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USERMANUAL DASPversion2.3 DASP:DistributiveAnalysisStataPackage By AbdelkrimAraar, Jean‐YvesDuclos UniversitéLaval PEP,CIRPÉEandWorldBank June2013 Tableofcontents Tableofcontents............................................................................................................................. 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 2 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:decompositionbypopulation subgroups(dmdafg) ................................................................................................................. 42 15.4 AlkireandFoster(2011:decompositionbydimensionsusingtheShapleyvalue (dmdafs) .................................................................................................................................... 42 15.5 FGTPoverty:decompositionbyincomecomponentsusingtheShapleyvalue (dfgts) 43 15.6 DecompositionofthevariationinFGTindicesintogrowthandredistribution components(dfgtgr)................................................................................................................. 45 15.7 DecompositionofchangeinFGTpovertybypovertyandpopulationgroup components–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:decompositionbypopulation subgroups(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 3 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 EstimatingFGTcurvesanddifferencesbetweenFGTcurveswithconfidence intervals ................................................................................................................................... 110 23.6 Testingpovertydominanceandestimatingcriticalvalues..................................... 114 4 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 5 ListofFigures Figure1:Ouputofnetdescribedasp..................................................................................................................9 Figure2:DASPsubmenu......................................................................................................................................10 Figure3:UsingDASPwithacommandwindow.......................................................................................11 Figure4:AccessinghelponDASP....................................................................................................................12 Figure5:EstimatingFGTpovertywithonedistribution......................................................................13 Figure6:EstimatingFGTpovertywithtwodistributions....................................................................13 Figure7:Povertyandthetargetingbypopulationgroups..................................................................21 Figure8:DecompositionoftheFGTindexbygroups.............................................................................39 Figure9:DecompositionofFGTbyincomecomponents......................................................................44 Figure10:SectoraldecompositionofFGT..................................................................................................48 Figure11:Decompositionofpovertyintotransientandchroniccomponents...........................50 Figure12:DecompositionoftheGiniindexbyincomesources(Shapleyapproach)..............52 Figure13:FGTcurves...........................................................................................................................................61 Figure14:Lorenzandconcentrationcurves..............................................................................................63 Figure15:Consumptiondominancecurves...............................................................................................66 Figure16:ungroupdialogbox..........................................................................................................................82 Figure17:Surveydatasettings........................................................................................................................87 Figure18:Settingsamplingweights..............................................................................................................88 Figure19:EstimatingFGTindices...................................................................................................................91 Figure20:EstimatingFGTindiceswithrelativepovertylines...........................................................92 Figure21:FGTindicesdifferentiatedbygender......................................................................................93 Figure22:EstimatingdifferencesbetweenFGTindices.......................................................................96 Figure23:EstimatingdifferencesinFGTindices.....................................................................................97 Figure24:FGTdifferencesacrossyearsbygenderandzone.............................................................98 Figure25:Estimatingmultidimensionalpovertyindices(A)..........................................................100 Figure26:Estimatingmultidimensionalpovertyindices(B)..........................................................101 Figure27:DrawingFGTcurves.....................................................................................................................103 Figure28:EditingFGTcurves........................................................................................................................103 Figure29:GraphofFGTcurves.....................................................................................................................104 Figure30:FGTcurvesbyzone.......................................................................................................................105 Figure31:GraphofFGTcurvesbyzone....................................................................................................106 Figure32:DifferencesofFGTcurves..........................................................................................................107 Figure33:Listingcoordinates.......................................................................................................................108 Figure34:DifferencesbetweenFGTcurves............................................................................................109 Figure35:DifferencesbetweenFGTcurves............................................................................................110 Figure36:DrawingFGTcurveswithconfidenceinterval.................................................................111 Figure37:FGTcurveswithconfidenceinterval....................................................................................112 Figure38:DrawingthedifferencebetweenFGTcurveswithconfidenceinterval.................113 Figure39:DifferencebetweenFGTcurveswithconfidenceinterval ( 0) ...........................113 Figure40:DifferencebetweenFGTcurveswithconfidenceinterval ( 1) ...........................114 Figure41:Testingforpovertydominance...............................................................................................115 Figure42:DecomposingFGTindicesbygroups....................................................................................116 Figure43:Lorenzandconcentrationcurves...........................................................................................119 6 Figure44:Lorenzcurves..................................................................................................................................120 Figure45:Drawingconcentrationcurves................................................................................................121 Figure46:Lorenzandconcentrationcurves...........................................................................................122 Figure47:DrawingLorenzcurves...............................................................................................................123 Figure48:Lorenzcurves..................................................................................................................................123 Figure49:EstimatingGiniandconcentrationindices.........................................................................125 Figure50:Estimatingconcentrationindices...........................................................................................126 Figure51:EstimatingdifferencesinGiniandconcentrationindices...........................................127 Figure52:Drawingdensities..........................................................................................................................128 Figure53:Densitycurves.................................................................................................................................129 Figure54:Drawingquantilecurves............................................................................................................130 Figure55:Quantilecurves...............................................................................................................................130 Figure56:Drawingnon‐parametricregressioncurves......................................................................131 Figure57:Non‐parametricregressioncurves........................................................................................132 Figure58:Drawingderivativesofnon‐parametricregressioncurves........................................133 Figure59:Derivativesofnon‐parametricregressioncurves...........................................................133 Figure60:Plottingjointdensityfunction.................................................................................................134 Figure61:Plottingjointdistributionfunction........................................................................................136 Figure62:Testingforbi‐dimensionalpovertydominance...............................................................138 Figure63:Testingthepro‐poorgrowth(primalapproach).............................................................141 Figure64:Testingthepro‐poorgrowth(dualapproach)‐A............................................................142 Figure65:Testingthepro‐poorgrowth(dualapproach)–B..........................................................144 Figure66:Benefitincidenceanalysis..........................................................................................................147 Figure67:BenefitIncidenceAnalysis(unitcostapproach).............................................................149 7 1 Introduction TheStatasoftwarehasbecomeaverypopulartooltotransformandprocessdata.Itcomeswitha largenumberofbasicdatamanagementmodulesthatarehighlyefficientfortransformationoflarge datasets.TheflexibilityofStataalsoenablesprogrammerstoprovidespecialized“.ado”routinesto addtothepowerofthesoftware.ThisisindeedhowDASPinteractswithStata.DASP,whichstands forDistributiveAnalysisStataPackage,ismainlydesignedtoassistresearchersandpolicyanalysts interestedinconductingdistributiveanalysiswithStata.Inparticular,DASPisbuiltto: Estimatethemostpopularstatistics(indices,curves)usedfortheanalysisofpoverty, inequality,socialwelfare,andequity; Estimatethedifferencesinsuchstatistics; Estimatestandarderrorsandconfidenceintervalsbytakingfullaccountofsurvey design; Supportdistributiveanalysisonmorethanonedatabase; Performthemostpopularpovertyanddecompositionprocedures; Checkfortheethicalrobustnessofdistributivecomparisons; Unifysyntaxandparameteruseacrossvariousestimationproceduresfordistributive analysis. ForeachDASPmodule,threetypesoffilesareprovided: *.ado: Thisfilecontainstheprogramofthemodule *.hlp: Thisfilecontainshelpmaterialforthegivenmodule *.dlg: Thisfileallowstheusertoperformtheestimationusingthe module’sdialogbox The *.dlg files in particular makes the DASP package very user friendly and easy to learn. When these dialog boxes are used, the associated program syntax is also generated and showed in the reviewwindow.Theusercansavethecontentsofthiswindowina*.dofiletobesubsequentlyused inanothersession. 