestimation.

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Figure5:EstimatingFGTpovertywithonedistribution

Forthesecondtypeofapplications,twodistributionsareneeded.Foreachofthesetwodistributions,theusercanspecifythecurrently‐loadeddatafile(theoneinmemory)oronesavedondisk.Figure6:EstimatingFGTpovertywithtwodistributions

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Notes:1. DASPconsiderstwodistributionstobestatisticallydependent(forstatisticalinference

purposes)ifthesamedatasetisused(thesameloadeddataordatawiththesamepathandfilename)forthetwodistributions.

2. IftheoptionDATAINFILEischosen,thekeyboardmustbeusedtotypethenameoftherequiredvariables.

9 BasicNotationThefollowingtablepresentsthebasicnotationusedinDASP’susermanual.

Symbol Indicationy variableofinteresti observationnumberyi valueofthevariableofinterestforobservationihw samplingweighthwi samplingweightforobservationihs sizevariablehsi sizeofobservationi(forexamplethesizeofhouseholdi)wi hwi*hsihg groupvariablehgi groupofobservationi.wik swik=swiifhgi=k,and0otherwise.n samplesize

Forexample,themeanofyisestimatedbyDASPas :

1

1

n

i ii

n

ii

w yˆ

w

10 DASPandpovertyindices

10.1 FGTandEDE‐FGTpovertyindices(ifgt).Thenon‐normalisedFoster‐Greer‐ThorbeckeorFGTindexisestimatedas

1

1

ni i

in

ii

w ( z y )

P( z; )

w

wherezisthepovertylineand max( ,0)x x .TheusualnormalisedFGTindexisestimatedas ( ; ) ( ; ) /( )P z P z z

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TheEDE‐FGTindexisestimatedas:

1/( ( ; )) ( ; )EDE P z P z for

> 0

Thereexistthreewaysoffixingthepovertyline:

1‐Settingadeterministicpovertyline;2‐Settingthepovertylinetoaproportionofthemean;3‐SettingthepovertylinetoaproportionofaquantileQ(p).

Theusercanchoosethevalueofparameter . Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan

estimatepovertybyusingsimultaneouslypercapitaconsumptionandpercapitaincome. Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa

groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged.

InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.1

10.2 DifferencebetweenFGTindices(difgt)ThismoduleestimatesdifferencesbetweentheFGTindicesoftwodistributions.Foreachofthetwodistributions: Thereexistthreewaysoffixingthepovertyline:

1‐Settingadeterministicpovertyline;2‐Settingthepovertylinetoaproportionofthemean;3‐SettingthepovertylinetoaproportionofaquantileQ(p)

Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Alevelfortheparameter canbechosenforeachofthetwodistributions.

InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.2.

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10.3 Wattspovertyindex(iwatts).TheWattspovertyindexisestimatedas

1

1

q

i ii

ni

i

w (ln( z / y )

P( z )

w

wherezisthepovertylineand q thenumberofpoor.

Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan




10.4 DifferencebetweenWattsindices(diwatts)ThismoduleestimatesdifferencesbetweentheWattsindicesoftwodistributions.Foreachofthetwodistributions: Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both


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10.5 Sen‐Shorrocks‐Thonpovertyindex(isst).TheSen‐Shorroks‐Thonpovertyindexisestimatedas:

1* *gP( z ) HP ( z, )[ G ]

wherezisthepovertylineH istheheadcount, *P ( z, ) thepovertygapestimatedatthelevelof

poorgroupand *gG theGiniindexofpovertygaps( z y ) / z .





10.6 DifferencebetweenSen‐Shorrocks‐Thonindices(disst)ThismoduleestimatesdifferencesbetweentheWattsindicesoftwodistributions.Foreachofthetwodistributions: Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both


10.7 DASPandmultidimensionalpovertyindicesThegeneralformofanadditivemultidimensionalpovertyindexis:

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1

1

( , )

( , )

n

i ii

n

ii

w p X Z

P X Z

w

where ( , )ip X Z isindividualI’spovertyfunction(withvectorofattributes ,1 ,,...,i i i JX x x and

vectorofpovertylines 1,..., JZ z z ),determiningI’scontributiontototalpoverty ( , )P X Z .

[1]Chakravartyetal(1998)index(imdp_cmr)

,

1

( , )J

j i ji j

jj

z xp X Z a

z

[2]ExtendedWattsindex(imdp_ewi)

,1

( , ) lnmin( ; )

Jj

i jj i jj

zp X Z a

z x

[3]MultiplicativeextendedFGTindex(imdp_mfi)

,

1

( , )jJ

j i ji

jj

z xp X Z

z

[4]Tsui(2002)index(imdp_twu)

,1

( , ) 1min( ; )

jbJ

ji

j i jj

zp X Z

z x

[5]Intersectionheadcountindex(imdp_ihi)

,1

( , )J

i j i jj

p X Z I z x

[6]Unionheadcountindex(imdp_uni)

,1

( , ) 1J

i j i jj

p X Z I z x

[7]BourguignonandChakravartybi‐dimensional(2003)index(imdp_bci)

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1 2( , )ip X Z C C

where:

1 ,11

1

iz xC

z

and 2 ,22

2

iz xC

z

[8]AlkireandFoster(2007)index(imdp_afi)

,

1

1 1( , , ) ( )

N Jj i j

i j i cji j

z xp X Z w I d d

N J z

Where1

J

jj

w J

and id denotesthenumberofdimensionsinwhichtheindividual i isdeprived.

cd denotesthenormativedimensionalcut‐off.

Thepresentedmodulesabovecanbeusedtoestimatethemultidimensionalpovertyindicesaswellastheirstandarderrors.

Theusercanselectamongthesevenmultidimensionalpovertyindices. Thenumberofdimensionscanbeselected(1to10). Ifapplicable,theusercanchooseparametervaluesrelevanttoachosenindex. Agroupvariablecanbeusedtoestimatetheselectedindexatthelevelofacategorical

group. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith3decimals;thiscanbealsochanged.

UsersareencouragedtoconsidertheexercisesthatappearinSection23.3

10.8 Multipleoverlappingdeprivationanalysis(MODA)indices TheimodaDASPmoduleproducesaseriesofmultidimensionalpovertyindicesinordertoshow the incidence of deprivation in each dimension. Further, this application estimates theincidenceofmulti‐deprivationinthedifferentcombinationsofdimensions.Inthisapplication,thenumberofdimensionsissupposedtobethree.Further,themultidimensionalpovertyismeasuredbytheheadcount(unionandintersectionheadcountindices)andtheAlkireandFoster(2007)M0indexfordifferentlevelsofthedimensionalcut‐off.

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Thenumberofdimensionsisthree. AgroupvariablecanbeusedtoestimatetheMODAindicesatthelevelofacategorical

group. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith3decimals;thiscanbealsochanged.

11 DASP,povertyandtargetingpolicies

11.1 Povertyandtargetingbypopulationgroups

Theper‐capitadollarimpactofamarginaladditionofaconstantamountofincometoeveryonewithinagroupk–calledLump‐SumTargeting(LST)–ontheFGTpovertyindexP(k, z; ) ,isasfollows:

P(k, z; 1) if 1LST

f (k, z) if 0

where z isthepovertyline,kisthepopulationsubgroupforwhichwewishtoassesstheimpactoftheincomechange,and ( , )f k z isthedensityfunctionofthegroup k atlevelofincome z .Theper‐capitadollarimpactofaproportionalmarginalvariationofincomewithinagroupk,calledInequalityNeutralTargeting,ontheFGTpovertyindexP(k, z; ) isasfollows:

P(k, z; ) zP(k, z; 1)if 1

(k)INT

zf (k, z)if 0

(k)

Themoduleitargetgallowsto: Estimatetheimpactofmarginalchangeinincomeofthegrouponpovertyofthegroupand

thatofthepopulation; Selectthedesignofchange,constantorproportionaltoincometokeepinequality

unchanged; Drawcurvesofimpactaccordingforarangeofpovertylines; Drawtheconfidenceintervalofimpactcurvesorthelowerorupperboundofconfidence

interval; Etc.

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Figure 7: Poverty and the targeting by population groups

Reference: DUCLOS,J.‐Y.ANDA.ARAAR(2006):PovertyandEquityMeasurement,Policy,andEstimationwithDAD,BerlinandOttawa:SpringerandIDRC.(sec.12.1)

11.2 Povertyandtargetingbyincomecomponents

Proportional change per 100% of component

Assume that total income Y is the sum of J income components, with 1

J

j jj

Y y

and where c is a factor

that multiplies income component jy and that can be subject to growth. The derivative of the normalized

FGT index with respect to j is given by

jj 1, j 1 Jj

P(z, )CD (z, )

where CDj is the Consumption dominance curve of component j. Change per $ of component

The per-capita dollar impact of growth in the thj component on the normalized FGT index of the thk group is as follows:

22

j j

jj

P(z, )

yCD (z, )

y

where j

CD is the normalized consumption dominance curve of the component j. Constant change per component Simply we assume that the change concerns the group with component level greater than zero. Thus, this is similar to targeting by the nonexclusive population groups. Themoduleitargetcallowsto: Estimatetheimpactofmarginalchangeinincomecomponentonpoverty; Selecttheoptionnormalisedornonnormalisedbytheaverageofcomponent; Selectthedesignofchange,constant(lumpsum)orproportionaltoincometokeep

inequalityunchanged; Drawcurvesofimpactaccordingforarangeofpovertylines; Drawtheconfidenceintervalofimpactcurvesorthelowerorupperboundofconfidence

interval; Etc.

Reference: DUCLOS,J.‐Y.ANDA.ARAAR(2006):PovertyandEquityMeasurement,Policy,andEstimationwithDAD,BerlinandOttawa:SpringerandIDRC.(sec.12)

12 Marginalpovertyimpactsandpovertyelasticities

12.1 FGTelasticity’swithrespecttotheaverageincomegrowth(efgtgr).

The overall growth elasticity (GREL) of poverty, when growth comes exclusively from growth within a group k (namely, within that group, inequality neutral), is given by:

( , )0

( )

( , ; ) ( , ; 1)1

( , )

zf k zif

F zGREL

P k z P k zif

P z

where z is the poverty line, k is the population subgroup in which growth takes place, ( , )f k z is the

density function at level of income z of group k , and ( )F z is the headcount.

23

Araar,AbdelkrimandJean‐YvesDuclos,(2007),Povertyandinequalitycomponents:a micro framework,Working Paper: 07‐35. CIRPEE, Department of Economics,UniversitéLaval.Kakwani, N. (1993) "Poverty and economic growth with application to CôteD’Ivoire",ReviewofIncomeandWealth,39(2):121:139.

ToestimatetheFGTelasticity’swithrespectaverageincomegrowththegrouporthewhole

population; Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan




12.2 FGTelasticity’swithrespecttotheaverageincomegrowthwithdifferentapproaches(efgtgro).

The overall growth elasticity of poverty estimated by using one approach among the following list:

The counterfactual approach; The marginal approach; The parameterized approach; The numerical approach;

The module efgtgroallowstheestimationofpovertyelasticityorsemi‐elasticitywithrespecttogrowthwiththedifferentproposedapproachesabove.Formoredetailsontheseapproaches,see:

Abdelkrim Araar, 2012. "Expected Poverty Changeswith Economic Growth andRedistribution,"Cahiersderecherche1222,CIRPEE.

ToestimatetheFGTelasticity’s–semi‐elasticity’s‐withrespectaverageincomegrowththe

grouporthewholepopulation; Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa

groupvariableisselected,onlythefirstvariableofinterestisthenused. Theresultsaredisplayedwith6decimals;thiscanbechanged.

12.3 FGTelasticity’swithrespecttoGiniinequality(efgtineq).

The overall growth elasticity (INEL) of poverty, when growth comes exclusively from change in inequality within a group k is given by:

24

( ) ( , ) ( ) ( ) ( ) ( )/ 0

( )

( , ; ) ( ) / ( , ; 1) ( ) ( ) ( )/ 1

( , )

k f k z k z k k C kif

F z IINEL

P k z k z z P k z k k C kif

P z I

where z is the poverty line, k is the population subgroup in which growth takes place, ( , )f k z is the

density function at level of income z of group k , and ( )F z is the headcount. ( )C k is the concentration

coefficient of group k when incomes of the complement group are preplaced by ( )k . I denotes the Gini index.

Araar,AbdelkrimandJean‐YvesDuclos,(2007),Povertyandinequalitycomponents:a micro framework,Working Paper: 07‐35. CIRPEE, Department of Economics,UniversitéLaval.Kakwani, N. (1993) "Poverty and economic growth with application to CôteD’Ivoire",ReviewofIncomeandWealth,39(2):121:139.

ToestimateFGTelasticity’swithrespectaverageincomegrowththegrouporthewhole

population; Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan




12.4 FGTelasticity’swithrespecttoGini‐inequalitywithdifferentapproaches(efgtine).

The overall Gini-inequality elasticity of poverty can be estimated by using one approach among the following list:

The counterfactual approach; The marginal approach; The parameterized approach; The numerical approach;

The module efgtineallowstheestimationofpovertyelasticityorsemi‐elasticitywithrespecttoinequalitywiththedifferentproposedapproachesabove.Formoredetailsontheseapproaches,see:

Abdelkrim Araar, 2012. "Expected Poverty Changeswith Economic Growth andRedistribution,"Cahiersderecherche1222,CIRPEE.

25

ToestimatetheFGTelasticity’s–semi‐elasticity’s‐withrespecttoinequality; Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa

groupvariableisselected,onlythefirstvariableofinterestisthenused. Theresultsaredisplayedwith6decimals;thiscanbechanged.

12.5 FGTelasticitieswithrespecttowithin/betweengroupcomponentsofinequality(efgtg).

ThismoduleestimatesthemarginalFGTimpactandFGTelasticitywithrespecttowithin/betweengroupcomponentsofinequality.Agroupvariablemustbeprovided.ThismoduleismostlybasedonAraarandDuclos(2007):

Araar,AbdelkrimandJean‐YvesDuclos,(2007),Povertyandinequalitycomponents:a micro framework,Working Paper: 07‐35. CIRPEE, Department of Economics,UniversitéLaval.

Toopenthedialogboxofthismodule,typethecommanddbefgtg.

