Distributive Analysis Stata Package V2.3

USERMANUAL

DASPversion2.3

DASP:DistributiveAnalysisStataPackageBy

AbdelkrimAraar,Jean‐YvesDuclos

UniversitéLavalPEP,CIRPÉEandWorldBank

June2013

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TableofcontentsTableofcontents............................................................................................................................. 2 ListofFigures .................................................................................................................................. 6 1 Introduction ............................................................................................................................ 8 2 DASPandStataversions ......................................................................................................... 8 3 InstallingandupdatingtheDASPpackage ........................................................................... 9 3.1 InstallingDASPmodules ................................................................................................ 9 3.2 AddingtheDASPsubmenutoStata’smainmenu ...................................................... 10

4 DASPanddatafiles ............................................................................................................... 10 5 Mainvariablesfordistributiveanalysis ............................................................................. 11 6 HowcanDASPcommandsbeinvoked? .............................................................................. 11 7 HowcanhelpbeaccessedforagivenDASPmodule? ....................................................... 12 8 ApplicationsandfilesinDASP ............................................................................................. 12 9 BasicNotation ....................................................................................................................... 14 10 DASPandpovertyindices ................................................................................................ 14 10.1 FGTandEDE‐FGTpovertyindices(ifgt). .................................................................... 14 10.2 DifferencebetweenFGTindices(difgt) ...................................................................... 15 10.3 Wattspovertyindex(iwatts). ...................................................................................... 16 10.4 DifferencebetweenWattsindices(diwatts) .............................................................. 16 10.5 Sen‐Shorrocks‐Thonpovertyindex(isst). ................................................................. 17 10.6 DifferencebetweenSen‐Shorrocks‐Thonindices(disst) ......................................... 17 10.7 DASPandmultidimensionalpovertyindices ............................................................. 17 10.8 Multipleoverlappingdeprivationanalysis(MODA)indices ..................................... 19

11 DASP,povertyandtargetingpolicies ............................................................................... 20 11.1 Povertyandtargetingbypopulationgroups ............................................................. 20 11.2 Povertyandtargetingbyincomecomponents ......................................................... 21

12 Marginalpovertyimpactsandpovertyelasticities ........................................................ 22 12.1 FGTelasticity’swithrespecttotheaverageincomegrowth(efgtgr). ..................... 22 12.2 FGTelasticitieswithrespecttoaverageincomegrowthwithdifferentapproaches(efgtgro). .................................................................................................................................... 23 12.3 FGTelasticitywithrespecttoGiniinequality(efgtineq). .......................................... 23 12.4 FGTelasticitywithrespecttoGini‐inequalitywithdifferentapproaches(efgtine). 24 12.5 FGTelasticitieswithrespecttowithin/betweengroupcomponentsofinequality(efgtg). ........................................................................................................................................ 25 12.6 FGTelasticitieswithrespecttowithin/betweenincomecomponentsofinequality(efgtc). ........................................................................................................................................ 26

13 DASPandinequalityindices ............................................................................................ 28 13.1 Giniandconcentrationindices(igini) ........................................................................ 28 13.2 DifferencebetweenGini/concentrationindices(digini) .......................................... 28 13.3 Generalisedentropyindex(ientropy) ........................................................................ 29 13.4 Differencebetweengeneralizedentropyindices(diengtropy) ................................ 29 13.5 Atkinsonindex(iatkinson) ......................................................................................... 30 13.6 DifferencebetweenAtkinsonindices(diatkinson) .................................................. 30

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13.7 Coefficientofvariationindex(icvar) .......................................................................... 31 13.8 Differencebetweencoefficientofvariation(dicvar) ................................................. 31 13.9 Quantile/shareratioindicesofinequality(inineq) .................................................. 31 13.10 DifferencebetweenQuantile/Shareindices(dinineq) .......................................... 32 13.11 TheAraar(2009)multidimensionalinequalityindex ........................................... 32

14 DASPandpolarizationindices ......................................................................................... 32 14.1 TheDERindex(ipolder) .............................................................................................. 32 14.2 DifferencebetweenDERpolarizationindices(dipolder) ......................................... 33 14.3 TheFosterandWolfson(1992)polarizationindex(ipolfw) ................................... 34 14.4 DifferencebetweenFosterandWolfson(1992)polarizationindices(dipolfw) .... 34 14.5 TheGeneralisedEsteban,GradinandRay(1999)polarisationindex(ipoger) ...... 34 14.6 TheInaki(2008)polarisationindex(ipoger) ............................................................ 36