2 DASPandStataversions DASPrequires o Stataversion10.0orhigher o adofilesmustbeupdated Toupdatetheexecutablefile(from10.0to10.2)andtheadofiles,see: http://www.stata.com/support/updates/ 8 3 InstallingandupdatingtheDASPpackage Ingeneral,the*.adofilesaresavedinthefollowingmaindirectories: Priority Directory Sources 1 UPDATES: OfficialupdatesofStata *.adofiles 2 BASE: *.adofilesthatcomewiththeinstalledStata software 3 SITE: *.adofilesdownloadedfromthenet 4 PLUS: .. 5 PERSONAL: Personal*.adofiles 3.1 InstallingDASPmodules a. Unzipthefiledasp.zipinthedirectoryc: b. Makesurethatyouhavec:/dasp/dasp.pkgorc:/dasp/stata.toc c. IntheStatacommandwindows,typethesyntax netfromc:/dasp Figure1:Ouputofnetdescribedasp d. Typethesyntax netinstalldasp_p1.pkg,forcereplace netinstalldasp_p2.pkg,forcereplace netinstalldasp_p3.pkg,forcereplace netinstalldasp_p4.pkg,forcereplace 9 3.2 AddingtheDASPsubmenutoStata’smainmenu WithStata9,submenuscanbeaddedtothemenuitemUser. Figure2:DASPsubmenu ToaddtheDASPsubmenus,thefileprofile.do(whichisprovidedwiththeDASPpackage)mustbe copiedintothePERSONALdirectory.Ifthefileprofile.doalreadyexists,addthecontentsofthe DASP–providedprofile.dofileintothatexistingfileandsaveit.Tocheckifthefileprofile.do alreadyexists,typethecommand:findfileprofile.do. 4 DASPanddatafiles DASP makes it possible to use simultaneously more than one data file. The user should, however, “initialize”eachdatafilebeforeusingitwithDASP.Thisinitializationisdoneby: 1. Labelingvariablesandvaluesforcategoricalvariables; 2. Initializingthesamplingdesignwiththecommandsvyset; 3. Savingtheinitializeddatafile. UsersarerecommendedtoconsultappendicesA,BandC, 10 5 Mainvariablesfordistributiveanalysis VARIABLEOFINTEREST.Thisisthevariablethatusuallycaptureslivingstandards.Itcanrepresent,for instance, income per capita, expenditures per adult equivalent, calorie intake, normalized height‐ for‐agescoresforchildren,orhouseholdwealth. SIZEVARIABLE.Thisreferstothe"ethical"orphysicalsizeoftheobservation.Forthecomputationof many statistics, we will indeed wish to take into account how many relevant individuals (or statisticalunits)arefoundinagivenobservation. GROUPVARIABLE.(ThisshouldbeusedincombinationwithGROUPNUMBER.)Itisoftenusefultofocus one’s analysis on some population subgroup. We might, for example, wish to estimate poverty withinacountry’sruralareaorwithinfemale‐headedfamilies.OnewaytodothisistoforceDASP to focus on a population subgroup defined as those for whom some GROUP VARIABLE (say, area of residence)equalsagivenGROUPNUMBER(say2,forruralarea). SAMPLINGWEIGHT.Samplingweightsaretheinverseofthesamplingprobability.Thisvariableshould besetupontheinitializationofthedataset. 6 HowcanDASPcommandsbeinvoked? Statacommandscanbeentereddirectlyintoacommandwindow: Figure3:UsingDASPwithacommandwindow 11 Analternativeistousedialogboxes.Forthis,thecommanddbshouldbetypedandfollowedbythe nameoftherelevantDASPmodule. Example: dbifgt 7 HowcanhelpbeaccessedforagivenDASPmodule? TypethecommandhelpfollowedbythenameoftherelevantDASPmodule. Example: helpifgt Figure4:AccessinghelponDASP 8 ApplicationsandfilesinDASP TwomaintypesofapplicationsareprovidedinDASP.Forthefirstone,theestimationprocedures requireonlyonedatafile.Insuchcases,thedatafileinmemoryistheonethatisused(or“loaded”); itisfromthatfilethattherelevantvariablesmustbespecifiedbytheusertoperformtherequired estimation. 12 Figure5:EstimatingFGTpovertywithonedistribution Forthesecondtypeofapplications,twodistributionsareneeded.Foreachofthesetwo distributions,theusercanspecifythecurrently‐loadeddatafile(theoneinmemory)oronesaved ondisk. Figure6:EstimatingFGTpovertywithtwodistributions 13 Notes: 1. DASPconsiderstwodistributionstobestatisticallydependent(forstatisticalinference purposes)ifthesamedatasetisused(thesameloadeddataordatawiththesamepathand filename)forthetwodistributions. 2. IftheoptionDATAINFILEischosen,thekeyboardmustbeusedtotypethenameofthe requiredvariables. 9 BasicNotation ThefollowingtablepresentsthebasicnotationusedinDASP’susermanual. Symbol Indication y variableofinterest i observationnumber yi valueofthevariableofinterestforobservationi hw samplingweight hwi samplingweightforobservationi hs sizevariable hsi sizeofobservationi(forexamplethesizeofhouseholdi) wi hwi*hsi hg groupvariable hgi groupofobservationi. wik swik=swiifhgi=k,and0otherwise. n samplesize Forexample,themeanofyisestimatedbyDASPas ̂ : n ˆ w y i 1 n i i w i i 1 10 DASPandpovertyindices 10.1 FGTandEDE‐FGTpovertyindices(ifgt). Thenon‐normalisedFoster‐Greer‐ThorbeckeorFGTindexisestimatedas n wi ( z yi ) z; ) i 1 P( n wi i 1 wherezisthepovertylineand x max( x, 0) .TheusualnormalisedFGTindexisestimatedas ( z; ) /( z ) P ( z; ) P 14 TheEDE‐FGTindexisestimatedas: ( P( z; )) P ( z; ) 1/ EDE for > 0 Thereexistthreewaysoffixingthepovertyline: 1‐Settingadeterministicpovertyline; 2‐Settingthepovertylinetoaproportionofthemean; 3‐SettingthepovertylinetoaproportionofaquantileQ(p). Theusercanchoosethevalueofparameter . Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepovertybyusingsimultaneouslypercapitaconsumptionandpercapitaincome. Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.1 10.2 DifferencebetweenFGTindices(difgt) ThismoduleestimatesdifferencesbetweentheFGTindicesoftwodistributions. Foreachofthetwodistributions: Thereexistthreewaysoffixingthepovertyline: 1‐Settingadeterministicpovertyline; 2‐Settingthepovertylinetoaproportionofthemean; 3‐SettingthepovertylinetoaproportionofaquantileQ(p) Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Alevelfortheparameter canbechosenforeachofthetwodistributions. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.2. 15 10.3 Wattspovertyindex(iwatts). TheWattspovertyindexisestimatedas q wi (ln( z / yi ) z ) i 1 P( n wi i 1 wherezisthepovertylineand q thenumberofpoor. Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepovertybyusingsimultaneouslypercapitaconsumptionandpercapitaincome. Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 10.4 DifferencebetweenWattsindices(diwatts) ThismoduleestimatesdifferencesbetweentheWattsindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Alevelfortheparameter canbechosenforeachofthetwodistributions. 16 10.5 Sen‐Shorrocks‐Thonpovertyindex(isst). TheSen‐Shorroks‐Thonpovertyindexisestimatedas: z ) HP* ( z, )[ 1 G* ] P( g wherezisthepovertyline H istheheadcount, P* ( z, ) thepovertygapestimatedatthelevelof poorgroupand G*g theGiniindexofpovertygaps ( z y ) / z . Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepovertybyusingsimultaneouslypercapitaconsumptionandpercapitaincome. Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 10.6 DifferencebetweenSen‐Shorrocks‐Thonindices(disst) ThismoduleestimatesdifferencesbetweentheWattsindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Alevelfortheparameter canbechosenforeachofthetwodistributions. 10.7 DASPandmultidimensionalpovertyindices Thegeneralformofanadditivemultidimensionalpovertyindexis: 17 n P( X , Z ) wi p( X i , Z ) i 1 n wi i 1 where p ( X i , Z ) isindividualI’spovertyfunction(withvectorofattributes X i xi ,1 ,..., xi , J and vectorofpovertylines Z z1 ,..., z J ),determiningI’scontributiontototalpoverty P ( X , Z ) . [1]Chakravartyetal(1998)index(imdp_cmr) z j xi , j p( X i , Z ) a j zj j 1 J [2]ExtendedWattsindex(imdp_ewi) J zj p( X i , Z ) a j ln min( z j ; xi , j ) j 1 [3]MultiplicativeextendedFGTindex(imdp_mfi) z j xi , j p( X i , Z ) zj j 1 [4]Tsui(2002)index(imdp_twu) J j b j zj p( X i , Z ) 1 j 1 min( z j ; xi , j ) J [5]Intersectionheadcountindex(imdp_ihi) J p( X i , Z ) I z j xi , j j 1 [6]Unionheadcountindex(imdp_uni) J p( X i , Z ) 1 I z j xi , j j 1 [7]BourguignonandChakravartybi‐dimensional(2003)index(imdp_bci) 18 p ( X i , Z ) C1 C2 where: z x z x C1 1 i ,1 and C2 2 i ,2 z1 z2 [8]AlkireandFoster(2011)index(imdp_afi) 1 p( , X i , Z ) N z j xi , j 1 J w J j z j i j 1 N I (di d c ) J where w j J and di denotesthenumberofdimensionsinwhichtheindividual i isdeprived. j 1 dc denotesthenormativedimensionalcut‐off. Themodulespresentedabovecanbeusedtoestimatethemultidimensionalpovertyindicesaswell astheirstandarderrors. Theusercanselectamongthesevenmultidimensionalpovertyindices. Thenumberofdimensionscanbeselected(1to10). Ifapplicable,theusercanchooseparametervaluesrelevanttoachosenindex. Agroupvariablecanbeusedtoestimatetheselectedindexatthelevelofacategorical group. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith3decimals;thiscanbealsochanged. UsersareencouragedtoconsidertheexercisesthatappearinSection23.3 10.8 Multipleoverlappingdeprivationanalysis(MODA)indices TheimodaDASPmoduleproducesaseriesofmultidimensionalpovertyindicesinorderto show the incidence of deprivation in each dimension. Further, this application estimates the incidenceofmulti‐deprivationinthedifferentcombinationsofdimensions.Inthisapplication,the number of dimensions is set to three. Further, the multidimensional poverty is measured by the headcount(union andintersectionheadcountindices)andtheAlkire andFoster(2011) M0index fordifferentlevelsofthedimensionalcut‐off. 19 Thenumberofdimensionsisthree. AgroupvariablecanbeusedtoestimatetheMODAindicesatthelevelofacategorical group. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith3decimals;thiscanbealsochanged. 11 DASP,povertyandtargetingpolicies 11.1 Povertyandtargetingbypopulationgroups Theper‐capitadollarimpactofamarginaladditionofaconstantamountofincometoeveryone withinagroupk–calledLump‐SumTargeting(LST)–ontheFGTpovertyindex P(k, z; ) ,isas follows: P(k, z; 1) if 1 LST if 0 f (k, z) where z isthepovertyline,kisthepopulationsubgroupforwhichwewishtoassesstheimpactof theincomechange,and f (k , z ) isthedensityfunctionofthegroup k atlevelofincome z .Theper‐ capitadollarimpactofaproportionalmarginalvariationofincomewithinagroupk,called InequalityNeutralTargeting,ontheFGTpovertyindex P(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. 20 Figure 7: Poverty and the targeting by population groups Reference: DUCLOS,J.‐Y.ANDA.ARAAR(2006):PovertyandEquityMeasurement,Policy,andEstimationwith DAD,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 Y J y j 1 j j and where c is a factor that multiplies income component y j and that can be subject to growth. The derivative of the normalized FGT index with respect to j is given by P(z, ) j CD j (z, ) j 1, j1J where CDj is the Consumption dominance curve of component j. Change per $ of component The per-capita dollar impact of growth in the jth component on the normalized FGT index of the k th group is as follows: 21 P(z, ) y j j j CD (z, ) y j j where 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,andEstimationwith DAD,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: zf (k , z ) if 0 F ( z ) GREL P (k , z; ) P (k , z; 1) if 1 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. 