AfterclickingonSUBMIT,thefollowingshouldbedisplayed:

26

(g)

12.6 FGT‐elasticitieswithrespecttowithin/betweenincomecomponentsofinequality(efgtc).

ThismoduleestimatesthemarginalFGTimpactandFGTelasticitywithrespecttowithin/betweenincomecomponentsofinequality.Alistofincomecomponentsmustbeprovided.ThismoduleismostlybasedonAraarandDuclos(2007):

Araar,AbdelkrimandJean‐YvesDuclos,(2007),Povertyandinequalitycomponents:a micro framework,Working Paper: 07‐35. CIRPEE, Department of Economics,UniversitéLaval.

Toopenthedialogboxofthismodule,typethecommanddbefgtc.

27


(k)

Incaseoneisinterestedinchangingsomeincomecomponentonlyamongthoseindividualsthatare

effectivelyactiveinsomeeconomicsectors(schemes * * *(k), and inthepapermentionedabove),theusershouldselecttheapproach“Truncatedincomecomponent”.

28

13 DASPandinequalityindices

13.1 Giniandconcentrationindices(igini)TheGiniindexisestimatedas

ˆˆ 1

ˆI

where

2 21

21 1

( ) ( )ˆn

i ii

i

V Vy

V

andn

i hh i

V w

and 11 2 n ny y y y .

TheconcentrationindexforthevariableTwhentherankingvariableisYisestimatedas

1 TT

T

ˆIC

ˆ

where ˆT istheaverageofvariableT,

2 21

21 1

( ) ( )ˆn

i iT i

i

V Vt

V

andwheren

i hh i

V w

and 11 2 n ny y y y .

Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecanestimateinequality,forinstancebyusingsimultaneouslypercapitaconsumptionandpercapitaincome.

Toestimateaconcentrationindex,theusermustselectarankingvariable.. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa




13.2 DifferencebetweenGini/concentrationindices(digini)ThismoduleestimatesdifferencesbetweentheGini/concentrationindicesoftwodistributions.Foreachofthetwodistributions:

Onevariableofinterestshouldbeselected; Toestimateaconcentrationindex,arankingvariablemustbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups;

29

Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Boththetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged.

Theresultsaredisplayedwith6decimals;thiscanbechanged.

13.3 Generalisedentropyindex(ientropy)Thegeneralizedentropyindexisestimatedas

i

ini

ii 1

ini i

ii 1

i i in

ii

i 1

y1w 1 if 0,1

ˆ1 w

ˆ1I w log if 0

yw

w y y1log if 1

ˆ ˆw


estimateinequalitysimultaneouslyforpercapitaconsumptionandforpercapitaincome. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa



13.4 Differencebetweengeneralizedentropyindices(diengtropy)Thismoduleestimatesdifferencesbetweenthegeneralizedentropyindicesoftwodistributions.Foreachofthetwodistributions:

Onevariableofinterestshouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both


30

13.5 Atkinsonindex(iatkinson)DenotetheAtkinsonindexofinequalityforthegroupkby I(ε) .Itcanbeexpressedasfollows:

ni i

i 1n

ii 1

w yˆˆ ( )ˆ ˆI(ε) whereˆ

w

TheAtkinsonindexofsocialwelfareisasfollows:

1

1n 1 ε

i ini 1

ii 1

ni in

i 1i

i 1

1w (y ) if 1 and 0

w

ξ(ε)

1Exp w ln(y ) ε 1

w





13.6 DifferencebetweenAtkinsonindices(diatkinson)ThismoduleestimatesdifferencesbetweentheAtkinsonindicesoftwodistributions.Foreachofthetwodistributions:



31

13.7 Coefficientofvariationindex(icvar)Denote the coefficient of variation index of inequality for the group k by CV. It can be expressed as follows:

1n n 22 2

i i ii 1 i 1

2

ˆw y / wCV

ˆ





13.8 Differencebetweencoefficientofvariation(dicvar)Thismoduleestimatesdifferencesbetweencoefficient of variation indicesoftwodistributions.Foreachofthetwodistributions:



13.9 Quantile/shareratioindicesofinequality(inineq)Thequantileratioisestimatedas

11 2

2

Q(p )QR(p , p )

Q(p )

whereQ(p) denotesap‐quantileand 1p and 2p arepercentiles.

Theshareratioisestimatedas

GL(p2)-GL(p1)

SR(p1,p2,p3,p4) GL(p4)-GL(p3)

32

whereGL(p) istheGeneralisedLorenzcurveand 1p , 2p , 3p and 4p arepercentiles.





13.10 DifferencebetweenQuantile/Shareindices(dinineq)ThismoduleestimatesdifferencesbetweentheQuantile/Shareindicesoftwodistributions.Foreachofthetwodistributions:


thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged; Theresultsaredisplayedwith6decimals;thiscanbechanged.

13.11 TheARAAR(2009)multidimensionalinequalityindexThe Araar (2009) the multidimensional inequality index for the K dimensions of wellbeing takes the following form:

1

1K

k k k k ki

I I C

Where k is the attributed weight to the dimension k (may take the same value across the

dimensions or can depend on the averages of the wellbeing dimensions. k kI C are respectively

the relative –absolute- Gini and concentration indices of component k . The normative parameter

k controls the sensitivity of index to the inter-correlation between dimensions. For more details,

see: Abdelkrim Araar, 2009. "The Hybrid Multidimensional Index of Inequality," Cahiers de recherche 0945,

CIRPEE: http://ideas.repec.org/p/lvl/lacicr/0945.html

14 DASPandpolarizationindices

14.1 TheDERindex(ipolder)

33

TheDuclos,EstebanandRay(2004)(DER)polarizationindexcanbeexpressedas:1DER( ) f (x) f (y) y x dydx

wheref(x)denotesthedensityfunctionatx.Thediscreteformulathatisusedtoestimatethisindexisasfollows:

ni i i

i 1n

ii 1

w f (y ) a(y )DER( )

w

ThenormalizedDERestimatedbythismoduleisdefinedas:

(1 )DER( )

DER( )2

where:i i 1

j i j j i ij 1 j 1

i i N Ni i

i 1 i 1

2 w w 2 w y w y

a(y ) y 1

w w

TheGaussiankernelestimatorisusedtoestimatethedensityfunction.


estimatepolarizationbyusingsimultaneouslypercapitaconsumptionandpercapitaincome.

Agroupvariablecanbeusedtoestimatepolarizationatthelevelofacategoricalgroup.Ifagroupvariableisselected,onlythefirstvariableofinterestisthenused.


Theresultsaredisplayedwith6decimals;thiscanbechanged.Mainreference

DUCLOS,J.‐Y.,J.ESTEBAN,ANDD.RAY(2004):“Polarization:Concepts,Measurement,Estimation,”Econometrica,72,1737–1772.

14.2 DifferencebetweenDERpolarizationindices(dipolder)ThismoduleestimatesdifferencesbetweentheDERindicesoftwodistributions.Foreachofthetwodistributions:

Onevariableofinterestshouldbeselected; Conditionscanbespecifiedsuchastofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged.

34

Theresultsaredisplayedwith6decimals;thiscanbechanged.

14.3 TheFosterandWolfson(1992)polarizationindex(ipolfw)TheFosterandWolfson(1992)polarizationindexcanbeexpressedas:

FW 2 2 0.5 Lorenz(p 0.5) Ginimedian

Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecanestimatepolarizationbyusingsimultaneouslypercapitaconsumptionandpercapitaincome.

Agroupvariablecanbeusedtoestimatepolarizationatthelevelofacategoricalgroup.Ifagroupvariableisselected,onlythefirstvariableofinterestisthenused.


Theresultsaredisplayedwith6decimals;thiscanbechanged.MainreferenceFOSTER,J.ANDM.WOLFSON(1992):“PolarizationandtheDeclineoftheMiddleClass:CanadaandtheU.S.”mimeo,VanderbiltUniversity.

14.4 DifferencebetweenFosterandWolfson(1992)polarizationindices(dipolfw)

ThismoduleestimatesdifferencesbetweentheFWindicesoftwodistributions.Foreachofthetwodistributions:

Onevariableofinterestshouldbeselected; Conditionscanbespecifiedsuchastofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both


14.5 TheGeneralisedEsteban,GardinandRay(1999)polarisationindex(ipoger)

The proposed measurement of polarisation by Esteban and Ray (1994) is defined asfollows:

35

m m

ER 1j k j k

j 1 k 1

P (f , ) p p

where, j and jp denote respectively the average income and the population share of

groupj.Theparameter 1,1.6reflelcssensitivityofthesocietytopolarisation.Thefirst

stepfortheestimationrequirestodefinetheexhaustiveandmutuallyexclusivegroups, .Thiswillinvolvesomedegreeoferror.However,asthisgroupingwillgeneratesomelossofinformation, depending on the degree of income dispersion in each of the groups con‐sidered. Taking intoaccount this idea, themeasureofgeneralisedpolarisation,proposedby Esteban et al. (1999), is obtained after correcting the ERP ( ) index applied to thesimplifiedrepresentationoftheoriginaldistributionwithameasureofthegroupingerror.Nonetheless, when dealing with personal or spatial income distributions, there are nounanimouscriteriaforestablishingtheprecisedemarcationbetweendifferentgroups. Toaddress this problem,Estebanet al. (1999) follow themethodologyproposedbyAghevliandMehran(1981)andDaviesandShorrocks(1989)inordertofindtheoptimalpartitionofthedistributioninagivennumberofgroups, * .Thismeansselectingthepartitionthat

minimises the Gini index value of within‐group inequality, *Error G f G (see

Estebanetal.,1999). ThemeasureofgeneralisedpolarisationproposedbyEstebanetal.(1999),therefore,isgivenby:

m m

EGR * 1 *j k j k

j 1 k 1

P (f , , , ) p p G f G

where, 0 isaparameterthatinformsabouttheassignedweighttotheerrorterm.(InthestudyofEstebanetal.(1999),theusedvaluewhere: 1 ).The Statamodule ipoegr.ado estimates the generalised form of the Esteban et al. (1999)polarisationindex.Inadditiontotheusualvariables,thattheusercanindicate,thisroutineofferstotheuserthreefollowingoptions:

1. Thenumberofgroups.Empiricalstudiesuse twoor threegroups.Theusercanselect thenumber of groups. According to this number, the program seeks for the optimal incomeinterval for each group and displays them. It also displays the error in percentage, ie:

*G f G*100

G f

;

2. Theparameter ;3. Theparameter .

Torespectthescaleinvarianceprinciple,wedividebeforehandallincomesbytheaverage

incomei.e. j j / .Inaddition,wedividetheindexbythescalar2tomakeitsinterval

between0and1whentheparameter 1 .

36

m m

EGR * 1 *j k j k

j 1 k 1

P (f , , , ) 0.5 p p G f G

14.6 TheInaki(2008)polarisationindex(ipoger)

Wehaveapopulationsplitinto N groups,eachoneofsize 0in .Thedensityfunction,themean

andthepopulationshareofgroup i aredenotedby ( )if x , i and i respectively. is theoverall

mean.Wethereforehavethat ( ) 1 if x ,1

N

i ii

and1

1

N

ii

.StartingfromInaki(2008),a

socialpolarisationindexcanbedefinedas:

1

( ) ( , ) ( , )N

W Bi

P F P i F P i F

where1 2 1( , ) ( ) ( )W i i iP i F f x f y x y dydx

and

1 1 1 1( , ) ( ) 1 ( ) B i i i i i iP i F f x dx f x xdx

ThemoduleStatadspolallowsperformingthedecompositionofthesocialpolarisationindex ( )P F intogroupcomponents. Theusercanselecttheparameteralpha; Theusercanselecttheuseofafasterapproachfortheestimationofthedensityfunction; Standarderrorsareprovidedforallestimatedindices.Theytakeintoaccountthefull

samplingdesign; Theresultsaredisplayedwith6decimalsbydefault;thiscanbechanged; TheusercansaveresultsinExcelformat.

Theresultsshow: Theestimatedpopulationshareofsubgroup i : i ;

Theestimatedincomeshareofsubgroup i : / i i ;

Theestimated ( , )WP i F indexofsubgroup i ;

Theestimated ( , )BP i F indexofsubgroup i ;

Theestimated ( , )W WiP P i F index;

Theestimated ( , )B BiP P i F index;

Theestimatedtotalindex FP

37

Toopenthedialogboxformoduledspol,typedbdspolinthecommandwindow.

Example:Forillustrativepurposes,weusea1996Cameroonianhouseholdsurvey,whichismadeofapproximately1700households.Thevariablesusedare:Variables:STRATA StratuminwhichahouseholdlivesPSU PrimarysamplingunitofthehouseholdWEIGHT SamplingweightSIZE HouseholdsizeINS_LEV Educationleveloftheheadofthehousehold

1. Primary;2. ProfessionalTraining,secondaryandsuperior;3. Notresponding.

We decompose the above social polarization index using the module dspol by splitting theCameroonian population into three exclusive groups, according to the education level of thehouseholdhead.Wefirstinitializethesamplingdesignofthesurveywiththedialogboxsvysetasshowninwhatfollows:

38

Afterthat,openthedialogboxbytypingdbdspol,andchoosevariablesandparametersasin:

AfterclickingSUBMIT,thefollowingresultsappear:

Mainreferences

1. DUCLOS,J.‐Y.,J.ESTEBAN,ANDD.RAY(2004):“Polarization:Concepts,Measurement,Estimation,”Econometrica,72,1737–1772.

2. TianZ.&all(1999)"FastDensityEstimationUsingCF‐kernelforVeryLargeDatabases".http://portal.acm.org/citation.cfm?id=312266

3. IñakiPermanyer,2008."TheMeasurementofSocialPolarizationinaMulti‐groupContext,"UFAEandIAEWorkingPapers736.08,UnitatdeFonamentsdel'AnàlisiEconòmica(UAB)andInstitutd'AnàlisiEconòmica(CSIC).