15 DASPanddecompositions ................................................................................................ 39 15.1 FGTPoverty:decompositionbypopulationsubgroups(dfgtg) ............................... 39 15.2 FGTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue(dfgts) 40 15.3 AlkireandFoster(2011)MDindexofpoverty:decompositionbypopulationsubgroups(dmdafg) ................................................................................................................. 42 15.4 AlkireandFoster(2011:decompositionbydimensionsusingtheShapleyvalue(dmdafs) .................................................................................................................................... 42 15.5 FGTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue(dfgts) 43 15.6 DecompositionofthevariationinFGTindicesintogrowthandredistributioncomponents(dfgtgr)................................................................................................................. 45 15.7 DecompositionofchangeinFGTpovertybypovertyandpopulationgroupcomponents–sectoraldecomposition‐(dfgtg2d). ................................................................. 46 15.8 DecompositionofFGTpovertybytransientandchronicpovertycomponents(dtcpov) ..................................................................................................................................... 49 15.9 Inequality:decompositionbyincomesources(diginis) ........................................... 51 15.10 Regression‐baseddecompositionofinequalitybyincomesources ..................... 52 15.11 Giniindex:decompositionbypopulationsubgroups(diginig). ........................... 58 15.12 Generalizedentropyindicesofinequality:decompositionbypopulationsubgroups(dentropyg). ........................................................................................................... 59 15.13 Polarization:decompositionoftheDERindexbypopulationgroups(dpolag) .. 59 15.14 Polarization:decompositionoftheDERindexbyincomesources(dpolas) ....... 60

16 DASPandcurves................................................................................................................ 60 16.1 FGTCURVES(cfgt). ....................................................................................................... 60 16.2 FGTCURVEwithconfidenceinterval(cfgts). ............................................................. 62 16.3 DifferencebetweenFGTCURVESwithconfidenceinterval(cfgts2d). .................... 62 16.4 LorenzandconcentrationCURVES(clorenz). ............................................................ 62 16.5 Lorenz/concentrationcurveswithconfidenceintervals(clorenzs). ....................... 63 16.6 DifferencesbetweenLorenz/concentrationcurveswithconfidenceinterval(clorenzs2d) .............................................................................................................................. 64 16.7 Povertycurves(cpoverty) ........................................................................................... 64 16.8 Consumptiondominancecurves(cdomc) .................................................................. 65 16.9 Difference/Ratiobetweenconsumptiondominancecurves(cdomc2d) ................. 66

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16.10 DASPandtheprogressivitycurves ......................................................................... 66 16.10.1 Checkingtheprogressivityoftaxesortransfers .............................................. 66 16.10.2 Checkingtheprogressivityoftransfervstax ................................................... 67

17 Dominance ......................................................................................................................... 67 17.1 Povertydominance(dompov) ..................................................................................... 67 17.2 Inequalitydominance(domineq) ................................................................................ 68 17.3 DASPandbi‐dimensionalpovertydominance(dombdpov) .................................... 68

18 Distributivetools .............................................................................................................. 69 18.1 Quantilecurves(c_quantile) ........................................................................................ 69 18.2 Incomeshareandcumulativeincomesharebygroupquantiles(quinsh) ............. 69 18.3 Densitycurves(cdensity) ............................................................................................ 69 18.4 Non‐parametricregressioncurves(cnpe) ................................................................. 71 18.4.1 Nadaraya‐Watsonapproach ................................................................................ 71 18.4.2 Locallinearapproach ........................................................................................... 71

18.5 DASPandjointdensityfunctions. ............................................................................... 71 18.6 DASPandjointdistributionfunctions ........................................................................ 72

19 DASPandpro‐poorgrowth .............................................................................................. 72 19.1 DASPandpro‐poorindices .......................................................................................... 72 19.2 DASPandpro‐poorcurves ........................................................................................... 73 19.2.1 Primalpro‐poorcurves ........................................................................................ 73 19.2.2 Dualpro‐poorcurves ........................................................................................... 74

20 DASPandBenefitIncidenceAnalysis .............................................................................. 75 20.1 Benefitincidenceanalysis ............................................................................................ 75