22 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 Côte D’Ivoire",ReviewofIncomeandWealth,39(2):121:139. ToestimatetheFGTelasticity’swithrespectaverageincomegrowththegrouporthewhole population; Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepovertybyusingsimultaneouslypercapitaconsumptionandpercapitaincome. Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 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‐elasticitywithrespectto growthwiththedifferentapproachesmentionedabove.Formoredetailsontheseapproaches,see: Abdelkrim Araar, 2012. "Expected Poverty Changes with Economic Growth and Redistribution,"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: 23 (k ) f (k , z ) (k ) z (k ) (k ) C (k ) if 0 / F ( z) I INEL P (k , z; ) (k ) z / z P (k , z; 1) (k ) (k ) C (k ) if 1 / 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 Côte D’Ivoire",ReviewofIncomeandWealth,39(2):121:139. 12.4 ToestimateaFGTelasticitywithrespecttoaverageincomegrowthinagrouporinan entirepopulation; Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepovertybyusingsimultaneouslypercapitaconsumptionandpercapitaincome. Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 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‐elasticitywithrespectto inequalitywiththedifferentapproachesmentionedabove.Formoredetailsontheseapproaches, see: Abdelkrim Araar, 2012. "Expected Poverty Changes with Economic Growth and Redistribution,"Cahiersderecherche1222,CIRPEE. 24 12.5 ToestimateaFGTelasticity–semi‐elasticity‐withrespecttoinequality; Agroupvariablecanbeusedtoestimatepovertyatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Theresultsaredisplayedwith6decimals;thiscanbechanged. FGTelasticitieswithrespecttowithin/betweengroupcomponentsofinequality (efgtg). ThismoduleestimatesthemarginalFGTimpactandFGTelasticitywithrespecttowithin/between groupcomponentsofinequality.Agroupvariablemustbeprovided.Thismoduleismostlybasedon AraarandDuclos(2007): Araar,AbdelkrimandJean‐YvesDuclos,(2007),Povertyandinequalitycomponents: a micro framework, Working Paper: 07‐35. CIRPEE, Department of Economics, UniversitéLaval. Toopenthedialogboxofthismodule,typethecommanddbefgtg. AfterclickingonSUBMIT,thefollowingshouldbedisplayed: 25 (g) 12.6 FGTelasticitieswithrespecttowithin/betweenincomecomponentsofinequality (efgtc). ThismoduleestimatesthemarginalFGTimpactandFGTelasticitywithrespecttowithin/between incomecomponentsofinequality.Alistofincomecomponentsmustbeprovided.Thismoduleis mostlybasedonAraarandDuclos(2007): Araar,AbdelkrimandJean‐YvesDuclos,(2007),Povertyandinequalitycomponents: a micro framework, Working Paper: 07‐35. CIRPEE, Department of Economics, UniversitéLaval. Toopenthedialogboxofthismodule,typethecommanddbefgtc. 26 AfterclickingonSUBMIT,thefollowingshouldbedisplayed: (k) Incaseoneisinterestedinchangingsomeincomecomponentonlyamongthoseindividualsthatare effectivelyactiveinsomeeconomicsectors(schemes * (k), *and * inthepapermentioned above),theusershouldselecttheapproach“Truncatedincomecomponent”. 27 13 DASPandinequalityindices 13.1 Giniandconcentrationindices(igini) TheGiniindexisestimatedas ˆ Iˆ 1 ˆ where n (V )2 (V ˆ i 1 i i 1 ) 2 V1 2 n yi and Vi wh and y y yn 1 yn . 1 2 h i TheconcentrationindexforthevariableTwhentherankingvariableisYisestimatedas where ˆT istheaverageofvariableT, ˆ T 1 T IC ˆ T n (V )2 (V ˆT i 1 i i 1 ) 2 V1 2 ti n andwhere Vi wh and y y yn 1 yn . 1 2 h i Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimateinequality,forinstancebyusingsimultaneouslypercapitaconsumptionandper capitaincome. Toestimateaconcentrationindex,theusermustselectarankingvariable.. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.9 13.2 DifferencebetweenGini/concentrationindices(digini) ThismoduleestimatesdifferencesbetweentheGini/concentrationindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected; Toestimateaconcentrationindex,arankingvariablemustbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; 28 Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.3 Generalisedentropyindex(ientropy) Thegeneralizedentropyindexisestimatedas y 1 w i i 1 if 0,1 ˆ 1 n w i i i 1 1 ˆ Î w i log if 0 n y i i w i 1 i y wy 1 if 1 i i log i n ˆ ˆ i wi i 1 Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimateinequalitysimultaneouslyforpercapitaconsumptionandforpercapitaincome. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.4 Differencebetweengeneralizedentropyindices(diengtropy) Thismoduleestimatesdifferencesbetweenthegeneralizedentropyindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 29 13.5 Atkinsonindex(iatkinson) DenotetheAtkinsonindexofinequalityforthegroupkby I(ε) .Itcanbeexpressedasfollows: n w i yi ˆ ˆI(ε) ˆ () where ˆ i 1 n ˆ wi TheAtkinsonindexofsocialwelfareisasfollows: i 1 1 1 1 n if 1 and 0 w i (yi )1 ε n i 1 wi i 1 ξ̂(ε) n 1 ε 1 Exp w ln(y ) i i n i 1 wi i 1 Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimateinequalitysimultaneouslyforpercapitaconsumptionandforpercapitaincome. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.6 DifferencebetweenAtkinsonindices(diatkinson) ThismoduleestimatesdifferencesbetweentheAtkinsonindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 30 13.7 Coefficientofvariationindex(icvar) Denote the coefficient of variation index of inequality for the group k by CV. It can be expressed as follows: 1 n n 2 2 w y / w i ˆ 2 i i i 1 i 1 CV ˆ 2 Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimateinequalitysimultaneouslyforpercapitaconsumptionandforpercapitaincome. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.8 Differencebetweencoefficientofvariation(dicvar) Thismoduleestimatesdifferencesbetweencoefficient of variation indicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.9 Quantile/shareratioindicesofinequality(inineq) Thequantileratioisestimatedas , p ) Q̂(p1 ) QR(p 1 2 Q̂(p 2 ) where Q(p) denotesap‐quantileand p1 and p 2 arepercentiles. Theshareratioisestimatedas GL(p2)-GL(p1) SR(p1,p2,p3,p4) GL(p4)-GL(p3) 31 where GL(p) istheGeneralisedLorenzcurveand p1 , p 2 , p3 and p 4 arepercentiles. Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimateinequalitysimultaneouslyforpercapitaconsumptionandforpercapitaincome. Agroupvariablecanbeusedtoestimateinequalityatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.10 DifferencebetweenQuantile/Shareindices(dinineq) ThismoduleestimatesdifferencesbetweentheQuantile/Shareindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged; Theresultsaredisplayedwith6decimals;thiscanbechanged. 13.11 TheAraar(2009)multidimensionalinequalityindex The Araar (2009) the multidimensional inequality index for the K dimensions of wellbeing takes the following form: K I k k I k 1 k Ck i 1 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). I k Ck 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) 32 TheDuclos,EstebanandRay(2004)(DER)polarizationindexcanbeexpressedas: DER() f (x)1 f (y) y x dydx wheref(x)denotesthedensityfunctionatx.Thediscreteformulathatisusedtoestimatethisindex isasfollows: n w i f (yi ) a(yi ) DER() i 1 n wi i 1 ThenormalizedDERestimatedbythismoduleisdefinedas: DER() DER() 2(1 ) where: i 2 w j wi j1 a(yi ) yi N wi i 1 i 1 2 w j y j w i yi 1 j1 N wi i 1 TheGaussiankernelestimatorisusedtoestimatethedensityfunction. Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepolarizationbyusingsimultaneouslypercapitaconsumptionandpercapita income. Agroupvariablecanbeusedtoestimatepolarizationatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. 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. 33 Theresultsaredisplayedwith6decimals;thiscanbechanged. 14.3 TheFosterandWolfson(1992)polarizationindex(ipolfw) TheFosterandWolfson(1992)polarizationindexcanbeexpressedas: FW 2 2 0.5 Lorenz(p 0.5) Gini median Theusercanselectmorethanonevariableofinterestsimultaneously.Forexample,onecan estimatepolarizationbyusingsimultaneouslypercapitaconsumptionandpercapita income. Agroupvariablecanbeusedtoestimatepolarizationatthelevelofacategoricalgroup.Ifa groupvariableisselected,onlythefirstvariableofinterestisthenused. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Mainreference FOSTER,J.ANDM.WOLFSON(1992):“PolarizationandtheDeclineoftheMiddleClass:Canadaand theU.S.”mimeo,VanderbiltUniversity. 14.4 DifferencebetweenFosterandWolfson(1992)polarizationindices (dipolfw) ThismoduleestimatesdifferencesbetweentheFWindicesoftwodistributions. Foreachofthetwodistributions: Onevariableofinterestshouldbeselected; Conditionscanbespecifiedsuchastofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. 14.5 TheGeneralisedEsteban,GradinandRay(1999)polarisationindex (ipoger) The proposed measurement of polarisation by Esteban and Ray (1994) is defined as follows: 34 P ER m m (f , ) p1j p k j k j1 k 1 where j and p j denote respectively the average income and the population share of group j. The parameter 1,1.6 reflects sensitivity of society to polarisation. The first stepfortheestimationrequirestodefineexhaustiveandmutuallyexclusivegroups, .This willinvolvesomedegreeoferror.Takingintoaccountthisidea,themeasureofpolarisation proposedbyEstebanetal.(1999)isobtainedaftercorrectingthe P ER () indexappliedto the simplified representation of the original distribution with a measure of the grouping error.Nonetheless,whendealingwithpersonalorspatialincomedistributions,thereare no unanimous criteria for establishing the precise demarcation between different groups. Toaddressthisproblem,Estebanetal.(1999)followthemethodologyproposedbyAghevli andMehran(1981)andDaviesandShorrocks(1989)inordertofindanoptimalpartition ofthedistributionforagivennumberofgroups, * .Thismeansselectingthepartitionthat minimises the Gini index value of within‐group inequality, Error G f G * (see Esteban et al., 1999). The measure of polarisation proposed by Esteban et al. (1999) is thereforegivenby: m m P EGR (f , , * , ) p1j p k j k G f G * j1 k 1 where, 0 isaparameterthatinformsabouttheweightassignedtotheerrorterm.(In thestudyofEstebanetal.(1999),thevalueusedis 1 ). The Stata module ipoegr.ado estimates the generalised form of the Esteban et al. (1999) polarisationindex.Inadditiontotheusualvariables,thisroutineoffersthethreefollowing options: 1. The number of groups. Empirical studies use two or three groups. The user can select the numberofgroups.Accordingtothisnumber,theprogramsearchesfortheoptimalincome interval for each group and displays them. It also displays the error in percentage, ie: *100 ; G f G * 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 . 