39

15 DASPanddecompositions

15.1 FGTPoverty:decompositionbypopulationsubgroups(dfgtg)ThedgfgtmoduledecomposestheFGTpovertyindexbypopulationsubgroups.Thisdecompositiontakestheform

1

( ; ) ( ) ( ; ; )G

gP z g P z g

whereG isthenumberofpopulationsubgroups.Theresultsshow:

TheestimatedFGTindexofsubgroup g : ( ; ; )P z g

Theestimatedpopulationshareofsubgroup g : ( )g

Theestimatedabsolutecontributionofsubgroupg tototalpoverty: ( ) ( ; ; )g P z g Theestimatedrelativecontributionofsubgroupg tototal

poverty: ( ) ( ; ; ) / ( ; )g P z g P z

Anasymptoticstandarderrorisprovidedforeachofthesestatistics.Toopenthedialogboxformoduledfgtg,typedbdfgtginthecommandwindow.Figure8:DecompositionoftheFGTindexbygroups

NotethattheusercansaveresultsinExcelformat.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.7

40

15.2 FGTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue(dfgts)

The dfgts module decomposes the total alleviation of FGT poverty into a sum of the contributions generated by separate income components. Total alleviation is maximal when all individuals have an income greater than or equal to the poverty line. A negative sign on a decomposition term indicates that

an income component reduces poverty. Assume that there exist K income sources and that ks denotes

income source .k The FGT index is defined as:

1

1

1

1; ;

niK

ik n

ki

i

yw zP z y s

w

where iw is the weight assigned to individual i and n is sample size. The dfgts Stata module estimates:

The share in total income of each income source k ;

The absolute contribution of each source k to the value of ( 1P );

The relative contribution of each source k to the value of ( 1P );

Note that the dfgts ado file requires the module shapar.ado, which is programmed to perform decompositions using the Shapley value algorithm developed by Araar and Duclos (2008).

Araar A and Duclos J‐Y (2008), “An algorithm for computing the Shapley Value”, PEP and CIRPEE. Tech.‐

Note: Novembre‐2008: http://dad.ecn.ulaval.ca/pdf_files/shap_dec_aj.pdf

Empirical illustration with the Nigerian household survey We use a survey of Nigerian households (NLSS, using 17764 observations) carried out between September 2003 and August 2004 to illustrate the use of the dfgts module. We use per capita total household income as a measure of individual living standards. Household observations are weighted by household size and sampling weights to assess poverty over all individuals. The six main income components are:

source_1: Employment income; source_2: Agricultural income; source_3: Fish-processing income; source_4: Non-farm business income; source_5: Remittances received; source_6: All other income;

The Stata data file is saved after initializing its sampling design with the command svyset.

To open the dialog box for module dfgts, type db dfgts in the command window.

41

Figure 1: Decomposition of the FGT index by income components

Indicate the varlist of the six income sources. Indicate that the poverty line is set to 15 000 $N. Set the variable HOUSEHOLD SIZE. Set the variable HOUSEHOLD WEIGHT. Click on the button SUBMIT. The following results appear:

42

15.3 AlkireandFoster(2007)MDindexofpoverty:decompositionbypopulationsubgroups(dmdafg)

ThedmdafgmoduledecomposestheMDAlkireandFosterindexofpovertyindexbypopulationsubgroups.Thisdecompositiontakestheform.Theresultsshow: TheestimatedAlkireandFosterindexofeachsubgroup: Theestimatedpopulationshareofsubgroup; Theestimatedabsolutecontributionofsubgroupg tototalpoverty; Theestimatedrelativecontributionofsubgroupg tototalpoverty;

Anasymptoticstandarderrorisprovidedforeachofthesestatistics.

15.4 AlkireandFoster(2007:decompositionbydimensionsusingtheShapleyvalue(dmdafs)

Thedmdafsmoduledecomposes theAlkire andFoster (2007)multidimensional poverty

indicesintoasumofthecontributionsgeneratedbyeachofthepovertydimensions.Theadoptedrule to define the Shapley characteristic function with the non‐presence of a given factor –dimension‐ is by setting its level to its specific poverty line. Of course, this ensures the non‐contributionof thisdimension to themulti‐derivation index–AF (2007) indices‐ . Note that thedmdafs ado file requires the module shapar.ado, which is programmed to performdecompositionsusingtheShapleyvaluealgorithmdevelopedbyAraarandDuclos(2008).

Araar A and Duclos J‐Y (2008), “An algorithm for computing the Shapley Value”, PEP andCIRPEE.Tech.‐Note:Novembre‐2008:http://dad.ecn.ulaval.ca/pdf_files/shap_dec_aj.pdf

43

15.5 F GTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue(dfgts)

The dfgts module decomposes the total alleviation of FGT poverty into a sum of the

contributions generated by separate income components. Total alleviation is maximal when allindividuals have an income greater than or equal to the poverty line. A negative sign on adecompositiontermindicatesthatanincomecomponentreducespoverty.AssumethatthereexistK incomesourcesandthat ks denotesincomesource .k TheFGTindexisdefinedas:

1

1

1

1; ;

niK

ik n

ki

i

yw zP z y s

w

where iw istheweightassignedtoindividual i and n issamplesize.ThedfgtsStatamodule

estimates: Theshareintotalincomeofeachincomesource k ;

Theabsolutecontributionofeachsource k tothevalueof( 1P );

Therelativecontributionofeachsource k tothevalueof( 1P );

Notethatthedfgtsadofilerequiresthemoduleshapar.ado,whichisprogrammedtoperformdecompositionsusingtheShapleyvaluealgorithmdevelopedbyAraarandDuclos(2008).

Araar A and Duclos J‐Y (2008), “An algorithm for computing the Shapley Value”, PEP andCIRPEE.Tech.‐Note:Novembre‐2008:http://dad.ecn.ulaval.ca/pdf_files/shap_dec_aj.pdf

EmpiricalillustrationwiththeNigerianhouseholdsurveyWe use a survey of Nigerian households (NLSS, using 17764 observations) carried out betweenSeptember2003andAugust2004toillustratetheuseofthedfgtsmodule.Weusepercapitatotalhousehold income as a measure of individual living standards. Household observations areweighted by household size and samplingweights to assess poverty over all individuals. The sixmainincomecomponentsare:

source_1:Employmentincome; source_2:Agriculturalincome; source_3:Fish‐processingincome; source_4:Non‐farmbusinessincome; source_5:Remittancesreceived; source_6:Allotherincome;

44

TheStatadatafileissavedafterinitializingitssamplingdesignwiththecommandsvyset.Toopenthedialogboxformoduledfgts,typedbdfgtsinthecommandwindow.Figure9:DecompositionofFGTbyincomecomponents

Indicatethevarlistofthesixincomesources. Indicatethatthepovertylineissetto15000$N. SetthevariableHOUSEHOLDSIZE. SetthevariableHOUSEHOLDWEIGHT. ClickonthebuttonSUBMIT.Thefollowingresultsappear:

45

15.6 DecompositionofthevariationinFGTindicesintogrowthandredistributioncomponents(dfgtgr)

DattandRavallion(1992)decomposethechangeintheFGTindexbetweentwoperiods,t1andt2,intogrowthandredistributioncomponentsasfollows:

t2 t1 t1 t1 t1 t2 t1 t12 1

var iation C1 C2

P P P( , ) P( , ) P( , ) P( , ) R / ref 1

t2 t2 t1 t2 t2 t2 t2 t1

2 1

var iation C1 C2

P P P( , ) P( , ) P( , ) P( , ) R / ref 2

wherevariation =differenceinpovertybetweent1andt2;C1 =growthcomponent;C2 =redistributioncomponent;R =residual;Ref =periodofreference.

1 1( , )t tP :theFGTindexofthefirstperiod1 1( , )t tP :theFGTindexofthesecondperiod

46

),(P 1t2t :theFGTindexofthefirstperiodwhenallincomes 1tiy ofthefirstperiodaremultiplied

by 1t2t /

),(P 2t1t :theFGTindexofthesecondperiodwhenallincomes2t

iy ofthesecondperiodare

multipliedby 2t1t / TheShapleyvaluedecomposesthevariationintheFGTIndexbetweentwoperiods,t1andt2,intogrowthandredistributioncomponentsasfollows:

21

Variation

12 CCPP

),(P),(P),(P),(P2

1C 2t1t2t2t1t1t1t2t

1 ),(P),(P),(P),(P

2

1C 1t2t2t2t1t1t2t1t

2

15.7 DecompositionofchangeinFGTpovertybypovertyandpopulationgroupcomponents–sectoraldecomposition‐(dfgtg2d).

For the additive poverty measurements, like the FGT indices, their level can be

expressed as a sumof the poverty contributions of the various subgroups of population.Eachsubgroupcontributesbytheirspopulationshareandpovertylevel.Thus,thechangeinpovertyacrosstimewilldependsonthechangeinthesetwocomponents.Ifwedenotethepopulationshareofgroup inperiod by , thechange inpovertybetweentwoperiodscanbeexpressedasfellows(seeHuppi(1991)andDuclosandAraar(2006)):

(06)

Aswecannote,forthelatterdecomposition,weassumethatthereferenceperiodis

theinitial.Ifthereferenceperiodisthefinal,thedecompositiontakesthefollwingform:

47

(06)

Toremovethearbitrarnessinselectingthereferenceperiod,wecanusetheShapley

decompositionapproachandwefindthen:

(07)

where is average population shares and

.TheDASPmoduledfgtg2dallowstoperformthesectoraldecomposition,andthisbyselectingthereferenceperiodoftheShapleyapproach(seethefollowingdialogbox):

48

Figure 10: Sectoral decomposition of FGT

=== === === Population 0.005650 0.002463 0.000000 0.002932 0.004317 0.000000 Inactive -0.001724 -0.011681 0.000000 0.016127 0.018726 0.000000 Subsistence farmer 0.012648 -0.014336 0.000000 0.005992 0.013963 0.000000 Crop farmer -0.010387 0.029328 0.000000 0.000700 0.000930 0.000000 Other type of earner 0.000610 -0.000234 0.000000 0.001380 0.002417 0.000000 Artisan or trader 0.001731 -0.000768 0.000000 0.000930 0.001222 0.000000 Wage-earner (private sector) 0.001224 0.000218 0.000000 0.001117 0.001931 0.000000 Wage-earner (public sector) 0.001548 -0.000064 0.000000 Component Component Component Group Poverty Population Interaction Decomposition components

0.000000 0.016124 0.000000 0.010927 0.019477 Population 1.000000 0.444565 1.000000 0.452677 0.008113 0.004839 0.035336 0.003354 0.032340 0.047901 Inactive 0.075856 0.414986 0.046719 0.386852 -0.028134 0.016403 0.021132 0.015083 0.011572 0.024093 Subsistence farmer 0.680885 0.514999 0.653552 0.533956 0.018957 0.014896 0.034911 0.014125 0.024457 0.042625 Crop farmer 0.104402 0.500707 0.167806 0.424391 -0.076316 0.001308 0.060817 0.000923 0.089680 0.108357 Other type of earner 0.006650 0.194481 0.005689 0.293404 0.098923 0.004288 0.014712 0.004666 0.018202 0.023404 Artisan or trader 0.062640 0.097548 0.055795 0.126776 0.029228 0.002164 0.024093 0.002624 0.023087 0.033369 Wage-earner (private sector) 0.026598 0.067271 0.029035 0.111283 0.044012 0.003790 0.012599 0.003927 0.023396 0.026573 Wage-earner (public sector) 0.042971 0.022406 0.041403 0.059094 0.036688 Pop. share FGT index Pop. share FGT index FGT index Group Initial Initial Final Final Difference in Population shares and FGT indices

Parameter alpha : 0.00 Group variable : gse Decomposition of the FGT index by groups

. dfgtg2d exppc exppcz, alpha(0) hgroup(gse) pline(41099) file1(C:\data\bkf94I.dta) hsize1(size) file2(C:\data\bkf98I.dta) hsize2(size) ref(0)

49

15.8 DecompositionofFGTpovertybytransientandchronicpovertycomponents(dtcpov)

Thisdecomposestotalpovertyacrosstimeintotransientandchroniccomponents.TheJalanandRavallion(1998)approach

Let tiy betheincomeofindividualiinperiodtand i beaverageincomeovertheTperiodsforthat

sameindividuali,i=1,…,N.Totalpovertyisdefinedas:

T N ti i

t 1i 1N

ii 1

w (z y )

TP( , z)

T w

Thechronicpovertycomponentisthendefinedas:N

i ii 1

Ni

i 1

w (z )CPC( , z)

w

Transientpovertyequals:

TPC( , z) TP( , z) CPC( , z)

Duclos,AraarandGiles(2006)approach

Let tiy be the incomeof individual i inperiodtand i beaverage incomeover theTperiods for

individuali.Let ( , z) bethe”equally‐distributed‐equivalent”(EDE)povertygapsuchthat:

1/( , z) TP( , z)

Transientpovertyisthendefinedas

Ni i

i 1N

ii 1

w ( , z)TPC( , z)

w

where i i i, z 1, z andB1/

T ti i

i t( , z) (z y ) / T

50

andchronicpovertyisgivenby

CPC( , z) ( , z) TPC( , z)

Notethatthenumberofperiodsavailableforthistypeofexerciseisgenerallysmall.Becauseofthis,abias‐correctionistypicallyuseful,usingeitherananalytical/asymptoticorbootstrapapproach.Toopenthedialogboxformoduledtcpov,typedbdtcpovinthecommandwindow.Figure11:Decompositionofpovertyintotransientandchroniccomponents

Theusercanselectmorethanonevariableofinterestsimultaneously,whereeachvariable

representsincomeforoneperiod. Theusercanselectoneofthetwoapproachespresentedabove. Small‐T‐bias‐correctionscanbeapplied,usingeitherananalytical/asymptoticora

bootstrapapproach. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both


MainreferencesJalan Jyotsna, and Martin Ravallion. (1998) "TransientPovertyinPostreformRuralChina" Journal of Comparative Economics, 26(2), pp. 338:57.

Jean‐YvesDuclos&AbdelkrimAraar& JohnGiles, 2006. "Chronic and Transient Poverty:MeasurementandEstimation,withEvidencefromChina,"WorkingPaper0611,CIRPEE.

51

15.9 Inequality:decompositionbyincomesources(diginis)AnalyticalapproachThediginismoduledecomposesthe(usual)relativeortheabsoluteGiniindexbyincomesources.Thethreeavailableapproachesare:

Rao’sapproach(1969) LermanandYitzhaki’sapproach(1985) Araar’sapproach(2006)

Reference(s)

Lerman, R. I., and S. Yitzhaki. "Income Inequality Effects by Income Source: A NewApproachandApplications to theUnitedStates."ReviewofEconomics andStatistics67(1985):151‐56.