21 Disaggregatinggroupeddata ........................................................................................... 79 22 Appendices ........................................................................................................................ 83 22.1 AppendixA:illustrativehouseholdsurveys ............................................................... 83 22.1.1 The1994BurkinaFasosurveyofhouseholdexpenditures(bkf94I.dta) ........ 83 22.1.2 The1998BurkinaFasosurveyofhouseholdexpenditures(bkf98I.dta) ........ 84 22.1.3 CanadianSurveyofConsumerFinance(asubsampleof1000observations–can6.dta) ................................................................................................................................ 84 22.1.4 PeruLSMSsurvey1994(Asampleof3623householdobservations‐PEREDE94I.dta) .................................................................................................................... 84 22.1.5 PeruLSMSsurvey1994(Asampleof3623householdobservations–PERU_A_I.dta) ........................................................................................................................ 85 22.1.6 The1995ColombiaDHSsurvey(columbiaI.dta) ............................................... 85 22.1.7 The1996DominicanRepublicDHSsurvey(Dominican_republic1996I.dta) . 85

22.2 AppendixB:labellingvariablesandvalues ................................................................ 86 22.3 AppendixC:settingthesamplingdesign .................................................................... 87

23 Examplesandexercises ................................................................................................... 89 23.1 EstimationofFGTpovertyindices .............................................................................. 89 23.2 EstimatingdifferencesbetweenFGTindices. ............................................................ 95 23.3 Estimatingmultidimensionalpovertyindices ........................................................... 99 23.4 EstimatingFGTcurves. ............................................................................................... 102 23.5 EstimatingFGTcurvesanddifferencesbetweenFGTcurveswithconfidenceintervals ................................................................................................................................... 110 23.6 Testingpovertydominanceandestimatingcriticalvalues. .................................... 114

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23.7 DecomposingFGTindices. ......................................................................................... 115 23.8 EstimatingLorenzandconcentrationcurves. .......................................................... 118 23.9 EstimatingGiniandconcentrationcurves ............................................................... 124 23.10 Usingbasicdistributivetools ................................................................................ 128 23.11 Plottingthejointdensityandjointdistributionfunction ................................... 134 23.12 Testingthebi‐dimensionalpovertydominance .................................................. 137 23.13 Testingforpro‐poornessofgrowthinMexico ..................................................... 140 23.14 BenefitincidenceanalysisofpublicspendingoneducationinPeru(1994). .... 146

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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..........................................................................................................................82Figure17:Surveydatasettings........................................................................................................................87Figure18:Settingsamplingweights..............................................................................................................88Figure19:EstimatingFGTindices...................................................................................................................91Figure20:EstimatingFGTindiceswithrelativepovertylines...........................................................92Figure21:FGTindicesdifferentiatedbygender......................................................................................93Figure22:EstimatingdifferencesbetweenFGTindices.......................................................................96Figure23:EstimatingdifferencesinFGTindices.....................................................................................97Figure24:FGTdifferencesacrossyearsbygenderandzone.............................................................98Figure25:Estimatingmultidimensionalpovertyindices(A)..........................................................100Figure26:Estimatingmultidimensionalpovertyindices(B)..........................................................101Figure27:DrawingFGTcurves.....................................................................................................................103Figure28:EditingFGTcurves........................................................................................................................103Figure29:GraphofFGTcurves.....................................................................................................................104Figure30:FGTcurvesbyzone.......................................................................................................................105Figure31:GraphofFGTcurvesbyzone....................................................................................................106Figure32:DifferencesofFGTcurves..........................................................................................................107Figure33:Listingcoordinates.......................................................................................................................108Figure34:DifferencesbetweenFGTcurves............................................................................................109Figure35:DifferencesbetweenFGTcurves............................................................................................110Figure36:DrawingFGTcurveswithconfidenceinterval.................................................................111Figure37:FGTcurveswithconfidenceinterval....................................................................................112Figure38:DrawingthedifferencebetweenFGTcurveswithconfidenceinterval.................113Figure39:DifferencebetweenFGTcurveswithconfidenceinterval ( 0) ...........................113Figure40:DifferencebetweenFGTcurveswithconfidenceinterval ( 1) ...........................114Figure41:Testingforpovertydominance...............................................................................................115Figure42:DecomposingFGTindicesbygroups....................................................................................116Figure43:Lorenzandconcentrationcurves...........................................................................................119