35 m m P EGR (f , , * , ) 0.5 p1j p k j k G f G * j1 k 1 14.6 TheInaki(2008)polarisationindex(ipoger) Letapopulationbesplitinto N groups,eachoneofsize ni 0 .Thedensityfunction,themeanand thepopulationshareofgroup i aredenotedby fi ( x) , i and i respectively. istheoverallmean. We therefore have that fi ( x) 1 , N i i and i 1 N i 1 i 1 . Using Inaki (2008), a social polarisationindexcanbedefinedas: N P ( F ) PW (i, F ) PB (i, F ) i 1 where PW (i, F ) 1 i2 f 1 i ( x) f i ( y ) x y dydx and PB (i, F ) 1 i1 i i fi 1 ( x) dx 1 i fi 1 ( x) xdx ThemoduleStatadspolallowsperformingthedecompositionofthesocialpolarisationindex P( F ) intogroupcomponents. Theusercanselecttheparameteralpha; Theusercanselecttheuseofafasterapproachfortheestimationofthedensityfunction; Standarderrorsareprovidedforallestimatedindices.Theytakeintoaccountthefull samplingdesign; Theresultsaredisplayedwith6decimalsbydefault;thiscanbechanged; TheusercansaveresultsinExcelformat. Theresultsshow: Theestimatedpopulationshareofsubgroup i : i ; Theestimatedincomeshareofsubgroup i : i i / ; Theestimated PW (i, F ) indexofsubgroup i ; Theestimated PB (i, F ) indexofsubgroup i ; Theestimated PW i W Theestimated PB i B Theestimatedtotalindex PF P (i, F ) index; P (i, F ) index; 36 Toopenthedialogboxformoduledspol,typedbdspolinthecommandwindow. Example: Forillustrativepurposes,weusea1996Cameroonianhouseholdsurvey,whichismadeof approximately1700households.Thevariablesusedare: Variables: Stratuminwhichahouseholdlives STRATA Primarysamplingunitofthehousehold PSU WEIGHT Samplingweight SIZE Householdsize INS_LEV Educationleveloftheheadofthehousehold 1. Primary; 2. ProfessionalTraining,secondaryandsuperior; 3. Notresponding. We decompose the above social polarization index using the module dspol by splitting the Cameroonian population into three exclusive groups, according to the education level of the householdhead.Wefirstinitializethesamplingdesignofthesurveywiththedialogboxsvysetas showninwhatfollows: 37 Afterthat,openthedialogboxbytypingdbdspol,andchoosevariablesandparametersasin: AfterclickingSUBMIT,thefollowingresultsappear: Mainreferences 1. 2. 3. DUCLOS,J.‐Y.,J.ESTEBAN,ANDD.RAY(2004):“Polarization:Concepts,Measurement,Estimation,” Econometrica,72,1737–1772. TianZ.&all(1999)"FastDensityEstimationUsingCF‐kernelforVeryLargeDatabases". http://portal.acm.org/citation.cfm?id=312266 IñakiPermanyer,2008."TheMeasurementofSocialPolarizationinaMulti‐groupContext,"UFAE andIAEWorkingPapers736.08,UnitatdeFonamentsdel'AnàlisiEconòmica(UAB)andInstitut d'AnàlisiEconòmica(CSIC). 38 15 DASPanddecompositions 15.1 FGTPoverty:decompositionbypopulationsubgroups(dfgtg) ThedgfgtmoduledecomposestheFGTpovertyindexbypopulationsubgroups.Thisdecomposition takestheform G ( z; ) ( g ) P (z; ; g ) P g 1 where G isthenumberofpopulationsubgroups.Theresultsshow: ( z; ; g ) TheestimatedFGTindexofsubgroup g : P Theestimatedpopulationshareofsubgroup g : ( g ) ( z; ; g ) Theestimatedabsolutecontributionofsubgroup g tototalpoverty: ( g ) P Theestimatedrelativecontributionofsubgroup g tototal ( z; ; g ) / P ( z; ) poverty: ( g ) P Anasymptoticstandarderrorisprovidedforeachofthesestatistics. Toopenthedialogboxformoduledfgtg,typedbdfgtginthecommandwindow. Figure8:DecompositionoftheFGTindexbygroups NotethattheusercansaveresultsinExcelformat. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.7 39 15.2 FGTPoverty:decompositionbyincomecomponentsusingtheShapley value(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 sk denotes income source k . The FGT index is defined as: n y wi 1 z K z; ; y s i 1 P k k 1 n wi i 1 where wi 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; 1 ); k to the value of ( P 1 ); The relative contribution of each source k to the value of ( P The absolute contribution of each source 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. 40 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: 41 15.3 AlkireandFoster(2011)MDindexofpoverty:decompositionby populationsubgroups(dmdafg) ThedmdafgmoduledecomposestheMDAlkireandFosterindexofpovertyindexbypopulation subgroups.Thisdecompositiontakestheform.Theresultsshow: TheestimatedAlkireandFosterindexofeachsubgroup: Theestimatedpopulationshareofsubgroup; Theestimatedabsolutecontributionofsubgroup g tototalpoverty; Theestimatedrelativecontributionofsubgroup g tototalpoverty; Anasymptoticstandarderrorisprovidedforeachofthesestatistics. 15.4 AlkireandFoster(2011:decompositionbydimensionsusingthe Shapleyvalue(dmdafs) The dmdafs module decomposes the Alkire and Foster (2011) multidimensional poverty indices into a sum of the contributions generated by each of the poverty dimensions. It uses the Shapley characteristic function. The non‐presence of a given factor –dimension‐ is obtained by settingthelevelofthatdimensiontoitsspecificpovertyline,thusensuringthenon‐contributionof this dimension to the AF (2011) indices. Note that the dmdafs ado file requires the module shapar.ado, which is programmed to perform decompositions using the Shapley value algorithm developedbyAraarandDuclos(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 42 15.5 FGTPoverty:decompositionbyincomecomponentsusingthe Shapleyvalue(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 decompositiontermindicatesthatanincomecomponentreducespoverty.Assumethatthereexist K incomesourcesandthat sk denotesincomesource k . TheFGTindexisdefinedas: n y wi 1 z K z; ; y s i 1 P k k 1 n wi i 1 where wi istheweightassignedtoindividual i and n issamplesize.ThedfgtsStatamodule estimates: Theshareintotalincomeofeachincomesource k ; 1 ); Theabsolutecontributionofeachsource k tothevalueof( P 1 ); Therelativecontributionofeachsource k tothevalueof( P Notethatthedfgtsadofilerequiresthemoduleshapar.ado,whichisprogrammedtoperform decompositionsusingtheShapleyvaluealgorithmdevelopedbyAraarandDuclos(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 EmpiricalillustrationwithaNigerianhouseholdsurvey We use a survey of Nigerian households (NLSS, using 17764 observations) carried out between September2003andAugust2004toillustratetheuseofthedfgtsmodule.Weusepercapitatotal 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 mainincomecomponentsare: source_1:Employmentincome; source_2:Agriculturalincome; source_3:Fish‐processingincome; source_4:Non‐farmbusinessincome; source_5:Remittancesreceived; source_6:Allotherincome; 43 TheStatadatafileissavedafterinitializingitssamplingdesignwiththecommandsvyset.Toopen thedialogboxformoduledfgts,typedbdfgtsinthecommandwindow. Figure9:DecompositionofFGTbyincomecomponents Indicatethevarlistofthesixincomesources. Indicatethatthepovertylineissetto15000$N. SetthevariableHOUSEHOLDSIZE. SetthevariableHOUSEHOLDWEIGHT. ClickonthebuttonSUBMIT.Thefollowingresultsappear: 44 15.6 DecompositionofthevariationinFGTindicesintogrowthand redistributioncomponents(dfgtgr) DattandRavallion(1992)decomposethechangeintheFGTindexbetweentwoperiods,t1andt2, intogrowthandredistributioncomponentsasfollows: P2 P1 P( t2 , t1 ) P( t1 , t1 ) P( t1 , t2 ) P( t1 , t1 ) R / ref 1 var iation C1 C2 P2 P1 P( t2 , t2 ) P( t1 , t2 ) P( t2 , t2 ) P( t2 , t1 ) R var iation where variation C1 C2 R Ref C1 C2 =differenceinpovertybetweent1andt2; =growthcomponent; =redistributioncomponent; =residual; =periodofreference. P ( t1 , t1 ) :theFGTindexofthefirstperiod P ( t1 , t1 ) :theFGTindexofthesecondperiod 45 / ref 2 P( t 2 , t1 ) :theFGTindexofthefirstperiodwhenallincomes y it1 ofthefirstperiodaremultiplied by t 2 / t1 P( t1 , t 2 ) :theFGTindexofthesecondperiodwhenallincomes y it 2 ofthesecondperiodare multipliedby t1 / t 2 TheShapleyvaluedecomposesthevariationintheFGTIndexbetweentwoperiods,t1andt2,into growthandredistributioncomponentsasfollows: P2 P1 C1 C 2 Variation 1 P ( t 2 , t1 ) P ( t1 , t1 ) P ( t 2 , t 2 ) P ( t1 , t 2 ) 2 1 t1 t 2 t1 t1 t2 t2 t 2 t1 C 2 P ( , ) P ( , ) P ( , ) P ( , ) 2 C1 15.7 DecompositionofchangeinFGTpovertybypovertyandpopulation groupcomponents–sectoraldecomposition‐(dfgtg2d). Additive poverty measures, like the FGT indices, can be expressed as a sum of the poverty contributions of the various subgroups of population. Each subgroup contributes byitspopulationshareandpovertylevel.Thus,thechangeinpovertyacrosstimedepends onthechangeinthesetwocomponents.Denotingthepopulationshareofgroup inperiod by , the change in poverty between two periods can be expressed as (see Huppi (1991)andDuclosandAraar(2006)): (06) This decomposition use the initial period as the one. If the reference period is the final,thedecompositiontakestheform: 46 (06) Toremovethearbitrarnessinselectingthereferenceperiod,wecanusetheShapley decompositionapproach,finding: (07) where is the average population share and .TheDASPmoduledfgtg2dperformsthissectoral decomposition,andthisbyselectingthereferenceperiodoftheShapleyapproach(seethe followingdialogbox): 47 Figure 10: Sectoral decomposition of FGT . 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) Decomposition of the FGT index by groups Group variable : gse Parameter alpha : 0.00 Population shares and FGT indices Group Wage-earner (public sector) Wage-earner (private sector) Artisan or trader Other type of earner Crop farmer Subsistence farmer Inactive Population Initial Pop. share Initial FGT index Final Pop. share Wage-earner (public sector) Wage-earner (private sector) Artisan or trader Other type of earner Crop farmer Subsistence farmer Inactive Population Difference in FGT index 0.042971 0.003790 0.026598 0.002164 0.062640 0.004288 0.006650 0.001308 0.104402 0.014896 0.680885 0.016403 0.075856 0.004839 0.022406 0.012599 0.067271 0.024093 0.097548 0.014712 0.194481 0.060817 0.500707 0.034911 0.514999 0.021132 0.414986 0.035336 0.041403 0.003927 0.029035 0.002624 0.055795 0.004666 0.005689 0.000923 0.167806 0.014125 0.653552 0.015083 0.046719 0.003354 0.059094 0.023396 0.111283 0.023087 0.126776 0.018202 0.293404 0.089680 0.424391 0.024457 0.533956 0.011572 0.386852 0.032340 0.036688 0.026573 0.044012 0.033369 0.029228 0.023404 0.098923 0.108357 -0.076316 0.042625 0.018957 0.024093 -0.028134 0.047901 1.000000 0.000000 0.444565 0.016124 1.000000 0.000000 0.452677 0.010927 0.008113 0.019477 Decomposition components Group Final FGT index Poverty Component Population Component Interaction Component 0.001548 0.001117 0.001224 0.000930 0.001731 0.001380 0.000610 0.000700 -0.010387 0.005992 0.012648 0.016127 -0.001724 0.002932 -0.000064 0.001931 0.000218 0.001222 -0.000768 0.002417 -0.000234 0.000930 0.029328 0.013963 -0.014336 0.018726 -0.011681 0.004317 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.005650 === 0.002463 === 0.000000 === 48 15.8 DecompositionofFGTpovertybytransientandchronicpoverty components(dtcpov) Thisdecomposestotalpovertyacrosstimeintotransientandchroniccomponents. TheJalanandRavallion(1998)approach Let yit betheincomeofindividualiinperiodtand i beaverageincomeovertheTperiodsforthat sameindividuali,i=1,…,N.Totalpovertyisdefinedas: T N t w i (z yi ) TP(, z) t 1i 1 N T wi i 1 Thechronicpovertycomponentisthendefinedas: N w i (z i ) CPC(, z) i 1 N wi i 1 Transientpovertyequals: TPC(, z) TP(, z) CPC(, z) Duclos,AraarandGiles(2006)approach Let yit be the income of individual i in period t and i be average income over the T periods for individuali.Let (, z) bethe”equally‐distributed‐equivalent”(EDE)povertygapsuchthat: 1/ (, z) TP(, z) Transientpovertyisthendefinedas N w i i (, z) TPC(, z) i 1 N wi i 1 1/ T where i i , z i 1, z andB i (, z) (z yit ) / T it 49 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 thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. References JalanJyotsna,andMartinRavallion.(1998)"TransientPovertyinPostreformRuralChina"Journalof ComparativeEconomics,26(2),pp.338:57. Jean‐Yves Duclos & Abdelkrim Araar & John Giles, 2006. "Chronic and Transient Poverty: MeasurementandEstimation,withEvidencefromChina,"WorkingPaper0611,CIRPEE. 