AraarAbdelkrim(2006).OntheDecompositionoftheGiniCoefficient:anExactApproach,withanIllustrationUsingCameroonianData,Workingpaper02‐06,CIRPEE.

ShapleyapproachThedsineqsmoduledecomposes inequality indices intoasumofthecontributionsgeneratedbyseparateincomecomponents.ThedsineqsStatamoduleestimates: Theshareintotalincomeofeachincomesource k ; Theabsolutecontributionofeachsource k totheGiniindex; Therelativecontributionofeachsource k totheGiniindex;

For theShapleydecomposition, therule that isused toestimate the inequality index fora

subset of components is by suppressing the inequality generated by the complement subset ofcomponents. For this,we generate a counterfactual vector of incomewhich equals to the sumofcomponents of the subset (factors of the coalition) plus the average of the complement subset.Notethatthedsineqsadofilerequiresthemoduleshapar.ado,whichisprogrammedtoperformdecompositionsusingtheShapleyvaluealgorithmdevelopedbyAraarandDuclos(2008).

Araar A and Duclos J‐Y (2008), “An algorithm for computing the Shapley Value”, PEP andCIRPEE.Tech.‐Note:Novembre‐2008::http://dad.ecn.ulaval.ca/pdf_files/shap_dec_aj.pdf

To open the dialog box for module dsginis, type db dsginis in the command window.

52

Figure 12: Decomposition of the Gini index by income sources (Shapley approach)

15.10 RegressionbaseddecompositionofinequalitybyincomesourcesA useful approach to show the contribution of income covariates to total inequality is bydecomposingthelatterbythepredictedcontributionsofcovariates.Formally,wedenotethetotal

incomeby y and the setof covariates 1, 2 , , KX x x x . Usinga linearmodel specification,we

havethen:

0 1 1 2 2ˆ ˆ ˆ ˆ ˆ ˆ

k k K Ky x x x x

where 0 and denoterespectivelytheestimatedconstantandtheresidual.In general, there are two main approaches for the decomposition of total inequality by incomesources:

1‐ The Shapleyapproach: This approach is based on the expectedmarginal contribution ofincomesourcestothetotalinequality.

2‐ TheAnalyticalapproach:Thisapproach isbasedonalgebraicallydevelopments todefineinequalityindexbyaformulathatshowsthecontributionofincomesources.

WiththeShapleyapproach: Theusercanselectamongthefollowingrelativeinequalityindices;

Giniindex Atkinsonindex Generalizedentropyindex Coefficientvariationindex

Theusercanselectamongthefollowingmodelspecifications;

53

Linear: 0 1 1 2 2ˆ ˆ ˆ ˆ ˆ

K Ky x x x

SemiLogLinear: 0 1 1 2 2ˆ ˆ ˆ ˆ ˆlog( ) K Ky x x x

WiththeAnalyticapproach: Theusercanselectamongthefollowingrelativeinequalityindices;

Giniindex Squaredcoefficientvariationindex

Themodelspecificationislinear.

Decomposingtotalinequalitywiththeanalyticalapproach:

The total income equals to 0 1 2 K Ry s s s s s where 0s is the estimated constant,

ˆk k ks X and Rs istheestimatedresidual.AsreportedbyWang2004,relativeinequalityindices

arenotdefinedwhentheaverageofthevariableofinterestequalstozero(thecaseoftheresidual).Also,inequalityindicesequalstozerowhenwethevariableofinterestisaconstant(thecaseoftheestimatedconstant).Todealwith these twoproblemsWang (2004)proposes the followingbasicrules:

Let 0 1 2ˆ Ky s s s s and 1 2 Ky s s s then: 0( ) ( ) rI y cs I y cs

Thecontributionoftheconstant: 0 ˆ( ) ( )cs I y I y

Thecontributionoftheresidual: ˆ( ) ( )Rcs I y I y TheGiniindex:UsingtheRao’s1969approach,therelativeGiniindexcanbedecomposedasfollows:

( ) kk

y

I y C

Where y istheaverageof y and kC istheconfidentofconcentrationof ks when y istherankingvariable.TheSquaredcoefficientofvariationindex:AsshownbyShorrocks1982,Thesquaredcoefficientofvariationindexcanbedecomposedasfollows:

21

ov( , )( )

Kk

k y

C y sI y

DecomposingwiththeShapleyapproach:

54

Asindicated,theShapleyapproachisbasedontheexpectedmarginalcontributionofthecomponent.Theusercanselectamongthetwomethodstodefinetheimpactofmissingagivencomponent.

Withoption:method(mean),whenacomponentismissingfromagivensetofcomponents,wereplaceitbyitsmean.

Withoption:method(zero),whenacomponentismissingfromagivensetofcomponents,wereplaceitbyzero.

Asindicatedabove,wecannotestimatetherelativeinequalityfortheresidualcomponent.

Forthelinearmodel,thedecompositiontakesthefollowingform: ˆ( ) ( ) rI y I y cs ,where

thecontributionoftheresidual: ˆ( ) ( )rcs I y I y .

FortheSemi‐loglinearmodeltheShapleydecompositionisappliedforallcomponentsincludingtheconstantandtheresidual.

WiththeShapleyapproach,theusercanusetheloglinearspecification.However,theusermustindicatethevariableincomeandnotitslog(DASPrunsautomaticallytheregressionwiththelog(y)asdependantvariable).Example1

57

Example2

Withthisspecification,wehave 0 1 2xp( )K Ry E s s s s s .Thenwecannot:

Wecannotestimatetheincomeshare(nolinearform);

58

Thecontributioniftheconstantisnil. 01

xp( ). xp( ). xp( )K

k Ek

y E s E s E s

.Thenaddingthe

constantwillhavenotanyimpact,sincewemultiplysimplybyascalar.

Example3

15.11 Giniindex:decompositionbypopulationsubgroups(diginig).The diginig module decomposes the (usual) relative or the absolute Gini index by populationsubgroups.Let therebeGpopulationsubgroups.Wewish todetermine thecontributionof everyoneofthosesubgroupstototalpopulationinequality.TheGiniindexcanbedecomposedasfollows:

G

g g gWithin Overlapg 1

Between

I I I R

where

59

g thepopulationshareofgroupg;

g theincomeshareofgroupg.

I

between‐groupinequality(wheneachindividualisassignedtheaverageincomeofhisgroup).

R

Theresidueimpliedbygroupincomeoverlap

15.12 Generalizedentropyindicesofinequality:decompositionbypopulationsubgroups(dentropyg).

TheGeneralisedEntropyindicesofinequalitycanbedecomposedasfollows:

K

k 1

ˆ (k)ˆˆ ˆI( ) (k) .I(k; ) I( )ˆ

where:B )k( istheproportionofthepopulationfoundinsubgroupk.

B )k( isthemeanincomeofgroupk.

B ;kI isinequalitywithingroupk.

B I is population inequality if each individual in subgroup k is given the meanincomeofsubgroupk, (k) .

15.13 Polarization:decompositionoftheDERindexbypopulationgroups(dpolag)

As proposed byAraar (2008), theDuclos, Esteban andRay index can be decomposed as

follows:

1 1

Between

Within

g g g gg

P R P P

where

1

1

( ) ( ) ( )

( ) ( )

g gg

g g g

a x x f x dxR

a x f x dx

g and g arerespectivelythepopulationandincomesharesofgroup g .

60

( )g x denotesthelocalproportionofindividualsbelongingtogroup g andhaving

income x ; P istheDERpolarizationindexwhenthewithin‐grouppolarizationorinequalityis

ignored;

ThedpolasmoduledecomposestheDERindexbypopulationsubgroups.Reference(s)

Abdelkrim Araar, 2008. "On the Decomposition of Polarization Indices: Illustrations with Chinese and Nigerian Household Surveys," Cahiers de recherche 0806, CIRPEE.

15.14 Polarization:decompositionoftheDERindexbyincomesources(dpolas)

As proposed byAraar (2008), theDuclos, Esteban andRay index can be decomposed as

follows:

k kk

P CP

Where1

1

( ) ( )kk

k k

f x a x dxCP

and k arerespectivelythepseudoconcentrationindexand

incomeshareofincomesource k .ThedpolasmoduledecomposestheDERindexbyincomesources.Reference(s)

Abdelkrim Araar, 2008. "On the Decomposition of Polarization Indices: Illustrations with Chinese and Nigerian Household Surveys," Cahiers de recherche 0806, CIRPEE.

16 DASPandcurves.

16.1 FGTCURVES(cfgt).FGTcurvesareusefuldistributivetoolsthatcaninteraliabeusedto:

1. Showhowthelevelofpovertyvarieswithdifferentpovertylines;2. Testforpovertydominancebetweentwodistributions;

61

3. Testpro‐poorgrowthconditions.FGTcurvesarealsocalledprimaldominancecurves.Thecfgtmoduledrawssuchcurveseasily.Themodulecan: drawmorethanoneFGTcurvesimultaneouslywhenevermorethanonevariableofinterest

isselected; drawFGTcurvesfordifferentpopulationsubgroupswheneveragroupvariableisselected; drawFGTcurvesthatarenotnormalizedbythepovertylines; drawdifferencesbetweenFGTcurves; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats:

o *.gph:Stataformat;o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments;o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments.

Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.Toopenthedialogboxofthemodulecfgt,typethecommanddbdfgtinthecommandwindow.Figure13:FGTcurves


62

FGTCURVEwithconfidenceinterval(cfgts).ThecfgtsmoduledrawsanFGTcurveanditsconfidenceintervalbytakingintoaccountsamplingdesign.Themodulecan: drawanFGTcurveandtwo‐sided,lower‐boundedorupper‐boundedconfidenceintervals

aroundthatcurve; conditiontheestimationonapopulationsubgroup; drawaFGTcurvethatisnotnormalizedbythepovertylines; listorsavethecoordinatesofthecurveandofitsconfidenceinterval; savethegraphsindifferentformats:


Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.5.

16.3 DifferencebetweenFGTCURVESwithconfidenceinterval(cfgts2d).Thecfgts2dmoduledrawsdifferencesbetweenFGTcurvesandtheirassociatedconfidenceintervalbytakingintoaccountsamplingdesign.Themodulecan: drawdifferencesbetweenFGTcurvesandtwo‐sided,lower‐boundedorupper‐bounded

confidenceintervalsaroundthesedifferences; normalizeornottheFGTcurvesbythepovertylines; listorsavethecoordinatesofthedifferencesbetweenthecurvesaswellastheconfidence

intervals; savethegraphsindifferentformats:


Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.


LorenzandconcentrationCURVES(clorenz).Lorenzandconcentrationcurvesareusefuldistributivetoolsthatcaninteraliabeusedto:

1. showthelevelofinequality;2. testforinequalitydominancebetweentwodistributions;3. testforwelfaredominancebetweentwodistributions;4. testforprogressivity.

TheclorenzmoduledrawsLorenzandconcentrationcurvessimultaneously.Themodulecan:

63

drawmorethanoneLorenzorconcentrationcurvesimultaneouslywhenevermorethan

onevariableofinterestisselected; drawmorethanonegeneralizedorabsoluteLorenzorconcentrationcurvesimultaneously

whenevermorethanonevariableofinterestisselected; drawmorethanonedeficitsharecurve; drawLorenzandconcentrationcurvesfordifferentpopulationsubgroupswheneveragroup

variableisselected; drawdifferencesbetweenLorenzandconcentrationcurves; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats:


Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.Toopenthedialogboxofthemoduleclorenz,typethecommanddbclorenzinthecommandwindow.Figure14:Lorenzandconcentrationcurves


16.5 Lorenz/concentrationcurveswithconfidenceintervals(clorenzs).TheclorenzsmoduledrawsaLorenz/concentrationcurveanditsconfidenceintervalbytakingsamplingdesignintoaccount.Themodulecan:

64

drawaLorenz/concentrationcurveandtwo‐sided,lower‐boundedorupper‐boundedconfidenceintervals;

conditiontheestimationonapopulationsubgroup; drawLorenz/concentrationcurvesandgeneralizedLorenz/concentrationcurves; listorsavethecoordinatesofthecurvesandtheirconfidenceinterval; savethegraphsindifferentformats:



16.6 DifferencesbetweenLorenz/concentrationcurveswithconfidenceinterval(clorenzs2d)

Theclorenz2dmoduledrawsdifferencesbetweenLorenz/concentrationcurvesandtheirassociatedconfidenceintervalsbytakingsamplingdesignintoaccount.Themodulecan: drawdifferencesbetweenLorenz/concentrationcurvesandassociatedtwo‐sided,lower‐

boundedorupper‐boundedconfidenceintervals; listorsavethecoordinatesofthedifferencesandtheirconfidenceintervals; savethegraphsindifferentformats:



16.7 Povertycurves(cpoverty)Thecpovertymoduledrawsthepovertygaporthecumulativepovertygapcurves.

o Thepovertygapatapercentile p is: ( ; ) ( ( ))G p z z Q p

o Thecumulativepovertygapatapercentile p ,notedby ( ; )CPG p z ,isgivenby:

1

1

( ) ( ( ))

( ; )

ni i i

in

ii

w z y I y Q p

CPG p z

w

Themodulecanthus: drawmorethanonepovertygaporcumulativepovertygapcurvessimultaneously

whenevermorethanonevariableofinterestisselected; drawpovertygaporcumulativepovertygapcurvesfordifferentpopulationsubgroups

wheneveragroupvariableisselected;

65

drawdifferencesbetweenpovertygaporcumulativepovertygapcurves; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats:



16.8 Consumptiondominancecurves(cdomc) Consumption dominance curves are useful tools for studying the impact of indirect tax fiscal reforms on

poverty. The thj Commodity or Component dominance (C-Dominance for short) curve is defined as follows:

2

1

1

1

1

( )

2

( , )

( )

| ( ) 1

n jsi i i

in

iij

n ji i i

j in

ii

w z y y

if s

w

CD z s

w K z y y

E y y z f z if s

w

where K( ) is a kernel function and yj is the thj commodity. Dominance of order s is checked by

setting s 1 . Thecdomcmoduledrawssuchcurveseasily.Themodulecan: drawmorethanoneCDcurvesimultaneouslywhenevermorethanonecomponentis

selected; drawtheCDcurveswithconfidenceintervals; estimatetheimpactofchangeinpriceofagivencomponentonFGTindex(CDcurve)fora

specifiedpovertyline; drawthenormalizedCDcurvesbytheaverageofthecomponent; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats:


Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.Toopenthedialogboxofthemodulecdomc,typethecommanddbcdomcinthecommandwindow.