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Figure44:Lorenzcurves..................................................................................................................................120Figure45:Drawingconcentrationcurves................................................................................................121Figure46:Lorenzandconcentrationcurves...........................................................................................122Figure47:DrawingLorenzcurves...............................................................................................................123Figure48:Lorenzcurves..................................................................................................................................123Figure49:EstimatingGiniandconcentrationindices.........................................................................125Figure50:Estimatingconcentrationindices...........................................................................................126Figure51:EstimatingdifferencesinGiniandconcentrationindices...........................................127Figure52:Drawingdensities..........................................................................................................................128Figure53:Densitycurves.................................................................................................................................129Figure54:Drawingquantilecurves............................................................................................................130Figure55:Quantilecurves...............................................................................................................................130Figure56:Drawingnon‐parametricregressioncurves......................................................................131Figure57:Non‐parametricregressioncurves........................................................................................132Figure58:Drawingderivativesofnon‐parametricregressioncurves........................................133Figure59:Derivativesofnon‐parametricregressioncurves...........................................................133Figure60:Plottingjointdensityfunction.................................................................................................134Figure61:Plottingjointdistributionfunction........................................................................................136Figure62:Testingforbi‐dimensionalpovertydominance...............................................................138Figure63:Testingthepro‐poorgrowth(primalapproach).............................................................141Figure64:Testingthepro‐poorgrowth(dualapproach)‐A............................................................142Figure65:Testingthepro‐poorgrowth(dualapproach)–B..........................................................144Figure66:Benefitincidenceanalysis..........................................................................................................147Figure67:BenefitIncidenceAnalysis(unitcostapproach).............................................................149

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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,forcereplacenetinstalldasp_p4.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

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Analternativeistousedialogboxes.Forthis,thecommanddbshouldbetypedandfollowedbythenameoftherelevantDASPmodule.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.

InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.1

10.2 DifferencebetweenFGTindices(difgt)ThismoduleestimatesdifferencesbetweentheFGTindicesoftwodistributions.Foreachofthetwodistributions: Thereexistthreewaysoffixingthepovertyline:

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

Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Alevelfortheparameter canbechosenforeachofthetwodistributions.

InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.2.

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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(2011)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.

Themodulespresentedabovecanbeusedtoestimatethemultidimensionalpovertyindicesaswellastheirstandarderrors.

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

group. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both

thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith3decimals;thiscanbealsochanged.

UsersareencouragedtoconsidertheexercisesthatappearinSection 23.3

10.8 Multipleoverlappingdeprivationanalysis(MODA)indices TheimodaDASPmoduleproducesaseriesofmultidimensionalpovertyindicesinordertoshow the incidence of deprivation in each dimension. Further, this application estimates theincidenceofmulti‐deprivationinthedifferentcombinationsofdimensions.Inthisapplication,thenumber of dimensions is set to three. Further, themultidimensional poverty ismeasured by theheadcount(unionandintersectionheadcount indices)andtheAlkireandFoster(2011)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 FGTelasticitieswithrespecttoaverageincomegrowthwithdifferentapproaches(efgtgro).

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

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

The module efgtgroallowstheestimationofapovertyelasticityorsemi‐elasticitywithrespecttogrowthwiththedifferentapproachesmentionedabove.Formoredetailsontheseapproaches,see:

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

ToestimateaFGTelasticity–semi‐elasticity‐withrespecttoaverageincomegrowthina

grouporinanentirepopulation; Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa

groupvariableisselected,onlythefirstvariableofinterestisthenused. Theresultsaredisplayedwith6decimals;thiscanbechanged.

12.3 FGTelasticitywithrespecttoGiniinequality(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 for group k , and ( )F z is the headcount. ( )C k is the concentration

coefficient of group k when incomes of the complement group are replaced 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.