50 15.9 Inequality:decompositionbyincomesources(diginis) Analyticalapproach Thediginismoduledecomposesthe(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 New Approach and Applications to the United States." Review of Economics and Statistics 67 (1985):151‐56. AraarAbdelkrim(2006).OntheDecompositionoftheGiniCoefficient:anExactApproach, withanIllustrationUsingCameroonianData,Workingpaper02‐06,CIRPEE. Shapleyapproach The dsineqs module decomposes inequality indices into a sum of the contributions generated by separateincomecomponents.ThedsineqsStatamoduleestimates: Theshareintotalincomeofeachincomesource k ; Theabsolutecontributionofeachsource k totheGiniindex; Therelativecontributionofeachsource k totheGiniindex; For the Shapley decomposition, the rule that is used to estimate the inequality index for a subset of components is by suppressing the inequality generated by the complement subset of components. For this, we generate a counterfactual vector of income that equals the sum of the componentsofthesubsetplustheaverageofthecomplementsubset.Notethatthedsineqsado file requires the module shapar.ado, which is programmed to perform decompositions using the ShapleyvaluealgorithmdevelopedbyAraarandDuclos(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 To open the dialog box for module dsginis, type db dsginis in the command window. 51 Figure 12: Decomposition of the Gini index by income sources (Shapley approach) 15.10 Regression‐baseddecompositionofinequalitybyincomesources A useful approach to show the contribution of income covariates to total inequality is by decomposingthelatterbythepredictedcontributionsofcovariates.Formally,denotetotalincome by y andthesetofcovariatesby X x1, x2 , , xK .Usingalinearmodelspecification,wehave: y ˆ0 ˆ1 x1 ˆ2 x2 ˆk xk ˆK xK ˆ where ̂ 0 and ˆ denoterespectivelytheestimatedconstanttermandtheresidual. Twoapproachesforthedecompositionoftotalinequalitybyincomesourcesareused: 1‐ The Shapley approach: This approach is based on the expected marginal contribution of incomesourcestototalinequality. 2‐ The Analytical approach: This approach is based on algebraic developments that express totalinequalityasasumofinequalitycontributionsofincomesources. WiththeShapleyapproach: Theusercanselectamongthefollowingrelativeinequalityindices; Giniindex Atkinsonindex Generalizedentropyindex Coefficientvariationindex Theusercanselectamongthefollowingmodelspecifications; Linear: y ˆ0 ˆ1 x1 ˆ2 x2 ˆK xK ˆ SemiLogLinear: log( y ) ˆ0 ˆ1 x1 ˆ2 x2 ˆK xK ˆ 52 WiththeAnalyticalapproach: Theusercanselectamongthefollowingrelativeinequalityindices; Giniindex Squaredcoefficientvariationindex Themodelspecificationislinear. Decomposingtotalinequalitywiththeanalyticalapproach: Total income equals y s0 s1 s2 sK sR where s0 is the estimated constant, sk ˆk X k and sR is the estimated residual. As reported by Wang 2004, relative inequality indices are not defined when the average of the variable of interest equals zero (the case of the residual). Also, inequalityindicesequalzerowhenthevariableofinterestisaconstant(thecaseoftheestimated constant).Todealwiththesetwoproblems,Wang(2004)proposesthefollowingbasicrules: Let yˆ s0 s1 s2 sK and y s1 s2 sK ,then: I ( y ) cs0 I ( y ) csr Thecontributionoftheconstant: cs0 I ( y ) I ( yˆ ) Thecontributionoftheresidual: csR I ( yˆ ) I ( y ) TheGiniindex: UsingRao1969’sapproach,therelativeGiniindexcanbedecomposedasfollows: I ( y ) k C y k where y istheaverageof y and C k isthecoefficientofconcentrationof sk when y istheranking variable. TheSquaredcoefficientofvariationindex: AsshownbyShorrocks1982,thesquaredcoefficientofvariationindexcanbedecomposedas: K I ( y ) C ov( y , sk ) k 1 y2 Shapleydecompositions: TheShapleyapproachisbuiltaroundtheexpectedmarginalcontributionofacomponent.Theuser canselectamongtwomethodstodefinetheimpactofmissingagivencomponent. Withoption:method(mean),whenacomponentismissingfromagivensetofcomponents, itisreplacedbyitsmean. Withoption:method(zero),whenacomponentismissingfromagivensetofcomponents,it isreplacedbyzero. Asindicatedabove,wecannotestimaterelativeinequalityfortheresidualcomponent. 53 Forthelinearmodel,thedecompositiontakesthefollowingform: I ( y ) I ( yˆ ) csr ,where thecontributionoftheresidualis csr I ( y ) I ( yˆ ) . FortheSemi‐loglinearmodel,theShapleydecompositionisappliedtoallcomponents includingtheconstantandtheresidual. WiththeShapleyapproach,theusercanusetheloglinearspecification.However,theusermust indicatetheincomevariableandnotthelogofthatvariable(DASPautomaticallyrunsthe regressionwithlog(y)asthedependentvariable). Example1 54 55 56 Example2 Withthisspecification,wehave y E xp( s0 s1 s2 sK sR ) .Then: Wecannotestimatetheincomeshare(nolinearform); 57 K E xp( s ).E xp( s Thecontributionoftheconstantisnil. y E xp( s0 ). k 1 k E ) .Addinga constantwillhavenotanyimpact. Example3 15.11 Giniindex:decompositionbypopulationsubgroups(diginig). The diginig module decomposes the (usual) relative or the absolute Gini index by population subgroups. Let there be G population subgroups. We wish to determine the contribution of every oneofthosesubgroupstototalpopulationinequality.TheGiniindexcanbedecomposedasfollows: G I g g Ig I R g 1 Within Overlap Between where 58 g g I R thepopulationshareofgroupg; theincomeshareofgroupg. between‐groupinequality(wheneachindividualisassignedtheaverageincome ofhisgroup). Theresidueimpliedbygroupincomeoverlap 15.12 Generalizedentropyindicesofinequality:decompositionby populationsubgroups(dentropyg). TheGeneralisedEntropyindicesofinequalitycanbedecomposedasfollows: K ˆI() ˆ (k) ˆ (k) .I(k; ˆ ) I() ˆ k 1 where: B ( k ) B ( k ) istheproportionofthepopulationfoundinsubgroupk. isthemeanincomeofgroupk. isinequalitywithingroupk. B Ik; is population inequality if each individual in subgroup k is given the mean B I incomeofsubgroupk, (k) . 15.13 Polarization:decompositionoftheDERindexbypopulationgroups (dpolag) As proposed by Araar (2008), the Duclos, Esteban and Ray index can be decomposed as follows: P 1g 1g Rg Pg P Between g Within where Rg a g ( x) g ( x) f ( x)1 dx g ag ( x) f g ( x) 1 dx g and g arerespectivelythepopulationandincomesharesofgroup g . 59 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 by Araar (2008), the Duclos, Esteban and Ray index can be decomposed as follows: P k CPk k f ( x) 1 where CPk ak ( x)dx k k 1 and k arerespectivelythepseudoconcentrationindexand incomeshareofincomesource k .ThedpolasmoduledecomposestheDERindexbyincome sources. 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; 60 3. Testpro‐poorgrowthconditions. FGTcurvesarealsocalledprimaldominancecurves.Thecfgtmoduledrawssuchcurveseasily.The modulecan: drawmorethanoneFGTcurvesimultaneouslywhenevermorethanonevariableofinterest isselected; drawFGTcurvesfordifferentpopulationsubgroupswheneveragroupvariableisselected; drawFGTcurvesthatarenotnormalizedbythepovertylines; drawdifferencesbetweenFGTcurves; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. Toopenthedialogboxofthemodulecfgt,typethecommanddbdfgtinthecommandwindow. Figure13:FGTcurves InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.4. 61 FGTCURVEwithconfidenceinterval(cfgts). ThecfgtsmoduledrawsanFGTcurveanditsconfidenceintervalbytakingintoaccountsampling design.Themodulecan: drawanFGTcurveandtwo‐sided,lower‐boundedorupper‐boundedconfidenceintervals aroundthatcurve; conditiontheestimationonapopulationsubgroup; drawaFGTcurvethatisnotnormalizedbythepovertylines; listorsavethecoordinatesofthecurveandofitsconfidenceinterval; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.5. 16.3 DifferencebetweenFGTCURVESwithconfidenceinterval(cfgts2d). Thecfgts2dmoduledrawsdifferencesbetweenFGTcurvesandtheirassociatedconfidenceinterval bytakingintoaccountsamplingdesign.Themodulecan: drawdifferencesbetweenFGTcurvesandtwo‐sided,lower‐boundedorupper‐bounded confidenceintervalsaroundthesedifferences; normalizeornottheFGTcurvesbythepovertylines; listorsavethecoordinatesofthedifferencesbetweenthecurvesaswellastheconfidence intervals; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.5. LorenzandconcentrationCURVES(clorenz). Lorenzandconcentrationcurvesareusefuldistributivetoolsthatcaninteraliabeusedto: 1. showthelevelofinequality; 2. testforinequalitydominancebetweentwodistributions; 3. testforwelfaredominancebetweentwodistributions; 4. testforprogressivity. TheclorenzmoduledrawsLorenzandconcentrationcurvessimultaneously.Themodulecan: 62 drawmorethanoneLorenzorconcentrationcurvesimultaneouslywhenevermorethan onevariableofinterestisselected; drawmorethanonegeneralizedorabsoluteLorenzorconcentrationcurvesimultaneously whenevermorethanonevariableofinterestisselected; drawmorethanonedeficitsharecurve; drawLorenzandconcentrationcurvesfordifferentpopulationsubgroupswheneveragroup variableisselected; drawdifferencesbetweenLorenzandconcentrationcurves; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. Toopenthedialogboxofthemoduleclorenz,typethecommanddbclorenzinthecommand window. Figure14:Lorenzandconcentrationcurves InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.8. 16.5 Lorenz/concentrationcurveswithconfidenceintervals(clorenzs). TheclorenzsmoduledrawsaLorenz/concentrationcurveanditsconfidenceintervalbytaking samplingdesignintoaccount.Themodulecan: 63 drawaLorenz/concentrationcurveandtwo‐sided,lower‐boundedorupper‐bounded confidenceintervals; conditiontheestimationonapopulationsubgroup; drawLorenz/concentrationcurvesandgeneralizedLorenz/concentrationcurves; listorsavethecoordinatesofthecurvesandtheirconfidenceinterval; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. 16.6 DifferencesbetweenLorenz/concentrationcurveswithconfidence interval(clorenzs2d) Theclorenz2dmoduledrawsdifferencesbetweenLorenz/concentrationcurvesandtheir associatedconfidenceintervalsbytakingsamplingdesignintoaccount.Themodulecan: drawdifferencesbetweenLorenz/concentrationcurvesandassociatedtwo‐sided,lower‐ boundedorupper‐boundedconfidenceintervals; listorsavethecoordinatesofthedifferencesandtheirconfidenceintervals; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. 16.7 Povertycurves(cpoverty) Thecpovertymoduledrawsthepovertygaporthecumulativepovertygapcurves. o Thepovertygapatapercentile p is: G ( p; z ) ( z Q( p )) o Thecumulativepovertygapatapercentile p ,notedby CPG ( p; z ) ,isgivenby: n wi ( z yi ) I ( yi Q( p )) CPG ( p; z ) i 1 n wi i 1 Themodulecanthus: drawmorethanonepovertygaporcumulativepovertygapcurvessimultaneously whenevermorethanonevariableofinterestisselected; drawpovertygaporcumulativepovertygapcurvesfordifferentpopulationsubgroups wheneveragroupvariableisselected; 64 drawdifferencesbetweenpovertygaporcumulativepovertygapcurves; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. 16.8 Consumptiondominancecurves(cdomc) Consumption dominance curves are useful tools for studying the impact of indirect tax fiscal reforms on poverty. The jth Commodity or Component dominance (C-Dominance for short) curve is defined as follows: n s 2 j wi ( z yi ) yi i 1 n wi i 1 j CD ( z , s ) n j wi K ( z yi ) yi j i 1 E y | y z f ( z ) n wi i 1 if s 2 if s 1 where K( ) is a kernel function and yj is the jth commodity. Dominance of order s is checked by setting s 1 . Thecdomcmoduledrawssuchcurveseasily.Themodulecan: drawmorethanoneCDcurvesimultaneouslywhenevermorethanonecomponentis selected; drawtheCDcurveswithconfidenceintervals; estimatetheimpactofchangeinpriceofagivencomponentonFGTindex(CDcurve)fora specifiedpovertyline; drawthenormalizedCDcurvesbytheaverageofthecomponent; listorsavethecoordinatesofthecurves; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. Toopenthedialogboxofthemodulecdomc,typethecommanddbcdomcinthecommandwindow. 65 Figure15:Consumptiondominancecurves 16.9 Difference/Ratiobetweenconsumptiondominancecurves (cdomc2d) Thecdomc2dmoduledrawsdifferenceorratiobetweenconsumptiondominancecurvesandtheir associatedconfidenceintervalsbytakingsamplingdesignintoaccount.Themodulecan: drawdifferencesbetweenconsumptiondominancecurvesandassociatedtwo‐sided,lower‐ boundedorupper‐boundedconfidenceintervals; listorsavethecoordinatesofthedifferencesandtheirconfidenceintervals; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. 