66

Figure15:Consumptiondominancecurves

16.9 Difference/Ratiobetweenconsumptiondominancecurves(cdomc2d)

Thecdomc2dmoduledrawsdifferenceorratiobetweenconsumptiondominancecurvesandtheirassociatedconfidenceintervalsbytakingsamplingdesignintoaccount.Themodulecan: drawdifferencesbetweenconsumptiondominancecurvesandassociatedtwo‐sided,lower‐

boundedorupper‐boundedconfidenceintervals; listorsavethecoordinatesofthedifferencesandtheirconfidenceintervals; savethegraphsindifferentformats:



16.10 DASPandtheprogressivitycurves

16.10.1 CheckingtheprogressivityoftaxesortransfersThe module cprog allows checking whether taxes or transfers are progressive. Let X be a gross income, T be a given tax and B be a given transfer. The tax T is Tax Redistribution (TR) progressive if :

PR p L p C p 0 0,1X T p

67

The transfer B is Tax Redistribution (TR) progressive if : PR p C p L p 0 0,1B X p

The tax T is Income Redistribution (IR) progressive if :

PR p C p L p 0 0,1X T X p

The transfer B is Income Redistribution (IR) progressive if :

PR p C p L p 0 0,1X B X p

16.10.2 CheckingtheprogressivityoftransfervstaxThe module cprogbt allows checking whether a given transfer is more progressive than a given tax. The transfer B is more Tax Redistribution (TR) progressive than a tax T if :

PR p C p C p 2L p 0 0,1B T X p

The transfer B is more Income Redistribution (TR) progressive than a tax T if :

PR p C p -C p 0 0,1X B X T p

17 Dominance

17.1 Povertydominance(dompov)

Distribution1dominatesdistribution2atordersovertherange z,z ifonlyif:

1 2( ; ) ( ; ) ,P P z z for 1s .

ThisinvolvescomparingstochasticdominancecurvesatordersorFGTcurveswith 1s .Thisapplicationestimatesthepointsatwhichthereisareversaloftherankingofthecurves.Saiddifferently,itprovidesthecrossingpointsofthedominancecurves,thatis,thevaluesof and

1 ( ; )P forwhich1 2( ; ) ( ; )P P when:

1 2 2 1( ( ; ) ( ; )) ( ( ; ) ( ; ))sign P P sign P P forasmall .Thecrossingpoints canalsobereferredtoas“criticalpovertylines”.Thedompovmodulecanbeusedtocheckforpovertydominanceandtocomputecriticalvalues.ThismoduleismostlybasedonAraar(2006):

68

Araar,Abdelkrim,(2006),Poverty,InequalityandStochasticDominance,TheoryandPractice:Illustration with Burkina Faso Surveys,Working Paper: 06‐34. CIRPEE, Department ofEconomics,UniversitéLaval.


17.2 Inequalitydominance(domineq)

Distribution1inequality‐dominatesdistribution2atthesecondorderifandonlyif:

1 2( ) ( ) 0,1 L p L p p

Themoduledomineqcanbeusedtocheckforsuchinequalitydominance.ItisbasedmainlyonAraar(2006):

Araar,Abdelkrim,(2006),Poverty,InequalityandStochasticDominance,TheoryandPractice:Illustration with Burkina Faso Surveys,Working Paper: 06‐34. CIRPEE, Department ofEconomics,UniversitéLaval.

Intersectionsbetweencurvescanbeestimatedwiththismodule.ItcanalsousedtocheckfortaxandtransferprogressivitybycomparingLorenzandconcentrationcurves.

17.3 DASPandbi‐dimensionalpovertydominance(dombdpov)Lettwodimensionsofwell‐beingbedenotedby 1,2k .Theintersectionbi‐dimensionalFGTindexfordistributionD isestimatedas

2

1 1

1

n k k ki i

i kD n

ii

w ( z y )

P ( Z; A )

w

where 1 2,Z z z and 1 2,A arevectorsofpovertylinesandparameters respectively,

and max( ,0)x x .

Distribution1dominatesdistribution2atorders 1 2,s s overtherange 0, Z ifandonlyif:

1 2 1 2( ; 1) ( ; 1) 0, 0,P Z A s P Z A s Z z z andfor 1 1 2 21, 1s s .

TheDASPdombdpovmodulecanbeusedtocheckforsuchdominance.Foreachofthetwodistributions:

Thetwovariablesofinterest(dimensions)shouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged.

69

Surfacesshowingthedifference,thelowerboundandtheupperboundoftheconfidencesurfacesareplottedinteractivelywiththeGnuPlottool.

Coordinatescanbelisted. CoordinatescanbesavedinStataorGnuPlot‐ASCIIformat.


18 Distributivetools

18.1 Quantilecurves(c_quantile)Thequantileatapercentilepofacontinuouspopulationisgivenby:

1( ) ( )Q p F p where ( )p F y isthecumulativedistributionfunctionaty.For a discrete distribution, let n observations of living standards be ordered such that

1 2 1i i ny y y y y . If )()( 1 ii yFpyF , we define1( ) iQ p y . The normalised

quantileisdefinedas ( ) ( ) /Q p Q p .InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.10.

18.2 Incomeshareandcumulativeincomesharebygroupquantiles(quinsh)

Thismodulecanbeusedtoestimate the incomeshares,aswellas, thecumulative incomesharesbyquantilegroups.Theusercanindicatethenumberofgrouppartition.Forinstance,ifthenumber is five, the quintile income shares are provided. We can also plot the graph bar of theestimatedincomeshares.

18.3 Densitycurves(cdensity)TheGaussiankernelestimatorofadensityfunction )(xf isdefinedby

2

1

( ) 1ˆ ( ) ( ) exp 0.5 ( ) ( )2

i ii ii i in

ii

w K x x xf x and K x x and x

hhw

wherehisabandwidththatactsasa“smoothing”parameter.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.10.Boundary bias correction: A problem occurs with kernel estimation when a variable of interest is bounded. It may be for instance that consumption is bounded between two bounds, a minimum and a maximum, and that we wish to estimate its density “close” to these two bounds. If the true value of the density at these two bounds is

70

positive, usual kernel estimation of the density close to these two bounds will be biased. A similar problem occurs with non-parametric regressions. Renormalisation approach: One way to alleviate these problems is to use a smooth “corrected” Kernel estimator, following a paper by Peter Bearse, Jose Canals and Paul Rilstone. A boundary-corrected Kernel density estimator can then be written as

*i i ii

ni

i 1

w K (x)K (x)f (x)

w

where

h

xx)x(and)x(5.0exp

2h

1)x(K i

i2

ii

and where the scalar )x(K*i is defined as

))x((P)x()x(K i*i

!1s!21)(P

1s2

)0001(l,h

minxB,

h

maxxA:ld)(P)(P)(KlM)x( ss

1B

As1

min is the minimum bound, and max is the maximum one. h is the usual bandwidth. This correction removes bias to order hs. DASP offers four options, without correction, and with correction of order 1, 2 and 3. Refs:

Jones,M.C.1993,simplyboundarycorrectionforKerneldensityestimation.StatisticsandComputing3:135‐146.

Bearse,P.,Canals,J.andRilstone,P.EfficientSemiparametricEstimationofDurationModelsWithUnobservedHeterogeneity,EconometricTheory,23,2007,281–308

Reflection approach: The reflection estimator approaches the boundary estimator by reflecting the data at the boundaries:

ri ii

ni

i 1

w K (x)f (x)

w

r x X x X 2min x X 2maxK (x) K K K

h h h

Refs: CwikandMielniczuk(1993),Data‐dependentBandwidthChoiceforaGradeDensityKernel

Estimate.StatisticsandprobabilityLetters16:397‐405

71

Silverman,B.W.(1986),DensityforStatisticsandDataAnalysis.LondonChapmanandHall(p30).

18.4 Non‐parametricregressioncurves(cnpe)Non‐parametric regression is useful to show the link between two variables without specifyingbeforehandafunctionalform.Itcanalsobeusedtoestimatethelocalderivativeofthefirstvariablewithrespecttothesecondwithouthavingtospecifythefunctionalformlinkingthem.Regressionswiththecnpemodulecanbeperformedwithoneofthefollowingtwoapproaches:

18.4.1 Nadaraya‐WatsonapproachAGaussiankernelregressionofyonxisgivenby:

( )( | )

( )i i ii

i ii

w K x yE y x y x

w K x

Fromthis,thederivativeof ( | )y x withrespecttoxisgivenby

( | )dy y xE x

dx x

18.4.2 LocallinearapproachThelocallinearapproachisbasedonalocalOLSestimationofthefollowingfunctionalform:

1 1 12 2 2( ) ( ) ( ) ( ) ( ) ( )i i i i iK x y x K x x K x x x v

or,alternatively,of:

1 1 12 2 2( ) ( ) ( ) ( )i i i i i iK x y K x K x x x v

Estimatesarethengivenby:

E y x ,

dyE x

dx


18.5 DASPandjointdensityfunctions.Themodulesjdensitycanbeusedtodrawajointdensitysurface.TheGaussiankernelestimatorofthejointdensityfunction ( , )f x y isdefinedas:

72

22ni i

ini 1 x y

x y ii 1

x x y y1 1f (x, y) w exp

2 h h2 h h w

Withthismodule: Thetwovariablesofinterest(dimensions)shouldbeselected; specificpopulationsubgroupcanbeselected; surfacesshowingthejointdensityfunctionareplottedinteractivelywiththeGnuPlottool; coordinatescanbelisted;c coordinatescanbesavedinStataorGnuPlot‐ASCIIformat.

InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.11???

18.6 DASPandjointdistributionfunctions

Themodulesjdistrubcanbeusedtodrawjointdistributionsurfaces.Thejointdistributionfunction ( , )F x y isdefinedas:

ni i i

i 1n

ii 1

w I(x x)I(y y)F(x, y)

w

Withthismodule: Thetwovariablesofinterest(dimensions)shouldbeselected; specificpopulationsubgroupscanbeselected; surfacesshowingthejointdistributionfunctionareplottedinteractivelywiththeGnuPlot

tool; coordinatescanbelisted; coordinatescanbesavedinStataorGnuPlot‐ASCIIformat.


19 DASPandpro‐poorgrowth

19.1 DASPandpro‐poorindicesThemoduleipropoorestimatessimultaneouslythethreefollowingpro‐poorindices:

1. TheChenandRavallionpro‐poorindex(2003):

73

1 2

1

W ( z ) W ( z )Index

F ( z )

where DW ( z ) istheWattsindexfordistribution 1 2D , and 1F ( z ) istheheadcountfor

indexforthefirstdistribution,bothwithpovertylinesz.

2. TheKakwaniandPerniapro‐poorindex(2000):

1 2

1 1 1 2

P ( z, ) P ( z )Index

P ( z, ) P ( z( / ), )

3. TheKakwani,KhandkerandSonpro‐poorindex(2003):

wheretheaveragegrowthis 2 1 1g( ) / andwhereasecondindexisgivenby:

2 1Index _ Index _ g

Onevariableofinterestshouldbeselectedforeachdistribution. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both


19.2 DASPandpro‐poorcurvesPro‐poorcurvescanbedrawnusingeithertheprimalorthedualapproach.Theformerusesincomelevels.Thelatterisbasedonpercentiles.

19.2.1 Primalpro‐poorcurvesThechangeinthedistributionfromstate1tostate2iss‐orderabsolutelypro‐poorwithstandardcons if:

+2 1( , ) ( , 1) ( , 1) <0 z 0,zz s P z cons s P z s

Thechangeinthedistributionfromstate1tostate2iss‐orderrelativelypro‐poorif:

+22 1

1

( , ) ( , 1) , 1 <0 z 0,zz s z P z s P z s

Themodule cpropoorp can be used to draw these primal pro‐poor curves and their associatedconfidenceintervalbytakingintoaccountsamplingdesign.Themodulecan:

1 2

1 1 1 21

P ( z, ) P ( z )Index _ g

P ( z, ) P ( z( / ), )

74

drawpro‐poorcurvesandtheirtwo‐sided,lower‐boundedorupper‐boundedconfidenceintervals;

listorsavethecoordinatesofthedifferencesbetweenthecurvesaswellasthoseoftheconfidenceintervals;

savethegraphsindifferentformats:o *.gph:Stataformat;o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments;o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments.

Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.13.

19.2.2 Dualpro‐poorcurvesLet:

( )Q p :quantileatpercentile p .( )GL p :GeneralisedLorenzcurveatpercentile p .

:averagelivingstandards.Thechangeinthedistributionfromstate1tostate2isfirst‐orderabsolutelypro‐poorwithstandardcons=0if:

+2 1( , ) ( ) ( )>0 0, p ( )z s Q p Q p p F z

orequivalentlyif:

+2 1

1

( ) ( )( , ) >0 0, p ( )

( )

Q p Q pz s p F z

Q p

Thechangeinthedistributionfromstate1tostate2isfirst‐orderrelativelypro‐poorif:

+2 2

1 1

( )( , ) - >0 0, p ( )

( )

Q pz s p F z

Q p

Thechangeinthedistributionfromstate1tostate2issecond‐orderabsolutelypro‐poorif:

+2 1( , ) ( ) ( )>0 0, p ( )z s GL p GL p p F z

orequivalentlyif:

+2 1

1

( ) ( )( , ) >0 0, p ( )

( )

GL p GL pz s p F z

GL p

Thechangeinthedistributionfromstate1tostate2isfirst‐orderrelativelypro‐poorif:

75

+2 2

1 1

( )( , ) - >0 0, p ( )

( )

GL pz s p F z

GL p

The module cpropoord can be used to draw these dual pro‐poor curves and their associatedconfidenceintervalbytakingintoaccountsamplingdesign.Themodulecan: drawpro‐poorcurvesandtheirtwo‐sided,lower‐boundedorupper‐boundedconfidence

intervals; listorsavethecoordinatesofthedifferencesbetweenthecurvesaswellasthoseofthe

confidenceintervals; savethegraphsindifferentformats:


Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.13

20 DASPandBenefitIncidenceAnalysis

20.1 Benefitincidenceanalysis

Themain objective of using a benefit incidence approach is to analyse the distribution ofbenefitsfromtheuseofpublicservicesaccordingtothedistributionoflivingstandards.