ToestimateaFGTelasticitywithrespecttoaverageincomegrowthinagrouporinan

entirepopulation; Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan




12.4 FGTelasticitywithrespecttoGini‐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 efgtineallowstheestimationofapovertyelasticityorsemi‐elasticitywithrespecttoinequalitywiththedifferentapproachesmentionedabove.Formoredetailsontheseapproaches,see:

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

25

ToestimateaFGTelasticity–semi‐elasticity‐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 FGTelasticitieswithrespecttowithin/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 weight attributed 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 the 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,GradinandRay(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

group j. Theparameter 1,1.6 reflectssensitivityof society topolarisation. The first

stepfortheestimationrequirestodefineexhaustiveandmutuallyexclusivegroups, .Thiswillinvolvesomedegreeoferror.Takingintoaccountthisidea,themeasureofpolarisationproposedbyEstebanetal.(1999)isobtainedaftercorrectingthe ERP ( ) indexappliedtothe simplified representation of the original distributionwith ameasure of the groupingerror. Nonetheless,whendealingwithpersonalorspatial incomedistributions, therearenounanimouscriteria forestablishing theprecisedemarcationbetweendifferentgroups.Toaddressthisproblem,Estebanetal.(1999)followthemethodologyproposedbyAghevliandMehran(1981)andDaviesandShorrocks(1989)inordertofindanoptimalpartitionofthedistributionforagivennumberofgroups, * .Thismeansselectingthepartitionthat

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

Esteban et al., 1999). Themeasureof polarisationproposedbyEsteban et al. (1999) isthereforegivenby:

m m

EGR * 1 *j k j k

j 1 k 1

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

where, 0 isaparameterthatinformsabouttheweightassignedtotheerrorterm.(InthestudyofEstebanetal.(1999),thevalueusedis 1 ).The Statamodule ipoegr.ado estimates the generalised form of the Esteban et al. (1999)polarisationindex.Inadditiontotheusualvariables,thisroutineoffersthethreefollowingoptions:

1. Thenumberofgroups.Empiricalstudiesuse twoor threegroups.Theusercanselect thenumberofgroups.Accordingtothisnumber,theprogramsearchesfortheoptimalincomeinterval for each group and displays them. It also displays the error in percentage, ie:

*G f G*100

G f

;

2. Theparameter ;3. Theparameter .

To respect the scale invariance principle, all incomes are divided by average income i.e.

j j / .Inaddition,wedividetheindexbythescalar2tomakeitsintervalliebetween

0and1when 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)

Letapopulationbesplitinto N groups,eachoneofsize 0in .Thedensityfunction,themeanand

thepopulationshareofgroup i aredenotedby ( )if x , i and i respectively. istheoverallmean.

We therefore have that ( ) 1 if x ,1

N

i ii

and1

1

N

ii

. Using Inaki (2008), a social

polarisationindexcanbedefinedas:

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.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.7

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 a 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(2011)MDindexofpoverty:decompositionbypopulationsubgroups(dmdafg)

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

Anasymptoticstandarderrorisprovidedforeachofthesestatistics.

15.4 AlkireandFoster(2011:decompositionbydimensionsusingtheShapleyvalue(dmdafs)

Thedmdafsmoduledecomposes theAlkire andFoster (2011)multidimensional poverty

indices intoa sumof the contributionsgeneratedbyeachof thepovertydimensions. Ituses theShapley characteristic function. The non‐presence of a given factor –dimension‐ is obtained bysettingthelevelofthatdimensiontoitsspecificpovertyline,thusensuringthenon‐contributionofthis dimension to the AF (2011) indices. Note that the dmdafs ado file requires the moduleshapar.ado,which isprogrammed toperformdecompositionsusing theShapleyvaluealgorithmdevelopedbyAraarandDuclos(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 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 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

EmpiricalillustrationwithaNigerianhouseholdsurveyWe 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).

Additivepovertymeasures, like theFGT indices, canbeexpressedasa sumof the

poverty contributionsof thevarious subgroupsofpopulation.Each subgroupcontributesbyitspopulationshareandpovertylevel.Thus,thechangeinpovertyacrosstimedependsonthechangeinthesetwocomponents.Denotingthepopulationshareofgroup inperiod by , the change in poverty between two periods can be expressed as (see Huppi(1991)andDuclosandAraar(2006)):

(06)

Thisdecompositionuse the initialperiodas theone. If the referenceperiod is the

final,thedecompositiontakestheform:

47

(06)

Toremovethearbitrarnessinselectingthereferenceperiod,wecanusetheShapley

decompositionapproach,finding:

(07)

where is the average population share and

.TheDASPmoduledfgtg2dperformsthissectoraldecomposition,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