16.10 DASPandtheprogressivitycurves 16.10.1 Checkingtheprogressivityoftaxesortransfers The 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 X p CT p 0 p 0,1 66 The transfer B is Tax Redistribution (TR) progressive if : PR p C B p L X p 0 p 0,1 The tax T is Income Redistribution (IR) progressive if : PR p C X T p L X p 0 p 0,1 The transfer B is Income Redistribution (IR) progressive if : PR p C X B p L X p 0 p 0,1 16.10.2 Checkingtheprogressivityoftransfervstax The 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 B p CT p 2L X p 0 p 0,1 The transfer B is more Income Redistribution (TR) progressive than a tax T if : PR p C X B p -C X T p 0 p 0,1 17 Dominance 17.1 Povertydominance(dompov) Distribution1dominatesdistribution2atordersovertherange z , z ifonlyif: P1 ( ; ) P2 ( ; ) z , z for s 1. ThisinvolvescomparingstochasticdominancecurvesatordersorFGTcurveswith s 1 .This applicationestimatesthepointsatwhichthereisareversaloftherankingofthecurves.Said differently,itprovidesthecrossingpointsofthedominancecurves,thatis,thevaluesof and P1 ( ; ) forwhich P1 ( ; ) P2 ( ; ) when: sign( P1 ( ; ) P2 ( ; )) sign( P2 ( ; ) P1 ( ; )) forasmall .Thecrossing points canalsobereferredtoas“criticalpovertylines”. Thedompovmodulecanbeusedtocheckforpovertydominanceandtocomputecriticalvalues. ThismoduleismostlybasedonAraar(2006): 67 Araar,Abdelkrim,(2006),Poverty,InequalityandStochasticDominance,TheoryandPractice: Illustration with Burkina Faso Surveys, Working Paper: 06‐34. CIRPEE, Department of Economics,UniversitéLaval. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.6. 17.2 Inequalitydominance(domineq) Distribution1inequality‐dominatesdistribution2atthesecondorderifandonlyif: L1 ( p ) L2 ( p ) p 0,1 Themoduledomineqcanbeusedtocheckforsuchinequalitydominance.Itisbasedmainlyon Araar(2006): Araar,Abdelkrim,(2006),Poverty,InequalityandStochasticDominance,TheoryandPractice: Illustration with Burkina Faso Surveys, Working Paper: 06‐34. CIRPEE, Department of Economics,UniversitéLaval. Intersectionsbetweencurvescanbeestimatedwiththismodule.Itcanalsousedtocheckfortax andtransferprogressivitybycomparingLorenzandconcentrationcurves. 17.3 DASPandbi‐dimensionalpovertydominance(dombdpov) Lettwodimensionsofwell‐beingbedenotedby k 1, 2 .Theintersectionbi‐dimensionalFGTindex fordistribution D isestimatedas n 2 k k wi ( z yi ) k D ( Z ; A ) i 1 k 1 P n wi i 1 where Z z1 , z2 and A 1 , 2 arevectorsofpovertylinesandparameters respectively, and x max( x, 0) . Distribution1dominatesdistribution2atorders s1 , s2 overtherange 0, Z ifandonlyif: P1 ( Z ; A s 1) P2 ( Z ; A s 1) Z 0, z1 0, z2 andfor 1 s1 1, 2 s2 1 . TheDASPdombdpovmodulecanbeusedtocheckforsuchdominance. Foreachofthetwodistributions: Thetwovariablesofinterest(dimensions)shouldbeselected; Conditionscanbespecifiedtofocusonspecificpopulationsubgroups; Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. 68 Surfacesshowingthedifference,thelowerboundandtheupperboundoftheconfidence surfacesareplottedinteractivelywiththeGnuPlottool. Coordinatescanbelisted. CoordinatescanbesavedinStataorGnuPlot‐ASCIIformat. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.12. 18 Distributivetools 18.1 Quantilecurves(c_quantile) Thequantileatapercentilepofacontinuouspopulationisgivenby: Q ( p ) F 1 ( p ) where p F ( y ) isthecumulativedistributionfunctionaty. For a discrete distribution, let n observations of living standards be ordered such that y1 y2 yi yi 1 yn . If F ( yi ) p F ( yi 1 ) , we define Q( p) yi 1 . The normalised quantileisdefinedas Q ( p ) Q ( p ) / . InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.10. 18.2 Incomeshareandcumulativeincomesharebygroupquantiles (quinsh) Thismodulecanbeusedtoestimate theincomeshares,aswellas,thecumulativeincome sharesbyquantilegroups.Theusercanindicatethenumberofgrouppartition.Forinstance,ifthe number is five, the quintile income shares are provided. We can also plot the graph bar of the estimatedincomeshares. 18.3 Densitycurves(cdensity) TheGaussiankernelestimatorofadensityfunction f ( x ) isdefinedby fˆ ( x) i wi Ki ( x) n w i 1 and Ki ( x) 1 exp 0.5 i ( x)2 and h 2 i ( x) x xi h i wherehisabandwidththatactsasa“smoothing”parameter. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.10. Boundary bias correction: A problem occurs with kernel estimation when a variable of interest is bounded. It may be for instance that consumption is bounded between two bounds, a minimum and a maximum, and that we wish to estimate its density “close” to these two bounds. If the true value of the density at these two bounds is 69 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 f̂ (x) * i w i K i (x)K i (x) n wi i 1 where K i (x) 1 h 2 exp 0.5 i ( x ) 2 and i ( x ) x xi h and where the scalar K *i ( x ) is defined as K *i ( x ) ( x ) P( i ( x )) P() 1 s 1 s 1! 2 2! 1 B x max x min , B , l s (1 0 0 0) ( x ) M l s K ()P()P() d l s : A A h h 1 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.Statisticsand Computing3:135‐146. Bearse,P.,Canals,J.andRilstone,P.EfficientSemiparametricEstimationofDuration ModelsWithUnobservedHeterogeneity,EconometricTheory,23,2007,281–308 Reflection approach: The reflection estimator approaches the boundary estimator by “reflecting” the data at the boundaries: f̂ (x) r i w i K i (x) n wi i 1 xX x X 2 min x X 2 max K r (x) K K K h h h Refs: CwikandMielniczuk(1993),Data‐dependentBandwidthChoiceforaGradeDensityKernel Estimate.StatisticsandprobabilityLetters16:397‐405 70 18.4 Silverman,B.W.(1986),DensityforStatisticsandDataAnalysis.LondonChapmanandHall (p30). Non‐parametricregressioncurves(cnpe) Non‐parametric regression is useful to show the link between two variables without specifying beforehandafunctionalform.Itcanalsobeusedtoestimatethelocalderivativeofthefirstvariable withrespecttothesecondwithouthavingtospecifythefunctionalformlinkingthem.Regressions withthecnpemodulecanbeperformedwithoneofthefollowingtwoapproaches: 18.4.1 Nadaraya‐Watsonapproach AGaussiankernelregressionofyonxisgivenby: E y x ( y | x) i wi Ki ( x) yi i wi Ki ( x) Fromthis,thederivativeof ( y | x ) withrespecttoxisgivenby dy ( y | x ) E x x dx 18.4.2 Locallinearapproach ThelocallinearapproachisbasedonalocalOLSestimationofthefollowingfunctionalform: 1 1 1 K i ( x) 2 yi ( x) K i ( x) 2 ( x) K i ( x) 2 ( xi x) v or,alternatively,of: 1 1 1 K i ( x) 2 yi K i ( x) 2 K i ( x) 2 ( xi x) vi Estimatesarethengivenby: dy x E y x , E dx InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.10. 18.5 DASPandjointdensityfunctions. Themodulesjdensitycanbeusedtodrawajointdensitysurface.TheGaussiankernelestimatorof thejointdensityfunction f ( x, y ) isdefinedas: 71 2 2 x x y y 1 i i f̂ (x, y) w i exp n 2 h h x y 2h x h y w i i 1 n 1 i 1 Withthismodule: Thetwovariablesofinterest(dimensions)shouldbeselected; specificpopulationsubgroupcanbeselected; surfacesshowingthejointdensityfunctionareplottedinteractivelywiththeGnuPlottool; coordinatescanbelisted;c coordinatescanbesavedinStataorGnuPlot‐ASCIIformat. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.11??? 18.6 DASPandjointdistributionfunctions Themodulesjdistrubcanbeusedtodrawjointdistributionsurfaces.Thejointdistribution function F ( x , y ) isdefinedas: n w i I(x i x)I(yi y) F̂(x, y) i 1 n wi i 1 Withthismodule: Thetwovariablesofinterest(dimensions)shouldbeselected; specificpopulationsubgroupscanbeselected; surfacesshowingthejointdistributionfunctionareplottedinteractivelywiththeGnuPlot tool; coordinatescanbelisted; coordinatescanbesavedinStataorGnuPlot‐ASCIIformat. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.11 19 DASPandpro‐poorgrowth 19.1 DASPandpro‐poorindices Themoduleipropoorestimatessimultaneouslythethreefollowingpro‐poorindices: 1. TheChenandRavallionpro‐poorindex(2003): 72 W ( z ) W2 ( z ) Index 1 F1( z ) where WD ( z ) istheWattsindexfordistribution D 1, 2 and F1( z ) istheheadcountfor indexforthefirstdistribution,bothwithpovertylinesz. 2. TheKakwaniandPerniapro‐poorindex(2000): Index P1( z, ) P2 ( z ) P1( z, ) P1( z( 1 / 2 ), ) 3. TheKakwani,KhandkerandSonpro‐poorindex(2003): Index _ 1 g P1( z, ) P2 ( z ) P1( z, ) P1( z( 1 / 2 ), ) wheretheaveragegrowthis g( 2 1 ) / 1 andwhereasecondindexisgivenby: Index _ 2 Index _ 1 g Onevariableofinterestshouldbeselectedforeachdistribution. Conditionscanbespecifiedtofocusonspecificpopulationsubgroups. Standarderrorsandconfidenceintervalswithaconfidencelevelof95%areprovided.Both thetypeofconfidenceintervalsprovidedandthelevelofconfidenceusedcanbechanged. Theresultsaredisplayedwith6decimals;thiscanbechanged. Alevelfortheparameter canbechosenforeachofthetwodistributions. 19.2 DASPandpro‐poorcurves Pro‐poorcurvescanbedrawnusingeithertheprimalorthedualapproach.Theformerusesincome levels.Thelatterisbasedonpercentiles. 19.2.1 Primalpro‐poorcurves Thechangeinthedistributionfromstate1tostate2iss‐orderabsolutelypro‐poorwithstandard cons if: ( z , s ) P2 ( z cons, s 1) P1 ( z, s 1) <0 z 0,z + Thechangeinthedistributionfromstate1tostate2iss‐orderrelativelypro‐poorif: ( z , s ) z P2 ( z 2 , s 1) P1 z , s 1 <0 z 0,z + 1 The module cpropoorp can be used to draw these primal pro‐poor curves and their associated confidenceintervalbytakingintoaccountsamplingdesign.Themodulecan: 73 drawpro‐poorcurvesandtheirtwo‐sided,lower‐boundedorupper‐boundedconfidence intervals; listorsavethecoordinatesofthedifferencesbetweenthecurvesaswellasthoseofthe confidenceintervals; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.13. 19.2.2 Dualpro‐poorcurves Let: Q( p) :quantileatpercentile p . GL( p) :GeneralisedLorenzcurveatpercentile p . :averagelivingstandards. Thechangeinthedistributionfromstate1tostate2isfirst‐orderabsolutelypro‐poorwith standardcons=0if: ( z , s ) Q2 ( p) Q1 ( p)>0 p 0, p + F ( z ) orequivalentlyif: ( z, s) Q2 ( p) Q1 ( p) >0 p 0, p + F ( z ) Q1 ( p ) Thechangeinthedistributionfromstate1tostate2isfirst‐orderrelativelypro‐poorif: ( z, s) Q2 ( p) 2 - >0 p 0, p + F ( z ) Q1 ( p) 1 Thechangeinthedistributionfromstate1tostate2issecond‐orderabsolutelypro‐poorif: ( z , s ) GL2 ( p ) GL1 ( p )>0 p 0, p + F ( z ) orequivalentlyif: ( z, s) GL2 ( p ) GL1 ( p) >0 p 0, p + F ( z ) GL1 ( p ) Thechangeinthedistributionfromstate1tostate2isfirst‐orderrelativelypro‐poorif: 74 ( z, s) GL2 ( p ) 2 - >0 p 0, p + F ( z ) GL1 ( p ) 1 The module cpropoord can be used to draw these dual pro‐poor curves and their associated confidenceintervalbytakingintoaccountsamplingdesign.Themodulecan: drawpro‐poorcurvesandtheirtwo‐sided,lower‐boundedorupper‐boundedconfidence intervals; listorsavethecoordinatesofthedifferencesbetweenthecurvesaswellasthoseofthe confidenceintervals; savethegraphsindifferentformats: o *.gph:Stataformat; o *.wmf:typicallyrecommendedtoinsertgraphsinWorddocuments; o *.eps:typicallyrecommendedtoinsertgraphsinTex/Latexdocuments. Manygraphicaloptionsareavailabletochangetheappearanceofthegraphs. InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.13 20 DASPandBenefitIncidenceAnalysis 20.1 Benefitincidenceanalysis Themainobjectiveofbenefitincidenceistoanalysethedistributionofbenefitsfromtheuse ofpublicservicesaccordingtothedistributionoflivingstandards. Two main sources of information are used. The first informs on access of household memberstopublicservices.Thisinformationcanbefoundinusualhouseholdsurveys.Thesecond deals with the amount of total public expenditures on each public service. This information is usuallyavailableatthenationallevelandsometimesinamoredisaggregatedformat,suchasatthe regionallevel.Thebenefitincidenceapproachcombinestheuseofthesetwosourcesofinformation toanalysethedistributionofpublicbenefitsanditsprogressivity. Formally,let bethesamplingweightofobservation i ; wi bethelivingstandardofmembersbelongingtoobservation i (i.e.,percapitaincome); yi s bethenumberof“eligible”membersofobservationi,i.e.,membersthat“need”the ei publicserviceprovidedbysectors.ThereareSsectors; s bethenumberofmembersofobservationi thateffectivelyusethepublicservice fi providedbysectors; bethesocio‐economicgroupofeligiblemembersofobservationi (typicallyclassified gi byincomepercentiles); 75 beasubgroupindicatorforobservationi (e.g.,1foraruralresident,and2foranurban resident).Eligiblememberscanthusbegroupedintopopulationexclusivesubgroups; betotalpublicexpendituresonsector s inarea r .ThereareRareas(theareahere referstothegeographicaldivisionwhichonecanhavereliableinformationontotal publicexpendituresonthestudiedpublicservice); ci E sr Es betotalpublicexpendituresonsector s E s R r 1 Esr . Herearesomeofthestatisticsthatcanbecomputed. 