Twomain sources of information are used. The first informs on the access of household

members to public services. This information can be found in the usual household surveys. Theseconddealswiththeamountoftotalpublicexpendituresoneachpublicservice.Thisinformationisusuallyavailableatthenationallevelandsometimesinamoredisaggregatedformat,suchasatthe regional level. The benefit incidence approach combines the use of these two sources ofinformationtoanalysethedistributionofpublicbenefitsanditsprogressivity.

Formally,let

iw bethesamplingweightofobservation i ;

iy bethelivingstandardofmembersbelongingtoobservation i (i.e.,percapitaincome);

sie bethenumberof“eligible”membersofobservationi,i.e.,membersthat“need”the

publicserviceprovidedbysectors.ThereareSsectors;

sif bethenumberofmembersofobservationi thateffectivelyusethepublicservice

providedbysectors;

ig bethesocio‐economicgroupofeligiblemembersofobservationi (typicallyclassifiedbyincomepercentiles);

76

ic beasubgroupindicatorforobservationi (e.g.,1foraruralresident,and2foranurbanresident).Eligiblememberscanthusbegroupedintopopulationexclusivesubgroups;

srE betotalpublicexpendituresonsector s inarea r .ThereareRareas(theareahere

referstothegeographicaldivisionwhichonecanhavereliableinformationontotalpublicexpendituresonthestudiedpublicservice);

sE betotalpublicexpendituresonsector s

Rs s

rr 1

E E

.

Herearesomeofthestatisticsthatcanbecomputed.

1. Theshareofaginsector s isdefinedasfollows:

ns

i is i 1g n

si i

i 1

w f I(i g)

SH

w f

Notethat:G

sg

g 1

SH 1

.

2. Therateofparticipationofagroupginsectors isdefinedasfollows:

ns

i is i 1g n

si i

i 1

w f I(i g)

CR

w e I(i g)

Thisratecannotexceed100%since s si if e i .

3. Theunitcostofabenefitinsectorsforobservation j ,whichreferstothehousehold

membersthatliveinarea r :

r

ss rj n

sj j

j 1

EUC

w f

where rn isthenumberofsampledhouseholdsinarear.

4. Thebenefitofobservationifromtheuseofpublicsector s is:

s s si i iB f UC

5. Thebenefitofobservation i fromtheuseoftheSpublicsectorsis:

77

Ss

i is 1

B B

6. Theaveragebenefitatthelevelofthoseeligibletoaservicefromsectorsandforthose

observationsthatbelongtoagroupg ,isdefinedas:n

si i

s i 1g n

si i

i 1

w B I(i g)

ABE

w e I(i g)

7. Theaveragebenefitforthosethatusetheservice s andbelongtoagroupg isdefinedas:

8. Theproportionofbenefitsfromtheservicefromsectors thataccruestoobservationsthat

belongtoagroupg isdefinedas:

sgs

g s

BPB

E

wheren

s sg i i

i 1

B w B I(i g)

.

These statistics can be restricted to specific socio‐demographic groups (e.g.,. rural/urban) byreplacing I(i g) by I(i c) .

.Thebian.adomoduleallowsthecomputationofthesedifferentstatistics.Somecharacteristicsofthemodule:

o Possibilityofselectingbetweenoneandsixsectors.o Possibilityofusingfrequencydataapproachwheninformationabouttheleveloftotalpublic

expendituresisnotavailable.o Generationofbenefit variablesby the typeofpublic services (ex:primary, secondaryand

tertiaryeducationlevels)andbysector.o Generationofunitcostvariablesforeachsector.o Possibilityofcomputingstatisticsaccordingtogroupsofobservations.o Generationofstatisticsaccordingtosocial‐demographicgroups,suchasquartiles,quintiles

ordeciles.

ns

i is i 1g n

si i

i 1

w B I(i g)

ABF

w f I(i g)

78

Generally,publicexpendituresonagivenservicecanvaryfromonegeographicaloradministrativeareatoanother.Whentheinformationaboutpublicexpendituresisavailableatthelevelofareas,thisinformationcanbeusedwiththebianmoduletoestimateunitcostmoreaccurately.Example1Observationi HH

sizeEligibleHHmembers

Frequency Areaindicator Totallevelofregionalpublicexpenditures

1 7 3 2 1 140002 4 2 2 1 140003 5 5 3 1 140004 6 3 2 2 120005 4 2 1 2 12000

Inthisexample,thefirstobservationcontainsinformationonhousehold1.

Thishouseholdcontains7individuals; Threeindividualsinthishouseholdareeligibletothepublicservice; Only2amongthe3eligibleindividualsbenefitfromthepublicservice; This household lives in area 1. In this area, the government spends a total of 14000 to

providethepublicserviceforthe7usersofthisarea(2+2+3).Theunitcostinarea1equals:14000/7=2000Theunitcostinarea2equals:12000/3=4000Bydefault, theareaindicator issetto1forallhouseholds.Whenthisdefault isused,thevariableRegionalpublicexpenditures(thefifthcolumnthatappearsinthedialogbox)shouldbesettototalpublic expenditures at the national level. This would occur when the information on publicexpendituresisonlyavailableatthenationallevel.Example2Observationi HH

sizeEligiblemembers

Frequency Areaindicator Regionalpublicexpenditures

1 7 3 2 1 280002 4 2 2 1 280003 5 5 3 1 280004 6 3 2 1 280005 4 2 1 1 28000

Theunitcostbenefit(atthenationallevel)equals:28000/10=2800

79


80

21 Disaggregatingthegroupeddata TheungroupDASPmodulegeneratesdisaggregateddatafromaggregatedistributiveinformation.Aggregate information isobtained fromcumulative incomeshares (orLorenzcurveordinates)atsomepercentiles.Forinstance:

Percentile(p) 0.10 0.30 0.50 0.60 0.90 1.00Lorenzvalues:L(p) 0.02 0.10 0.13 0.30 0.70 1.00

Theusermustspecifythetotalnumberofobservationstobegenerated.Theusercanalsoindicatethe number of observations to be generated specifically at the top and/or at the bottom of thedistribution,inwhichcasetheproportion(in%)ofthepopulationfoundatthetoporatthebottommustalsobespecified.Remarks:

If only the total number of observations is set, the generated data are self weighted (oruniformlydistributedoverpercentiles).

Ifanumberofobservationsissetforthebottomand/ortoptails,thegenerateddataarenotselfweightedandaweightvariableisprovidedinadditiontothegeneratedincomevariable. Example:Assumethatthetotalnumberofobservationstobegeneratedissetto1900,

but thatwewould like the bottom 10%of the population to be represented by 1000observations.Inthiscase,weightswillequal1/1000forthebottom1000observationsand1/100fortheremainingobservations(thesumofweightsbeingnormalizedtoone).

Thegeneratedincomevectortakesthenameof_yandthevectorweight,_w. The number of observations to be generated does not have to equal the number of

observations of the sample thatwas originally used to generate the aggregated data. Theungroup module cannot in itself serve to estimate the sampling errors that would haveoccurred had the original sample data been used to estimate poverty and/or inequalityestimates.

Theusercanselectanysamplesizethatexceeds(number_of_classes+1),butitmaybemoreappropriateforstatisticalbias‐reductionpurposestoselectrelativelylargesizes.

STAGEIGeneratinganinitialdistributionofincomesandpercentiles

S.1.1:GeneratingavectorofpercentilesStarting from information on the importance of bottom and top groups and on the number ofobservationstobegenerated,wefirstgenerateavectorofpercentiles.

81

Examples:Notations:NOBS:numberoftotalobservationsF:vectorofpercentilesB_NOBS:numberofobservationsforthebottomgroupT_NOBS:numberofobservationsforthetopgroup.

ForNOBS=1000spreadequallyacrossallpercentiles,F=0.001,0.002...0.999,1.Toavoidthe value F=1 for the last generated observation, we can simply replace F by F‐(0.5/NOBS).

For NOBS=2800, B_NOBS=1000 and T_NOBS=1000, with the bottom and top groupsbeingthefirstandlastdeciles:

a. F=0.0001,0.0002,...,0.0999,0.1000in0001/1000b. F=0.1010,0.1020,…,0.8990,0.9000in1001/1800c. F=0.9001,0.9002,...,0.9999,1.0000in1801/2800

AdjustmentscanalsobemadetoavoidthecaseofF(1)=1.Theweightvectorcaneasilybegenerated.S.1.2:GeneratinganinitialdistributionofincomesTheusermustindicatetheformofdistributionofthedesegregateddata.‐Normalandlognormaldistributions:Assumethat x followsalognormaldistributionwithmean andvariance 2 ,theLorenzcurveisdefinedasfollows:

2( ) ( )( )

Ln xL p

and

( )Ln xp

Weassumethat 1 andweestimatethevarianceusingtheproceduresuggestedbyShorrrocksandWan(2008):avalueforthestandarddeviationoflogincomes,σ,isobtainedbyaveragingthe

1m estimatesof 1 1 ( ) 1, , 1kk kp L p k m

wherem isthenumberofclassesandΦisthestandardnormaldistributionfunction(AitchisonandBrown1957;KolenikovandShorrocks2005,Appendix).‐GeneralizedQuadraticLorenzCurve:Itisassumedthat:

2(1 ) ( ) ( 1) ( )L L a p L bL p c p L

Wecanregress (1 )L L on 2( )p L , ( 1)L p and ( )p L withoutanintercept,droppingthelastobservationsincethechosenfunctionalformforcesthecurvetogothrough(1,1).

Wehave 0.52 22

( )2 4

mp n mp np ebQ p

2

1

4

2 4

e a b c

m b a

n be c

82

‐BetaLorenzCurve:Itisassumedthat:

log log( ) log( ) log(1 )p L p p

Afterestimatingtheparameters,wecangeneratequantilesasfollows

1

1Q p p p

p p

SeealsoDatt(1998).‐TheSingh‐MaddaladistributionThedistributionfunctionproposedbySinghandMaddala(1976)takesthefollowingform:

1( ) 1

1 ( / )a

q

F xx b

0, 0, 1/a b q a areparameters tobeestimated. The income ( x ) is assumed tobeequalorgreaterthanzero.Thedensityfunctionisdefinedasfollows:

( 1) 1( ) / 1 / /

qa af x aq b x b x b

Thequantileisdefinedasfollows:

1/1/( ) 1 1

aqQ p b p

We follow Jenkins’ (2008)approach for theestimationofparameters. For this,wemaximize thelikelihood function, which is simply the product of density functions evaluated at the averageincomeofclasses:http://stata‐press.com/journals/stbcontents/stb48.pdfSTAGEIIAdjustingtheinitialdistributiontomatchtheaggregateddata(optional).

ThisstageadjuststheinitialvectorofincomesusingtheShorrocksandWan(2008)procedure.Thisprocedureproceedswithtwosuccessiveadjustments:

Adjustment1:Correctingtheinitialincomevectortoensurethateachincomegrouphasitsoriginalmeanincome.

Adjustment2:Smoothingtheinter–classdistributions.

The generated sample is saved automatically in a new Stata data file (called by defaultungroup_data.dta;namesanddirectoriescanbechanged).TheusercanalsoplottheLorenzcurvesoftheaggregated(whenweassumethateachindividualhastheaverageincomeofhisgroup)andgenerateddata.Dialogboxoftheungroupmodule

83

Figure 16: ungroup dialog box

IllustrationwithBurkinaFasohouseholdsurveydataIn thisexample,weusedisaggregateddata togenerateaggregated information.Then,wecomparethedensitycurveoftruedatawiththoseofthegeneratedwiththedisaggregationofaggregateddata.genfw=size*weightgeny=exppc/r(mean)clorenzy,hs(size)lres(1)

Aggregatedinformation:pL(p).1.0233349.2.0576717.3.0991386.4.1480407.5.2051758.6.2729623.7.3565971.8.4657389.9.621357111.00000

0.5

11.

5

0 2 4 6Normalised per capita expenditures

True distribition Log NormalUniform Beta LCGeneralized Quadratic LC SINGH & MADALLA

(without adjustment)Density functions

0.5

11.

5

0 2 4 6Normalised per capita expenditures

True distribition Log NormalUniform Beta LCGeneralized Quadratic LC SINGH & MADALLA

(with adjustment)Density functions

84

22 Appendices

22.1 AppendixA:illustrativehouseholdsurveys

22.1.1 The1994BurkinaFasosurveyofhouseholdexpenditures(bkf94I.dta)Thisisanationallyrepresentativesurvey,withsampleselectionusingtwo‐stagestratifiedrandomsampling.Sevenstratawereformed.Fiveof thesestratawereruralandtwowereurban.Primarysamplingunitsweresampledfromalistdrawnfromthe1985census.Thelastsamplingunitswerehouseholds.Listofvariablesstrata Stratuminwhichahouseholdlives

psu Primarysamplingunit

weight Samplingweight

size Householdsize

exp Totalhouseholdexpenditures

expeq Totalhouseholdexpendituresperadultequivalent

expcp Totalhouseholdexpenditurespercapita

gse Socio‐economicgroupofthehouseholdhead

1wage‐earner(publicsector)2wage‐earner(privatesector)3Artisanortrader4Othertypeofearner5Cropfarmer6Subsistencefarmer7Inactive

sex Sexofhouseholdhead1Male2Female

zone Residentialarea1Rural2Urban

85

22.1.2 The1998BurkinaFasosurveyofhouseholdexpenditures(bkf98I.dta)Thissurveyissimilartothe1994one,althoughtenstratawereusedinsteadofsevenfor1994.Toexpress 1998 data in 1994 prices, two alternative procedures have been used. First, 1998expendituredataweremultipliedbytheratioof the1994officialpoverty linetothe1998officialpoverty line: z_1994/z_1998. Second, 1998 expenditure data weremultiplied by the ratio of the1994consumerpriceindextothe1998consumerpriceindex:ipc_1994/ipc_1998.Listofnewvariablesexpcpz Totalhouseholdexpenditurespercapitadeflatedby(z_1994/z_1998)

expcpi Totalexpenditurespercapitadeflatedby(ipc_1994/ipc_1998)

22.1.3 CanadianSurveyofConsumerFinance(asubsampleof1000observations–can6.dta)

ListofvariablesX Yearlygrossincomeperadultequivalent.

T Incometaxesperadultequivalent.

B1 Transfer1peradultequivalent.