References JalanJyotsna,andMartinRavallion.(1998)"TransientPovertyinPostreformRuralChina"JournalofComparativeEconomics,26(2),pp.338:57.Jean‐Yves Duclos & Abdelkrim Araar & John Giles, 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 intoa sumof thecontributionsgeneratedbyseparateincomecomponents.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 income that equals the sum of thecomponentsofthesubsetplustheaverageofthecomplementsubset. Notethatthedsineqsadofile requires themoduleshapar.ado,which isprogrammed toperformdecompositionsusing theShapleyvaluealgorithmdevelopedbyAraarandDuclos(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 Regression‐baseddecompositionofinequalitybyincomesourcesA useful approach to show the contribution of income covariates to total inequality is bydecomposingthelatterbythepredictedcontributionsofcovariates.Formally,denotetotalincome

by y andthesetofcovariatesby 1, 2 , , KX x x x .Usingalinearmodelspecification,wehave:

0 1 1 2 2ˆ ˆ ˆ ˆ ˆ ˆ

k k K Ky x x x x

where 0 and denoterespectivelytheestimatedconstanttermandtheresidual.Twoapproachesforthedecompositionoftotalinequalitybyincomesourcesareused:

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

2‐ TheAnalyticalapproach:Thisapproach isbasedonalgebraicdevelopments that expresstotalinequalityasasumofinequalitycontributionsofincomesources.

WiththeShapleyapproach: Theusercanselectamongthefollowingrelativeinequalityindices;

Giniindex Atkinsonindex Generalizedentropyindex Coefficientvariationindex

Theusercanselectamongthefollowingmodelspecifications;

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

53

WiththeAnalyticalapproach: Theusercanselectamongthefollowingrelativeinequalityindices;

Giniindex Squaredcoefficientvariationindex

Themodelspecificationislinear.

Decomposingtotalinequalitywiththeanalyticalapproach:

Total income equals 0 1 2 K Ry s s s s s where 0s is the estimated constant, ˆk k ks X

and Rs is the estimated residual. As reported byWang 2004, relative inequality indices are not

definedwhen the average of the variable of interest equals zero (the case of the residual). Also,inequalityindicesequalzerowhenthevariableof interestisaconstant(thecaseoftheestimatedconstant).Todealwiththesetwoproblems,Wang(2004)proposesthefollowingbasicrules:

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:UsingRao1969’sapproach,therelativeGiniindexcanbedecomposedasfollows:

( ) kk

y

I y C

where y istheaverageof y and kC isthecoefficientofconcentrationof ks when y istherankingvariable.TheSquaredcoefficientofvariationindex:AsshownbyShorrocks1982,thesquaredcoefficientofvariationindexcanbedecomposedas:

21

ov( , )( )

Kk

k y

C y sI y

Shapleydecompositions:TheShapleyapproachisbuiltaroundtheexpectedmarginalcontributionofacomponent.Theusercanselectamongtwomethodstodefinetheimpactofmissingagivencomponent.

Withoption:method(mean),whenacomponentismissingfromagivensetofcomponents,itisreplacedbyitsmean.

Withoption:method(zero),whenacomponentismissingfromagivensetofcomponents,itisreplacedbyzero.

Asindicatedabove,wecannotestimaterelativeinequalityfortheresidualcomponent.

54

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

thecontributionoftheresidualis ˆ( ) ( )rcs I y I y .

FortheSemi‐loglinearmodel,theShapleydecompositionisappliedtoallcomponentsincludingtheconstantandtheresidual.

WiththeShapleyapproach,theusercanusetheloglinearspecification.However,theusermustindicatetheincomevariableandnotthelogofthatvariable(DASPautomaticallyrunstheregressionwithlog(y)asthedependentvariable).Example1

57

Example2

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

Wecannotestimatetheincomeshare(nolinearform);

58

Thecontributionoftheconstantisnil. 01

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

k Ek

y E s E s E s

.Addinga

constantwillhavenotanyimpact.

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.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.5.

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 .InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.10.

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.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.10.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.

InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.11???

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.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.13.

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.InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.13

20 DASPandBenefitIncidenceAnalysis

20.1 Benefitincidenceanalysis

Themainobjectiveofbenefitincidenceistoanalysethedistributionofbenefitsfromtheuseofpublicservicesaccordingtothedistributionoflivingstandards.