1. Theshareofaginsector s isdefinedasfollows: n SH sg w ifis I(i g) i 1 n w i fis i 1 G Notethat: SHsg 1 . g 1 2. Therateofparticipationofagroupginsector s isdefinedasfollows: n CR sg w ifis I(i g) i 1 n i 1 Thisratecannotexceed100%since f is w i esi I(i g) esi i . 3. Theunitcostofabenefitinsectorsforobservation j ,whichreferstothehousehold membersthatliveinarea r : UCsj Esr nr j1 w jf js where n r isthenumberofsampledhouseholdsinarear. 4. Thebenefitofobservation ifromtheuseofpublicsector s is: Bsi fis UCsi 5. Thebenefitofobservation i fromtheuseoftheSpublicsectorsis: 76 S Bi Bsi s 1 6. Theaveragebenefitatthelevelofthoseeligibletoaservicefromsectorsandforthose observationsthatbelongtoagroup g ,isdefinedas: n ABEsg w i Bsi I(i g) i 1 n i 1 w i esi I(i g) 7. Theaveragebenefitforthosethatusetheservice s andbelongtoagroup g isdefinedas: n s ABFgs w i Bi I(i g) i 1 n w i fis I(i g) i 1 8. Theproportionofbenefitsfromtheservicefromsector s thataccruestoobservationsthat belongtoagroup g isdefinedas: PBsg where Bsg Bsg Es n w i Bsi I(i g) . i 1 These statistics can be restricted to specific socio‐demographic groups (e.g.,. rural/urban) by replacing I(i g) by I(i c) . . Thebian.adomoduleallowsthecomputationofthesedifferentstatistics. Somecharacteristicsofthemodule: o Possibilityofselectingbetweenoneandsixsectors. o Possibilityofusingfrequencydataapproachwheninformationabouttheleveloftotalpublic expendituresisnotavailable. o Generation of benefit variables by the type of public services (ex: primary, secondary and tertiaryeducationlevels)andbysector. o Generationofunitcostvariablesforeachsector. o Possibilityofcomputingstatisticsaccordingtogroupsofobservations. o Generationofstatisticsaccordingtosocial‐demographicgroups,suchasquartiles,quintiles ordeciles. 77 Publicexpendituresonagivenserviceoftenvaryfromonegeographicaloradministrativeareato another. When information about public expenditures is available at the level of areas, this informationcanbeusedwiththebianmoduletoestimateunitcostmoreaccurately. Example1 Observationi 1 2 3 4 5 HH EligibleHH size members 7 4 5 6 4 3 2 5 3 2 Frequency Areaindicator 2 2 3 2 1 1 1 1 2 2 Totallevel of regionalpublic expenditures 14000 14000 14000 12000 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=2000 Theunitcostinarea2equals:12000/3=4000 Bydefault,theareaindicatorissetto1forallhouseholds.Whenthisdefaultisused,thevariable Regionalpublicexpenditures(thefifthcolumnthatappearsinthedialogbox)shouldbesettototal public expenditures at the national level. This would occur when the information on public expendituresisonlyavailableatthenationallevel. Example2 Observationi HH Eligible Frequency Areaindicator Regionalpublic size members expenditures 1 7 3 2 1 28000 2 4 2 2 1 28000 3 5 5 3 1 28000 4 6 3 2 1 28000 5 4 2 1 1 28000 Theunitcostbenefit(atthenationallevel)equals:28000/10=2800 InterestedusersareencouragedtoconsidertheexercisesthatappearinSection23.14 78 21 Disaggregatinggroupeddata TheungroupDASPmodulegeneratesdisaggregateddatafromaggregatedistributiveinformation. Aggregate information is obtained from cumulative income shares (or Lorenz curve ordinates) at somepercentiles.Forinstance: Percentile(p) 0.10 0.30 0.50 0.60 0.90 1.00 Lorenzvalues:L(p) 0.02 0.10 0.13 0.30 0.70 1.00 Theusermustspecifythetotalnumberofobservationstobegenerated.Theusercanalsoindicate the number of observations to be generated specifically at the top and/or at the bottom of the distribution,inwhichcasetheproportion(in%)ofthepopulationfoundatthetoporatthebottom mustalsobespecified. Remarks: If only the total number of observations is set, the generated data are self weighted (or uniformlydistributedoverpercentiles). Ifanumberofobservationsissetforthebottomand/ortoptails,thegenerateddataarenot selfweightedandaweightvariableisprovidedinadditiontothegeneratedincomevariable. Example:Assumethatthetotalnumberofobservationstobegeneratedissetto1900, but that we would like the bottom 10% of the population to be represented by 1000 observations.Inthiscase,weightswillequal1/1000forthebottom1000observations and1/100fortheremainingobservations(thesumofweightsbeingnormalizedtoone). Thegeneratedincomevectortakesthenameof_yandthevectorweight,_w. The number of observations to be generated does not have to equal the number of observations of the sample that was originally used to generate the aggregated data. The ungroup module cannot in itself serve to estimate the sampling errors that would have occurred had the original sample data been used to estimate poverty and/or inequality estimates. Theusercanselectanysamplesizethatexceeds(number_of_classes+1),butitmaybemore appropriateforstatisticalbias‐reductionpurposestoselectrelativelylargesizes. STAGEIGeneratinganinitialdistributionofincomesandpercentiles S.1.1:Generatingavectorofpercentiles Starting from information on the importance of bottom and top groups and on the number of observationstobegenerated,wefirstgenerateavectorofpercentiles. 79 Examples: Notations: NOBS:numberoftotalobservations F:vectorofpercentiles B_NOBS:numberofobservationsforthebottomgroup T_NOBS:numberofobservationsforthetopgroup. ForNOBS=1000spreadequallyacrossallpercentiles,F=0.001,0.002...0.999,1.Toavoid the 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 groups beingthefirstandlastdeciles: a. F=0.0001,0.0002,...,0.0999,0.1000in0001/1000 b. F=0.1010,0.1020,…,0.8990,0.9000in1001/1800 c. F=0.9001,0.9002,...,0.9999,1.0000in1801/2800 AdjustmentscanalsobemadetoavoidthecaseofF(1)=1. Theweightvectorcaneasilybegenerated. S.1.2:Generatinganinitialdistributionofincomes Theusermustindicatetheformofdistributionofthedisaggregateddata. ‐Normalandlognormaldistributions: Assumethat x followsalognormaldistributionwithmean andvariance 2 .TheLorenzcurveis definedasfollows: Ln( x) ( 2 ) Ln( x ) L( p ) and p Weassumethat 1 andweestimatethevarianceusingtheproceduresuggestedbyShorrrocks andWan(2008):avalueforthestandarddeviationoflogincomes,σ,isobtainedbyaveragingthe m 1 estimatesof k 1 pk 1 L( pk ) k 1, , m 1 where m isthenumberofclassesandΦisthestandardnormaldistributionfunction(Aitchisonand Brown1957;KolenikovandShorrocks2005,Appendix). ‐GeneralizedQuadraticLorenzCurve: Itisassumedthat: L(1 L) a( p 2 L) bL( p 1) c( p L) Wecanregress L (1 L ) on ( p L) , L( p 1) and ( p L ) withoutanintercept,droppingthelast observationsincethechosenfunctionalformforcesthecurvetogothrough(1,1). 2 2 2 b 2mp n mp np e Wehave Q ( p ) 2 4 0.5 80 e a b c 1 m n b 2 4a 2be 4c ‐BetaLorenzCurve: Itisassumedthat: log p L log( ) log( p ) log(1 p ) Afterestimatingtheparameters,wecangeneratequantilesasfollows Q p p 1 p p 1 p SeealsoDatt(1998). ‐TheSingh‐Maddaladistribution ThedistributionfunctionproposedbySinghandMaddala(1976)takesthefollowingform: q 1 F ( x) 1 a 1 ( x / b) where a 0, b 0, q 1/ a are parameters to be estimated. The income ( x ) is assumed to be equaltoorgreaterthanzero.Thedensityfunctionisdefinedasfollows: f ( x ) aq / b 1 x / b Quantilesaredefinedasfollows: Q ( p ) b 1 p a ( q 1) 1/ q x / b 1 1/ a a 1 WefollowJenkins(2008)’sapproachfortheestimationofparameters.Forthis,wemaximizethe likelihood function, which is simply the product of density functions evaluated at the average incomeofeachclass: http://stata‐press.com/journals/stbcontents/stb48.pdf STAGEIIAdjustingtheinitialdistributiontomatchtheaggregateddata(optional). ThisstageadjuststheinitialvectorofincomesusingtheShorrocksandWan(2008)procedure.This procedureproceedswithtwosuccessiveadjustments: Adjustment1:Correctingtheinitialincomevectortoensurethateachincomegrouphasits originalmeanincome. Adjustment2:Smoothingtheinter–classdistributions. The generated sample is saved automatically in a new Stata data file (called by default ungroup_data.dta;namesanddirectoriescanbechanged).TheusercanalsoplottheLorenzcurves oftheaggregated(whenweassumethateachindividualhastheaverageincomeofhisgroup)and generateddata. Dialogboxoftheungroupmodule 81 Figure 16: ungroup dialog box IllustrationwithBurkinaFasohouseholdsurveydata In this example, we use disaggregated data to generate aggregated information. Then, we compare the density curve of the true data with those of the data generated through disaggregationofthepreviouslyaggregateddata. genfw=size*weight Aggregatedinformation: geny=exppc/r(mean) clorenzy,hs(size)lres(1) pL(p) .1.0233349 .2.0576717 .3.0991386 .4.1480407 .5.2051758 .6.2729623 .7.3565971 .8.4657389 .9.6213571 11.00000 Density functions (without adjustment) (with adjustment) 0 0 .5 .5 1 1 1.5 1.5 Density functions 0 2 4 Normalised per capita expenditures True distribition Uniform Generalized Quadratic LC 6 0 2 4 Normalised per capita expenditures True distribition Uniform Generalized Quadratic LC Log Normal Beta LC SINGH & MADALLA 82 Log Normal Beta LC SINGH & MADALLA 6 22 Appendices 22.1 AppendixA:illustrativehouseholdsurveys 22.1.1 The1994BurkinaFasosurveyofhouseholdexpenditures(bkf94I.dta) Thisisanationallyrepresentativesurvey,withsampleselectionusingtwo‐stagestratifiedrandom sampling.Sevenstratawereformed.Fiveofthesestratawereruralandtwowereurban.Primary samplingunitsweresampledfromalistdrawnfromthe1985census.Thelastsamplingunitswere households. Listofvariables strata Stratuminwhichahouseholdlives psu Primarysamplingunit weight Samplingweight size Householdsize exp Totalhouseholdexpenditures expeq Totalhouseholdexpendituresperadultequivalent expcp Totalhouseholdexpenditurespercapita gse Socio‐economicgroupofthehouseholdhead 1wage‐earner(publicsector) 2wage‐earner(privatesector) 3Artisanortrader 4Othertypeofearner 5Cropfarmer 6Subsistencefarmer 7Inactive sex Sexofhouseholdhead 1Male 2Female zone Residentialarea 1Rural 2Urban 83 22.1.2 The1998BurkinaFasosurveyofhouseholdexpenditures(bkf98I.dta) Thissurveyissimilartothe1994one,althoughtenstratawereusedinsteadofsevenfor1994.To express 1998 data in 1994 prices, two alternative procedures have been used. First, 1998 expendituredataweremultipliedbytheratioofthe1994officialpovertylinetothe1998official poverty line: z_1994/z_1998. Second, 1998 expenditure data were multiplied by the ratio of the 