B SumoftransfersB1,B2andB3

N Yearlynetincomeperadultequivalent(XminusT plusB)

22.1.4 PeruLSMSsurvey1994(Asampleof3623householdobservations‐PEREDE94I.dta)

Listofvariablesexppc

Totalexpenditures,percapita(constantJune1994solesperyear).

weight Samplingweight

86

size Householdsize

npubprim

Numberofhouseholdmembersinpublicprimaryschool

npubsec

Numberofhouseholdmembersinpublicsecondaryschool

npubuniv

Numberofhouseholdmembersinpublicpost‐secondaryschool

22.1.5 PeruLSMSsurvey1994(Asampleof3623householdobservations–PERU_A_I.dta)

Listofvariableshhid

HouseholdId.

exppc

Totalexpenditures,percapita(constantJune1994solesperyear).

size Householdsize

literate Numberofliteratehousehold members

pliterate literate/size

22.1.6 The1995ColombiaDHSsurvey(columbiaI.dta)ThissampleisapartoftheDatafromtheDemographicandHealthSurveys(Colombia_1995)witchcontainsthefollowinginformationforchildrenaged0‐59monthsListofvariableshid Householdidhaz height‐for‐agewaz weight‐for‐agewhz weight‐for‐heightsprob survivalprobabilitywght samplingweightAsset assetindex

22.1.7 The1996DominicanRepublicDHSsurvey(Dominican_republic1996I.dta)

ThissampleisapartoftheDatafromtheDemographicandHealthSurveys(RepublicDominican_1996)witchcontainsthefollowinginformationforchildrenaged0‐59months

87

Listofvariableshid Householdidhaz height‐for‐agewaz weight‐for‐agewhz weight‐for‐heightsprob survivalprobabilitywght samplingweightAsset assetindex

22.2 AppendixB:labellingvariablesandvalues Thefollowing.dofilecanbeusedtosetlabelsforthevariablesinbkf94.dta. Formoredetailsontheuseoflabelcommand,typehelplabelinthecommandwindow.

=================================lab_bkf94.do==================================#delim;/*Todropalllabelvalues*/labeldrop_all;/*Toassignlabels*/labelvarstrata"Stratuminwhichahouseholdlives";labelvarpsu"Primarysamplingunit";labelvarweight"Samplingweight";labelvarsize"Householdsize";labelvartotexp"Totalhouseholdexpenditures";labelvarexppc"Totalhouseholdexpenditurespercapita";labelvarexpeq"Totalhouseholdexpendituresperadultequivalent";labelvargse"Socio‐economicgroupofthehouseholdhead";/*Todefinethelabelvaluesthatwillbeassignedtothecategoricalvariablegse*/labeldefinelvgse1"wage‐earner(publicsector)"2"wage‐earner(privatesector)"3"Artisanortrader"4"Othertypeofearner"5"Cropfarmer"6"Subsistencefarmer"7"Inactive";/*Toassignthelabelvalues"lvgse"tothevariablegse*/labelvalgselvgse;labelvarsex"Sexofhouseholdhead";

88

labeldeflvsex1Male2Female;labelvalsexlvsex;labelvarzone"Residentialarea";labeldeflvzone1Rural2Urban;labelvalzonelvzone;====================================End======================================

22.3 AppendixC:settingthesamplingdesignTosetthesamplingdesignforthedatafilebkf94.dta,openthedialogboxforthecommandsvysetbytypingthesyntaxdbsvysetinthecommandwindow.IntheMainpanel,setSTRATAandSAMPLINGUNITSasfollows:Figure17:Surveydatasettings

IntheWeightspanel,setSAMPLINGWEIGHTVARIABLEasfollows:

89

Figure18:Settingsamplingweights

ClickonOKandsavethedatafile.Tocheckifthesamplingdesignhasbeenwellset,typethecommandsvydes.Thefollowingwillbedisplayed:

90

23 Examplesandexercises

23.1 EstimationofFGTpovertyindices“HowpoorwasBurkinaFasoin1994?”

1. Open the bkf94.dta file and label variables and values using the information of Section22.1.1.Typethedescribecommandandthenlabellisttolistlabels.

2. UsetheinformationofSection22.1.1.tosetthesamplingdesignandthensavethefile.3. Estimatetheheadcountindexusingvariablesofinterestexpccandexpeq.

a. You should set SIZE to household size in order to estimate poverty over thepopulationofindividuals.

b. Usetheso‐called1994officialpovertylineof41099FrancsCFAperyear.4. Estimate theheadcount indexusing the sameprocedureas aboveexcept that thepoverty

lineisnowsetto60%ofthemedian.5. Usingtheofficialpovertyline,howdoestheheadcountindexformale‐andfemale‐headed

householdscompare?6. Can you draw a 99% confidence interval around the previous comparison? Also, set the

numberofdecimalsto4.AnswerQ.1Ifbkf94.dtaissavedinthedirectoryc:/data,typethefollowingcommandtoopenit:use"C:\data\bkf94.dta",clearIflab_bkf94.doissavedinthedirectoryc:/do_files,typethefollowingcommandtolabelvariablesandlabels:do"C:\do_files\lab_bkf94.do"Typingthecommanddescribe,weobtain:obs: 8,625 vars: 9 31Oct200613:48size: 285,087(99.6%of memoryfree) storagedisplay valuevariable name type format label variablelabel weight float %9.0g Samplingweightsize byte %8.0g Householdsizestrata byte %8.0g Stratuminwhichahouseholdlivespsu byte %8.0g Primarysamplingunitgse byte %29.0g gse Socio‐economicgroupofthehouseholdheadsex byte %8.0g sex Sexofhouseholdheadzone byte %8.0g zone Residentialareaexp double %10.0g Totalhouseholdexpendituresexpeq double %10.0g Totalhouseholdexpendituresperadultequivalentexppc float %9.0g Totalhouseholdexpenditurespercapita Typinglabellist,wefind:zone: 1 Rural 2 Urban

91

sex: 1 Male 2 Femalegse: 1 wage‐earner(publicsector) 2 wage‐earner(privatesector) 3 Artisanortrader 4 Othertypeofearner 5 Cropfarmer 6 Foodfarmer 7 InactiveQ.2Youcansetthesamplingdesignwithadialogbox,asindicatedinSection22.3,orsimplybytypingsvysetpsu[pweight=weight],strata(strata)vce(linearized)Typingsvydes,weobtain

Q.3TypebdifgttoopenthedialogboxfortheFGTpovertyindexandchoosevariablesandparametersasindicatedinthefollowingwindow.ClickonSUBMIT.

92

Figure19:EstimatingFGTindices

Thefollowingresultsshouldthenbedisplayed:

Q.4SelectRELATIVEforthepovertylineandsettheotherparametersasabove.

93

Figure20:EstimatingFGTindiceswithrelativepovertylines

AfterclickingonSUBMIT,thefollowingresultsshouldbedisplayed:

Q.5Setthegroupvariabletosex.

94

Figure21:FGTindicesdifferentiatedbygender

ClickingonSUBMIT,thefollowingshouldappear:

Q.6UsingthepanelCONFIDENCEINTERVAL,settheconfidencelevelto99%andsetthenumberofdecimalsto4intheRESULTSpanel.

96

23.2 EstimatingdifferencesbetweenFGTindices.“HaspovertyBurkinaFasodecreasedbetween1994and1998?”

1. OpenthedialogboxforthedifferencebetweenFGTindices.2. Estimatethedifferencebetweenheadcountindiceswhen

a. Distribution1isyear1998anddistribution2isyear1994;b. Thevariableofinterestisexppcfor1994andexppczfor1998.c. You should set size to household size in order to estimate poverty over the

populationofindividuals.d. Use41099FrancsCFAperyearasthepovertylineforbothdistributions.

3. Estimatethedifferencebetweenheadcountindiceswhena. Distribution1isruralresidentsinyear1998anddistribution2isruralresidentsin

year1994;b. Thevariableofinterestisexppcfor1994andexppczfor1998.c. You should set size to household size in order to estimate poverty over the

populationofindividuals.d. Use41099FrancsCFAperyearasthepovertylineforbothdistributions.

4. Redothelastexerciseforurbanresidents.5. Redothelastexerciseonlyformembersofmale‐headedhouseholds.6. Testiftheestimateddifferenceinthelastexerciseissignificantlydifferentfromzero.Thus,

test:

0 1: ( 41099, 0) 0 : ( 41099, 0) 0H P z against H P z

Set the significance level to 5% and assume that the test statistics follows a normaldistribution.

AnswersQ.1OpenthedialogboxbytypingdbdifgtQ.2 Fordistribution1,choosetheoptionDATAINFILEinsteadofDATAINMEMORYandclickon

BROWSEtospecifythelocationofthefilebkf98I.dta. Followthesameprocedurefordistribution2tospecifythelocationofbkf94I.dta. Choosevariablesandparametersasfollows:

97

Figure22:EstimatingdifferencesbetweenFGTindices


98

Q.3 Restricttheestimationtoruralresidentsasfollows:

o SelecttheoptionCondition(s)o WriteZONEinthefieldnexttoCONDITION(1)andtype1inthenextfield.

Figure23:EstimatingdifferencesinFGTindices

AfterclickingonSUBMIT,weshouldsee:

Q.4

99

Onecanseethatthechangeinpovertywassignificantonlyforurbanresidents.Q.5Restricttheestimationtomale‐headedurbanresidentsasfollows:

o SetthenumberofCondition(s)to2;o SetsexinthefieldnexttoCondition(2)andtype1inthenextfield.

Figure24:FGTdifferencesacrossyearsbygenderandzone


Q.6

100

Wehavethat:LowerBound:=0.0222UpperBound:=0.1105Thenullhypothesisisrejectedsincethelowerboundofthe95%confidenceintervalisabovezero.

23.3 Estimatingmultidimensionalpovertyindices“Howmuchisbi‐dimensionalpoverty(totalexpendituresandliteracy)inPeruin1994?”Usingtheperu94I.dtafile,

1. EstimatetheChakravartyetal(1998)indexwithparameteralpha=1and

Var.ofinterest Pov.line a_jDimension1 exppc 400 1Dimension2 pliterate 0.90 1

2. EstimatetheBourguignonandChakravarty(2003)indexwithparameters alpha=beta=gamma=1and

Var.ofinterest Pov.lineDimension1 exppc 400Dimension2 literate 0.90

Q.1Steps: Type

use"C:\data\ peru94_A_I.dta",clear Toopentherelevantdialogbox,type

dbimdp_bci

Choosevariablesandparametersasin

101

Figure25:Estimatingmultidimensionalpovertyindices(A)

AfterclickingSUBMIT,thefollowingresultsappear.

Q.2 Toopentherelevantdialogbox,type

dbimdp_cmrSteps: Choosevariablesandparametersasin

102

Figure26:Estimatingmultidimensionalpovertyindices(B)

AfterclickingSUBMIT,thefollowingresultsappear.

103

23.4 EstimatingFGTcurves.“Howsensitivetothechoiceofapovertylineistherural‐urbandifferenceinpoverty?”

1. Openbkf94I.dta2. OpentheFGTcurvesdialogbox.3. DrawFGTcurvesforvariablesofinterestexppcandexpeqwith

a. parameter 0 ;b. povertylinebetween0and100,000FrancCFA;c. sizevariablesettosize;d. subtitleofthefiguresetto“Burkina1994”.

4. DrawFGTcurvesforurbanandruralresidentswitha. variableofinterestsettoexpcap;b. parameter 0 ;c. povertylinebetween0and100,000FrancCFA;d. sizevariablesettosize.

5. Drawthedifferencebetweenthesetwocurvesanda. save the graph in *.gph format to be plotted in Stata and in *.wmf format to be

insertedinaWorddocument.b. Listthecoordinatesofthegraph.

6. Redothelastgraphwith 1 .AnswersQ.1Openthefilewithuse"C:\data\bkf94I.dta",clearQ.2OpenthedialogboxbytypingdbdifgtQ.3Choosevariablesandparametersasfollows:

104

Figure27:DrawingFGTcurves

Tochangethesubtitle,selecttheTitlepanelandwritethesubtitle.Figure28:EditingFGTcurves

AfterclickingSUBMIT,thefollowinggraphappears:

105

Figure29:GraphofFGTcurves

106

Q.4Choosevariablesandparametersasinthefollowingwindow:Figure30:FGTcurvesbyzone

AfterclickingSUBMIT,thefollowinggraphappears:

107

Figure31:GraphofFGTcurvesbyzone

108

Q.5 ChoosetheoptionDIFFERENCEandselect:WITHTHEFIRSTCURVE; Indicatethatthegroupvariableiszone; SelecttheResultspanelandchoosetheoptionLISTintheCOORDINATESquadrant. IntheGRAPHquadrant,selectthedirectoryinwhichtosavethegraphingphformatandto

exportthegraphinwmfformat.

Figure32:DifferencesofFGTcurves

109

Figure33:Listingcoordinates

110

AfterclickingSUBMIT,thefollowingappears:

Figure34:DifferencesbetweenFGTcurves

Q.6

111

Figure35:DifferencesbetweenFGTcurves

23.5 EstimatingFGTcurvesanddifferencesbetweenFGTcurveswithconfidenceintervals

“Isthepovertyincreasebetween1994and1998inBurkinaFasostatisticallysignificant?”1) Using the filebkf94I.dta, draw the FGT curve and its confidence interval for the variable of

interestexppcwith:a) parameter 0 ;b) povertylinebetween0and100,000FrancCFA;c) sizevariablesettosize.

2) Using simultaneously the files bkf94I.dta and bkf98I.dta, draw the difference between FGTcurvesandassociatedconfidenceintervalswith:a) Thevariableofinterestexppcfor1994andexppczfor1998.b) parameter 0 ;c) povertylinebetween0and100,000FrancCFA;d) sizevariablesettosize.