Two main sources of information are used. The first informs on access of household

memberstopublicservices.Thisinformationcanbefoundinusualhouseholdsurveys.Theseconddeals with the amount of total public expenditures on each public service. This information isusuallyavailableatthenationallevelandsometimesinamoredisaggregatedformat,suchasattheregionallevel.Thebenefitincidenceapproachcombinestheuseofthesetwosourcesofinformationtoanalysethedistributionofpublicbenefitsanditsprogressivity.

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

Publicexpendituresonagivenserviceoftenvaryfromonegeographicaloradministrativeareatoanother. When information about public expenditures is available at the level of areas, thisinformationcanbeusedwiththebianmoduletoestimateunitcostmoreaccurately.Example1Observationi HH

sizeEligibleHHmembers

Frequency Areaindicator Totallevel ofregionalpublicexpenditures

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=2800InterestedusersareencouragedtoconsidertheexercisesthatappearinSection 23.14

79

21 Disaggregatinggroupeddata 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.

80

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:GeneratinganinitialdistributionofincomesTheusermustindicatetheformofdistributionofthedisaggregateddata.‐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

81

2

1

4

2 4

e a b c

m b a

n be c

‐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

where 0, 0, 1/a b q a are parameters to be estimated. The income ( x ) is assumed to beequaltoorgreaterthanzero.Thedensityfunctionisdefinedasfollows:

( 1) 1( ) / 1 / /

qa af x aq b x b x b

Quantilesaredefinedasfollows:

1/1/( ) 1 1

aqQ p b p

WefollowJenkins(2008)’sapproachfortheestimationofparameters. Forthis,wemaximizethelikelihood function, which is simply the product of density functions evaluated at the averageincomeofeachclass: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

82

Figure 16: ungroup dialog box

IllustrationwithBurkinaFasohouseholdsurveydataIn thisexample,weusedisaggregateddata togenerateaggregated information.Then,wecompare the density curve of the true data with those of the data generated throughdisaggregationofthepreviouslyaggregateddata.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

83

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

84

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

85

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 Numberofliteratehouseholdmembers

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

86

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";

87

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

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

IntheWeightspanel,setSAMPLINGWEIGHTVARIABLEasfollows:

88

Figure18:Settingsamplingweights

ClickonOKandsavethedatafile.Tocheckifthesamplingdesignhasbeenwellset,typethecommandsvydes.Thefollowingwillbedisplayed:

89

23 Examplesandexercises

23.1 EstimationofFGTpovertyindices“HowpoorwasBurkinaFasoin1994?”

1. Open the bkf94.dta file and label variables and values using the information of Section 22.1.1.Typethedescribecommandandthenlabellisttolistlabels.

2. UsetheinformationofSection 22.1.1.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

90

sex: 1 Male 2 Femalegse: 1 wage‐earner(publicsector) 2 wage‐earner(privatesector) 3 Artisanortrader 4 Othertypeofearner 5 Cropfarmer 6 Foodfarmer 7 InactiveQ.2Youcansetthesamplingdesignwithadialogbox,asindicatedinSection 22.3,orsimplybytypingsvysetpsu[pweight=weight],strata(strata)vce(linearized)Typingsvydes,weobtain

Q.3TypebdifgttoopenthedialogboxfortheFGTpovertyindexandchoosevariablesandparametersasindicatedinthefollowingwindow.ClickonSUBMIT.

91

Figure19:EstimatingFGTindices

Thefollowingresultsshouldthenbedisplayed:

Q.4SelectRELATIVEforthepovertylineandsettheotherparametersasabove.

92

Figure20:EstimatingFGTindiceswithrelativepovertylines

AfterclickingonSUBMIT,thefollowingresultsshouldbedisplayed:

Q.5Setthegroupvariabletosex.

93

Figure21:FGTindicesdifferentiatedbygender

ClickingonSUBMIT,thefollowingshouldappear:

Q.6UsingthepanelCONFIDENCEINTERVAL,settheconfidencelevelto99%andsetthenumberofdecimalsto4intheRESULTSpanel.