3) Redo2)withparameter 1 .AnswersQ.1

112

Steps: Type

use"C:\data\bkf94I.dta",clear Toopentherelevantdialogbox,type

dbcfgts


Figure36:DrawingFGTcurveswithconfidenceinterval


113

Figure37:FGTcurveswithconfidenceinterval

0.2

.4.6

.8

0 20000 40000 60000 80000 100000Poverty line (z)

Confidence interval (95 %) Estimate

Burkina FasoFGT curve (alpha = 0)

Q.2Steps: Toopentherelevantdialogbox,type

dbcfgtsd2 Choosevariablesandparametersasin

114

Figure38:DrawingthedifferencebetweenFGTcurveswithconfidenceinterval

Figure39:DifferencebetweenFGTcurveswithconfidenceinterval ( 0)

-.1

-.0

50

.05

0 20000 40000 60000 80000 100000Poverty line (z)

Confidence interval (95 %) Estimated difference

(alpha = 0)Difference between FGT curves

115

Figure40:DifferencebetweenFGTcurveswithconfidenceinterval ( 1)

-.0

4-.

02

0.0

2

0 20000 40000 60000 80000 100000Poverty line (z)

Confidence interval (95 %) Estimated difference

(alpha = 1)Difference between FGT curves

23.6 Testingpovertydominanceandestimatingcriticalvalues.“HasthepovertyincreaseinBurkinaFasobetween1994and1998beenstatisticallysignificant?”1) Usingsimultaneouslyfilesbkf94I.dtaandbkf98I.dta,checkforsecond‐orderpovertydominance

andestimatethevaluesofthepovertylineatwhichthetwoFGTcurvescross.a) Thevariableofinterestisexppcfor1994andexppczfor1998;b) Thepovertylineshouldvarybetween0and100,000FrancCFA;c) Thesizevariableshouldbesettosize.

AnswersQ.1Steps: Toopentherelevantdialogbox,type

dbdompov Choosevariablesandparametersasin

116

Figure41:Testingforpovertydominance


23.7 DecomposingFGTindices.“WhatisthecontributionofdifferenttypesofearnerstototalpovertyinBurkinaFaso?”

1. Openbkf94I.dtaanddecomposetheaveragepovertygapa. withvariableofinterestexppc;b. withsizevariablesettosize;c. attheofficialpovertylineof41099FrancsCFA;d. andusingthegroupvariablegse(Socio‐economicgroups).

2. Dotheaboveexercisewithoutstandarderrorsandwiththenumberofdecimalssetto4.

117

AnswersQ.1Steps: Type

use"C:\data\bkf94I.dta",clear Toopentherelevantdialogbox,type

dbdfgtg


Figure42:DecomposingFGTindicesbygroups

AfterclickingSUBMIT,thefollowinginformationisprovided:

118

Q.2UsingtheRESULTSpanel,changethenumberofdecimalsandunselecttheoptionDISPLAYSTANDARDERRORS.

AfterclickingSUBMIT,thefollowinginformationisobtained:

119

23.8 EstimatingLorenzandconcentrationcurves.“HowmuchdotaxesandtransfersaffectinequalityinCanada?”Byusingthecan6.dtafile,

1. Draw the Lorenz curves for gross income X and net income N. How can you see theredistributionofincome?

2. Draw Lorenz curves for gross income X and concentration curves for each of the threetransfersB1,B2 andB3 and the taxT. Whatcanyousayabout theprogressivityof theseelementsofthetaxandtransfersystem?

“WhatistheextentofinequalityamongBurkinaFasoruralandurbanhouseholdsin1994?”Byusingthebkf94I.dtafile,

3. DrawLorenzcurvesforruralandurbanhouseholds

a. withvariableofinterestexppc;b. withsizevariablesettosize;c. andusingthegroupvariablezone(asresidentialarea).

Q.1Steps: Type

use"C:\data\can6.dta",clear Toopentherelevantdialogbox,type

dbclorenz


120

Figure43:Lorenzandconcentrationcurves


121

Figure44:Lorenzcurves

Q.2Steps:


122

Figure45:Drawingconcentrationcurves

AfterclickingonSUBMIT,thefollowingappears:

123

Figure46:Lorenzandconcentrationcurves

Q.3Steps: Type

use"C:\data\bkf94I.dta",clear


124

Figure47:DrawingLorenzcurves

Figure48:Lorenzcurves

125

23.9 EstimatingGiniandconcentrationcurves“ByhowmuchdotaxesandtransfersaffectinequalityinCanada?”Usingthecan6.dtafile,

1. EstimatetheGiniindicesforgrossincomeXandnetincomeN.2. Estimate the concentration indices for variables T andN when the ranking variable is

grossincomeX.“ByhowmuchhasinequalitychangedinBurkinaFasobetween1994and1998?”Usingthebkf94I.dtafile,

3. EstimatethedifferenceinBurkinaFaso’sGiniindexbetween1998and1994

a. withvariableofinterestexpeqzfor1998andexpeqfor1994;b. withsizevariablesettosize.

Q.1Steps: Type


dbigini


126

Figure49:EstimatingGiniandconcentrationindices

AfterclickingSUBMIT,thefollowingresultsareobtained:

Q.2Steps:


127

Figure50:Estimatingconcentrationindices

AfterclickingSUBMIT,thefollowingresultsareobtained:


dbdigini


128

Figure51:EstimatingdifferencesinGiniandconcentrationindices

AfterclickingSUBMIT,thefollowinginformationisobtained:

129

23.10 Usingbasicdistributivetools“WhatdoesthedistributionofgrossandnetincomeslooklikeinCanada?”Usingthecan6.dtafile,

1. DrawthedensityforgrossincomeXandnetincomeN.‐ Therangeforthexaxisshouldbe[0,60000].

2. DrawthequantilecurvesforgrossincomeXandnetincomeN.‐ Therangeofpercentilesshouldbe[0,0.8]

3. Drawtheexpectedtax/benefitaccordingtogrossincomeX.‐ Therangeforthexaxisshouldbe[0,60000]‐ Usealocallinearestimationapproach.

4. EstimatemarginalratesfortaxesandbenefitsaccordingtogrossincomeX.‐ Therangeforthexaxisshouldbe[0,60000]‐ Usealocallinearestimationapproach.

Q.1Steps: Type


dbcdensity Choosevariablesandparametersasin

Figure52:Drawingdensities

130


Figure53:Densitycurves

0.0

0001

.000

02.0

0003

.000

04.0

0005

f(y)

0 12000 24000 36000 48000 60000y

X N

Density Curves


dbc_quantile Choosevariablesandparametersasin

131

Figure54:Drawingquantilecurves

AfterclickingSUBMIT,thefollowingappears:Figure55:Quantilecurves

01

0000

200

003

0000

Q(p

)

0 .2 .4 .6 .8Percentiles (p)

X N

Quantile Curves

132


dbcnpe Choosevariablesandparametersasin

Figure56:Drawingnon‐parametricregressioncurves


133

Figure57:Non‐parametricregressioncurves

05

000

100

001

5000

200

00E

(Y|X

)

0 12000 24000 36000 48000 60000X values

t b

(Linear Locally Estimation Approach | Bandwidth = 3699.26 )

Non parametric regression

Q.4Steps: Choosevariablesandparametersasin

134

Figure58:Drawingderivativesofnon‐parametricregressioncurves


Figure59:Derivativesofnon‐parametricregressioncurves

-1-.

50

.51

dE

[Y|X

]/dX

0 12000 24000 36000 48000 60000X values

t b

(Linear Locally Estimation Approach | Bandwidth = 3699.26 )

Non parametric derivative regression

135

23.11 Plottingthejointdensityandjointdistributionfunction“WhatdoesthejointdistributionofgrossandnetincomeslooklikeinCanada?”Usingthecan6.dtafile,

4. EstimatethejointdensityfunctionforgrossincomeXandnetincomeN.o Xrange:[0,60000]o Nrange:[0,60000]

5. EstimatethejointdistributionfunctionforgrossincomeXandnetincomeN.o Xrange:[0,60000]o Nrange:[0,60000]

Q.1Steps: Type


dbsjdensity


Figure60:Plottingjointdensityfunction

AfterclickingSUBMIT,thefollowinggraphisplottedinteractivelywithGnuPlot4.2:

136

0

10000

20000

30000

40000

50000

60000

0 10000

20000 30000

40000 50000

60000

0 5e-010 1e-009

1.5e-009 2e-009

2.5e-009 3e-009

Joint Density Function

f(x,y)

Dimension 1

Dimension 2


dbsjdistrub


137

Figure61:Plottingjointdistributionfunction


0 10000

20000 30000

40000 50000

60000

0 10000

20000 30000

40000 50000

60000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Joint Distribution Function

F(x,y)

Dimension 1Dimension 2

138

23.12 Testingthebi‐dimensionalpovertydominanceUsing the columbia95I.dta (distribution_1) and the dominican_republic95I.dta (distribution_2)files,

1. Draw the difference between the bi‐dimensional multiplicative FGT surfaces and theconfidenceintervalofthatdifferencewhen

Var.ofinterest Range alpha_jDimension1 haz:height‐for‐age ‐3.0/6.0 0Dimension2 sprob:survival

probability0.7/1.0 0

2. Testforbi‐dimensionalpovertyusingtheinformationabove.

Answer:Q.1Steps: Toopentherelevantdialogbox,type

dbdombdpov


139

Figure62:Testingforbi‐dimensionalpovertydominance


-3-2

-1 0

1 2

3 4

5 6

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5

Bi-dimensional poverty dominance

DifferenceLower-boundedUpper-bounded

Dimension 1

Dimension 2

Q.2

140

To make a simple test of multidimensional dominance, one should check if the lower‐boundedconfidenceintervalsurfaceisalwaysabovezeroforallcombinationsofrelevantpovertylines–orconversely.

o Forthis,clickonthepanel“Confidenceinterval”andselecttheoptionlower‐bounded.o ClickagainonthebuttonSubmit.


-3-2

-1 0

1 2

3 4

5 6 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

-0.35-0.3

-0.25-0.2

-0.15-0.1

-0.05 0

0.05

Bi-dimensional poverty dominance

Lower-bounded

Dimension 1

Dimension 2

141

23.13 Testingforpro‐poornessofgrowthinMexicoThe three sub‐samples used in these exercises are sub‐samples of 2000 observations drawnrandomlyfromthethreeENIGHMexicanhouseholdsurveysfor1992,1998and2004.Eachofthesethreesub‐samplescontainsthefollowingvariables:strata Thestratumpsu Theprimarysamplingunitweight Samplingweightinc Incomehhsz Householdsize

1. Usingthefilesmex_92_2mI.dtaandmex_98_2mI.dta,testforfirst‐orderrelativepro‐poornessofgrowthwhen:

Theprimalapproachisused. Therangeofpovertylinesis[0,3000].

2. Repeatwiththedualapproach.3. Byusingthefilesmex_98_2mI.dtaandmex_04_2mI.dta,testforabsolutesecond‐orderpro‐

poornesswiththedualapproach.

4. Usingmex_98_2mI.dtaandmex_04_2mI.dta,estimatethepro‐poorindicesofmodule

ipropoor. Parameteralphasetto1. Povertylineequalto600.

Answer:Q.1Steps: Toopentherelevantdialogbox,type

dbcpropoorp

142

Choosevariablesandparametersasin(selecttheupper‐boundedoptionfortheconfidenceinterval):

Figure63:Testingthepro‐poorgrowth(primalapproach)

AfterclickingSUBMIT,thefollowinggraphappears

143

-.15

-.1

-.05

0.0

5

0 600 1200 1800 2400 3000Poverty line (z)

Difference Upper bound of 95% confidence intervalNull horizontal line

(Order : s=1 | Dif. = P_2( (m2/m1)z, a=s-1) - P_1(z,a=s-1))

Relative propoor curve


dbcpropoord Choosevariablesandparametersas in(with the lower‐boundedoption for theconfidence

interval):Figure64:Testingthepro‐poorgrowth(dualapproach)‐A

144


-.4

-.2

0.2

.4

0 .184 .368 .552 .736 .92Percentiles (p)

Difference Lower bound of 95% confidence intervalNull horizontal line

(Order : s=1 | Dif. = Q_2(p) /Q_1(p) - mu_2/mu_1 )

Absolute propoor curves

Q.2Steps:

145

Toopentherelevantdialogbox,typedbcpropoord Choosevariablesandparametersas in(with the lower‐boundedoption for theconfidence

interval):Figure65:Testingthepro‐poorgrowth(dualapproach)–B


146

02

46

0 .184 .368 .552 .736 .92Percentiles (p)

Difference Lower bound of 95% confidence intervalNull horizontal line

(Order : s=2 | Dif. = (GL_2(p) - GL_1(p) ) / GL_2(p) )

Absolute propoor curves


dbipropoor Choosevariablesandparametersas.

147


23.14 BenefitincidenceanalysisofpublicspendingoneducationinPeru(1994).

1. Usingtheperedu94I.dtafile,estimateparticipationandcoverageratesoftwotypesofpublicspendingoneducationwhen:

‐ Thestandardoflivingisexppc‐ Thenumberofhouseholdmembers thatbenefit fromeducation is fr_prim for the

primarysectorandfr_secforthesecondaryone.‐ Thenumberof eligiblehouseholdmembers isel_prim for theprimary sector and

el_secforthesecondaryone.‐ Socialgroupsarequintiles.

148

Answer:Typedbbianinthewindowscommandandsetvariablesandoptionsasfollows:Figure66:Benefitincidenceanalysis

AfterclickingonSubmit,thefollowingappears:

149

2. To estimate total public expenditures on education by sector at the national level, the

followingmacroinformationwasused:‐ Pre‐primary and primary public education expenditure (as% of all levels), 1995:

35.2%‐ Secondarypubliceducationexpenditure(as%ofalllevels),1995:21.2%‐ Tertiarypubliceducationexpenditure(as%ofalllevels),1995:16%‐ Publiceducationexpenditure(as%ofGNP),1995=3%‐ GDPpercapita:about3800.

Usingthisinformation,thefollowingvariablesaregenerated

capdrop_var1;gen_var1=size*weight*3800;quisum_var1;quigenpri_pub_exp=0.03*0.352*`r(sum)';quigensec_pub_exp=0.03*0.212*`r(sum)';quigenuni_pub_exp=0.03*0.160*`r(sum)';capdrop_var1;

‐ Totalpublicexpendituresonprimarysector:pri_pub_exp‐ Totalpublicexpendituresonsecondarysector:sec_sec_exp‐ Totalpublicexpendituresonuniversitysector:uni_pub_exp

Estimatetheaveragebenefitsperquintileandgeneratethebenefitvariables.

Answer:Setvariablesandoptionsasfollows:

150

Figure67:BenefitIncidenceAnalysis(unitcostapproach)

AfterclickingonSubmit,thefollowingappears:

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