95

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:

96

Figure22:EstimatingdifferencesbetweenFGTindices


97

Q.3 Restricttheestimationtoruralresidentsasfollows:

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

Figure23:EstimatingdifferencesinFGTindices

AfterclickingonSUBMIT,weshouldsee:

Q.4

98

Onecanseethatthechangeinpovertywassignificantonlyforurbanresidents.Q.5Restricttheestimationtomale‐headedurbanresidentsasfollows:

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

Figure24:FGTdifferencesacrossyearsbygenderandzone


Q.6

99

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

100

Figure25:Estimatingmultidimensionalpovertyindices(A)

AfterclickingSUBMIT,thefollowingresultsappear.

Q.2 Toopentherelevantdialogbox,type

dbimdp_cmrSteps: Choosevariablesandparametersasin

101

Figure26:Estimatingmultidimensionalpovertyindices(B)

AfterclickingSUBMIT,thefollowingresultsappear.

102

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:

103

Figure27:DrawingFGTcurves

Tochangethesubtitle,selecttheTitlepanelandwritethesubtitle.Figure28:EditingFGTcurves

AfterclickingSUBMIT,thefollowinggraphappears:

104

Figure29:GraphofFGTcurves

105

Q.4Choosevariablesandparametersasinthefollowingwindow:Figure30:FGTcurvesbyzone

AfterclickingSUBMIT,thefollowinggraphappears:

106

Figure31:GraphofFGTcurvesbyzone

107

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

exportthegraphinwmfformat.

Figure32:DifferencesofFGTcurves

108

Figure33:Listingcoordinates

109

AfterclickingSUBMIT,thefollowingappears:

Figure34:DifferencesbetweenFGTcurves

Q.6

110

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

111

Steps: Type

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

dbcfgts


Figure36:DrawingFGTcurveswithconfidenceinterval


112

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

113

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

114

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

115

Figure41:Testingforpovertydominance


23.7 DecomposingFGTindices.“WhatisthecontributionofdifferenttypesofearnerstototalpovertyinBurkinaFaso?”

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

2. Dotheaboveexercisewithoutstandarderrorsandwiththenumberofdecimalssetto4.

116

AnswersQ.1Steps: Type

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

dbdfgtg


Figure42:DecomposingFGTindicesbygroups

AfterclickingSUBMIT,thefollowinginformationisprovided:

117

Q.2UsingtheRESULTSpanel,changethenumberofdecimalsandunselecttheoptionDISPLAYSTANDARDERRORS.

AfterclickingSUBMIT,thefollowinginformationisobtained:

118

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


119

Figure43:Lorenzandconcentrationcurves


120

Figure44:Lorenzcurves

Q.2Steps:


121

Figure45:Drawingconcentrationcurves

AfterclickingonSUBMIT,thefollowingappears:

122

Figure46:Lorenzandconcentrationcurves

Q.3Steps: Type

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


123

Figure47:DrawingLorenzcurves

Figure48:Lorenzcurves

124

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


125

Figure49:EstimatingGiniandconcentrationindices

AfterclickingSUBMIT,thefollowingresultsareobtained:

Q.2Steps:


126

Figure50:Estimatingconcentrationindices

AfterclickingSUBMIT,thefollowingresultsareobtained:


dbdigini


127

Figure51:EstimatingdifferencesinGiniandconcentrationindices

AfterclickingSUBMIT,thefollowinginformationisobtained:

128

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

129


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

130

Figure54:Drawingquantilecurves

AfterclickingSUBMIT,thefollowingappears:Figure55:Quantilecurves

01

0000

200

003

0000

Q(p

)

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

X N

Quantile Curves

131


dbcnpe Choosevariablesandparametersasin

Figure56:Drawingnon‐parametricregressioncurves


132

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

133

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

134

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:

135

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


136

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

137

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


138

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

139

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

140

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

141

Choosevariablesandparametersasin(selecttheupper‐boundedoptionfortheconfidenceinterval):

Figure63:Testingthepro‐poorgrowth(primalapproach)

AfterclickingSUBMIT,thefollowinggraphappears

142

-.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

143


-.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:

144

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

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


145

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.

146


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.

147

Answer:Typedbbianinthewindowscommandandsetvariablesandoptionsasfollows:Figure66:Benefitincidenceanalysis

AfterclickingonSubmit,thefollowingappears:

148

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:

149

Figure67:BenefitIncidenceAnalysis(unitcostapproach)

AfterclickingonSubmit,thefollowingappears:

Distributive Analysis Stata Package V2.3

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