1994consumerpriceindextothe1998consumerpriceindex:ipc_1994/ipc_1998. Listofnewvariables expcpz Totalhouseholdexpenditurespercapitadeflatedby(z_1994/z_1998) Totalexpenditurespercapitadeflatedby(ipc_1994/ipc_1998) expcpi 22.1.3 CanadianSurveyofConsumerFinance(asubsampleof1000 observations–can6.dta) Listofvariables X Yearlygrossincomeperadultequivalent. T Incometaxesperadultequivalent. B1 Transfer1peradultequivalent. B2 Transfer2peradultequivalent. B3 Transfer3peradultequivalent. B SumoftransfersB1,B2andB3 N Yearlynetincomeperadultequivalent(X minusT plusB) 22.1.4 PeruLSMSsurvey1994(Asampleof3623householdobservations‐ PEREDE94I.dta) Listofvariables exppc Totalexpenditures,percapita(constantJune1994solesperyear). weight Samplingweight 84 size npubprim npubsec npubuniv Householdsize Numberofhouseholdmembersinpublicprimaryschool Numberofhouseholdmembersinpublicsecondaryschool Numberofhouseholdmembersinpublicpost‐secondaryschool 22.1.5 PeruLSMSsurvey1994(Asampleof3623householdobservations– PERU_A_I.dta) Listofvariables hhid HouseholdId. exppc Totalexpenditures,percapita(constantJune1994solesperyear). size Householdsize literate Numberofliteratehouseholdmembers pliterate literate/size 22.1.6 The1995ColombiaDHSsurvey(columbiaI.dta) ThissampleisapartoftheDatafromtheDemographicandHealthSurveys(Colombia_1995)witch containsthefollowinginformationforchildrenaged0‐59months Listofvariables hid Householdid haz height‐for‐age waz weight‐for‐age whz weight‐for‐height sprob survivalprobability wght samplingweight Asset assetindex 22.1.7 The1996DominicanRepublicDHSsurvey (Dominican_republic1996I.dta) ThissampleisapartoftheDatafromtheDemographicandHealthSurveys(Republic Dominican_1996)witchcontainsthefollowinginformationforchildrenaged0‐59months 85 Listofvariables hid Householdid haz height‐for‐age waz weight‐for‐age whz weight‐for‐height sprob survivalprobability wght samplingweight Asset 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*/ labeldefinelvgse 1"wage‐earner(publicsector)" 2"wage‐earner(privatesector)" 3"Artisanortrader" 4"Othertypeofearner" 5"Cropfarmer" 6"Subsistencefarmer" 7"Inactive" ; /*Toassignthelabelvalues"lvgse"tothevariablegse*/ labelvalgselvgse; labelvarsex"Sexofhouseholdhead"; 86 labeldeflvsex 1Male 2Female ; labelvalsexlvsex; labelvarzone"Residentialarea"; labeldeflvzone 1Rural 2Urban ; labelvalzonelvzone; ====================================End====================================== 22.3 AppendixC:settingthesamplingdesign Tosetthesamplingdesignforthedatafilebkf94.dta,openthedialogboxforthecommandsvyset bytypingthesyntaxdbsvysetinthecommandwindow.IntheMainpanel,setSTRATAandSAMPLING UNITSasfollows: Figure17:Surveydatasettings IntheWeightspanel,setSAMPLINGWEIGHTVARIABLEasfollows: 87 Figure18:Settingsamplingweights ClickonOKandsavethedatafile. Tocheckifthesamplingdesignhasbeenwellset,typethecommandsvydes.Thefollowingwillbe displayed: 88 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. UsetheinformationofSection22.1.1.tosetthesamplingdesignandthensavethefile. 3. Estimatetheheadcountindexusingvariablesofinterestexpccandexpeq. a. You should set SIZE to household size in order to estimate poverty over the populationofindividuals. b. Usetheso‐called1994officialpovertylineof41099FrancsCFAperyear. 4. Estimate the headcount index using the same procedure as above except that the poverty lineisnowsetto60%ofthemedian. 5. Usingtheofficialpovertyline,howdoestheheadcountindexformale‐andfemale‐headed householdscompare? 6. Can you draw a 99% confidence interval around the previous comparison? Also, set the numberofdecimalsto4. Answer Q.1 Ifbkf94.dtaissavedinthedirectoryc:/data,typethefollowingcommandtoopenit: use"C:\data\bkf94.dta",clear Iflab_bkf94.doissavedinthedirectoryc:/do_files,typethefollowingcommandtolabelvariables andlabels: do"C:\do_files\lab_bkf94.do" Typingthecommanddescribe,weobtain: obs: 8,625 vars: 9 31Oct200613:48 size: 285,087(99.6%of memoryfree) storagedisplay value variable name type format label weight float %9.0g size byte %8.0g strata byte %8.0g psu byte %8.0g gse byte %29.0g gse sex byte %8.0g sex zone byte %8.0g zone exp double %10.0g expeq double %10.0g exppc float %9.0g Typinglabellist,wefind: zone: 1 2 variablelabel Samplingweight Householdsize Stratuminwhichahouseholdlives Primarysamplingunit Socio‐economicgroupofthehouseholdhead Sexofhouseholdhead Residentialarea Totalhouseholdexpenditures Totalhouseholdexpendituresperadultequivalent Totalhouseholdexpenditurespercapita Rural Urban 89 sex: gse: 1 2 Male Female 1 2 3 4 5 6 7 wage‐earner(publicsector) wage‐earner(privatesector) Artisanortrader Othertypeofearner Cropfarmer Foodfarmer Inactive Q.2 Youcansetthesamplingdesignwithadialogbox,asindicatedinSection22.3,orsimplybytyping svysetpsu[pweight=weight],strata(strata)vce(linearized) Typingsvydes,weobtain Q.3 TypebdifgttoopenthedialogboxfortheFGTpovertyindexandchoosevariablesandparameters asindicatedinthefollowingwindow.ClickonSUBMIT. 90 Figure19:EstimatingFGTindices Thefollowingresultsshouldthenbedisplayed: Q.4 SelectRELATIVEforthepovertylineandsettheotherparametersasabove. 91 Figure20:EstimatingFGTindiceswithrelativepovertylines AfterclickingonSUBMIT,thefollowingresultsshouldbedisplayed: Q.5 Setthegroupvariabletosex. 92 Figure21:FGTindicesdifferentiatedbygender ClickingonSUBMIT,thefollowingshouldappear: Q.6 UsingthepanelCONFIDENCEINTERVAL,settheconfidencelevelto99%andsetthenumberof decimalsto4intheRESULTSpanel. 93 94 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. Estimatethedifferencebetweenheadcountindiceswhen a. 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: H 0 : P( z 41099, 0) 0 against H1 : P( z 41099, 0) 0 Set the significance level to 5% and assume that the test statistics follows a normal distribution. Answers Q.1 Openthedialogboxbytyping dbdifgt Q.2 Fordistribution1,choosetheoptionDATAINFILEinsteadofDATAINMEMORYandclickon BROWSEtospecifythelocationofthefilebkf98I.dta. Followthesameprocedurefordistribution2tospecifythelocationofbkf94I.dta. Choosevariablesandparametersasfollows: 95 Figure22:EstimatingdifferencesbetweenFGTindices AfterclickingonSUBMIT,thefollowingshouldbedisplayed: 96 Q.3 Restricttheestimationtoruralresidentsasfollows: o SelecttheoptionCondition(s) o WriteZONEinthefieldnexttoCONDITION(1)andtype1inthenextfield. Figure23:EstimatingdifferencesinFGTindices AfterclickingonSUBMIT,weshouldsee: Q.4 97 Onecanseethatthechangeinpovertywassignificantonlyforurbanresidents.Q.5 Restricttheestimationtomale‐headedurbanresidentsasfollows: o SetthenumberofCondition(s)to2; o SetsexinthefieldnexttoCondition(2)andtype1inthenextfield. Figure24:FGTdifferencesacrossyearsbygenderandzone AfterclickingonSUBMIT,thefollowingshouldbedisplayed: Q.6 98 Wehavethat: LowerBound:=0.0222 UpperBound:=0.1105 Thenullhypothesisisrejectedsincethelowerboundofthe95%confidenceintervalisabovezero. 23.3 Estimatingmultidimensionalpovertyindices “Howmuchisbi‐dimensionalpoverty(totalexpendituresandliteracy)inPeruin1994?” Usingtheperu94I.dtafile, 1. EstimatetheChakravartyetal(1998)indexwithparameteralpha=1and Pov.line a_j Var.ofinterest Dimension1 exppc 400 1 Dimension2 pliterate 0.90 1 2. EstimatetheBourguignonandChakravarty(2003)indexwithparameters alpha=beta=gamma=1and Pov.line Var.ofinterest Dimension1 exppc 400 Dimension2 literate 0.90 Q.1 Steps: Type use"C:\data\ peru94_A_I.dta",clear Toopentherelevantdialogbox,type dbimdp_bci Choosevariablesandparametersasin 99 Figure25:Estimatingmultidimensionalpovertyindices(A) AfterclickingSUBMIT,thefollowingresultsappear. Q.2 Toopentherelevantdialogbox,type dbimdp_cmr Steps: Choosevariablesandparametersasin 100 Figure26:Estimatingmultidimensionalpovertyindices(B) AfterclickingSUBMIT,thefollowingresultsappear. 101 23.4 EstimatingFGTcurves. “Howsensitivetothechoiceofapovertylineistherural‐urbandifferenceinpoverty?” 1. Openbkf94I.dta 2. OpentheFGTcurvesdialogbox. 3. DrawFGTcurvesforvariablesofinterestexppcandexpeqwith a. parameter 0 ; b. povertylinebetween0and100,000FrancCFA; c. sizevariablesettosize; d. subtitleofthefiguresetto“Burkina1994”. 4. DrawFGTcurvesforurbanandruralresidentswith a. variableofinterestsettoexpcap; b. parameter 0 ; c. povertylinebetween0and100,000FrancCFA; d. sizevariablesettosize. 5. Drawthedifferencebetweenthesetwocurvesand a. save the graph in *.gph format to be plotted in Stata and in *.wmf format to be insertedinaWorddocument. b. Listthecoordinatesofthegraph. 6. Redothelastgraphwith 1 . Answers Q.1 Openthefilewith use"C:\data\bkf94I.dta",clear Q.2 Openthedialogboxbytyping dbdifgt Q.3 Choosevariablesandparametersasfollows: 102 Figure27:DrawingFGTcurves Tochangethesubtitle,selecttheTitlepanelandwritethesubtitle. Figure28:EditingFGTcurves AfterclickingSUBMIT,thefollowinggraphappears: 103 Figure29:GraphofFGTcurves 104 Q.4 Choosevariablesandparametersasinthefollowingwindow: Figure30:FGTcurvesbyzone AfterclickingSUBMIT,thefollowinggraphappears: 105 Figure31:GraphofFGTcurvesbyzone 106 Q.5 ChoosetheoptionDIFFERENCEandselect:WITHTHEFIRSTCURVE; Indicatethatthegroupvariableiszone; SelecttheResultspanelandchoosetheoptionLISTintheCOORDINATESquadrant. IntheGRAPHquadrant,selectthedirectoryinwhichtosavethegraphingphformatandto exportthegraphinwmfformat. Figure32:DifferencesofFGTcurves 107 Figure33:Listingcoordinates 108 AfterclickingSUBMIT,thefollowingappears: Figure34:DifferencesbetweenFGTcurves Q.6 109 Figure35:DifferencesbetweenFGTcurves 23.5 EstimatingFGTcurvesanddifferencesbetweenFGTcurveswith confidenceintervals “Isthepovertyincreasebetween1994and1998inBurkinaFasostatisticallysignificant?” 1) Using the file bkf94I.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 FGT curvesandassociatedconfidenceintervalswith: a) Thevariableofinterestexppcfor1994andexppczfor1998. b) parameter 0 ; c) povertylinebetween0and100,000FrancCFA; d) sizevariablesettosize. 3) Redo2)withparameter 1 . Answers Q.1 110 Steps: Type use"C:\data\bkf94I.dta",clear Toopentherelevantdialogbox,type dbcfgts Choosevariablesandparametersasin Figure36:DrawingFGTcurveswithconfidenceinterval AfterclickingSUBMIT,thefollowingappears: 111 Figure37:FGTcurveswithconfidenceinterval FGT curve (alpha = 0) 0 .2 .4 .6 .8 Burkina Faso 0 20000 40000 60000 80000 100000 Poverty line (z) Confidence interval (95 %) Estimate Q.2 Steps: Toopentherelevantdialogbox,type dbcfgtsd2 Choosevariablesandparametersasin 112 Figure38:DrawingthedifferencebetweenFGTcurveswithconfidenceinterval Figure39:DifferencebetweenFGTcurveswithconfidenceinterval ( 0) Difference between FGT curves -.1 -.05 0 .05 (alpha = 0) 0 20000 40000 60000 80000 100000 Poverty line (z) Confidence interval (95 %) Estimated difference 113 Figure40:DifferencebetweenFGTcurveswithconfidenceinterval ( 1) Difference between FGT curves -.04 -.02 0 .02 (alpha = 1) 0 20000 40000 60000 80000 100000 Poverty line (z) Confidence interval (95 %) Estimated difference 23.6 Testingpovertydominanceandestimatingcriticalvalues. “HasthepovertyincreaseinBurkinaFasobetween1994and1998beenstatisticallysignificant?” 1) Usingsimultaneouslyfilesbkf94I.dtaandbkf98I.dta,checkforsecond‐orderpovertydominance andestimatethevaluesofthepovertylineatwhichthetwoFGTcurvescross. a) Thevariableofinterestisexppcfor1994andexppczfor1998; b) Thepovertylineshouldvarybetween0and100,000FrancCFA; c) Thesizevariableshouldbesettosize. Answers Q.1 Steps: Toopentherelevantdialogbox,type dbdompov Choosevariablesandparametersasin 114 Figure41:Testingforpovertydominance AfterclickingSUBMIT,thefollowingresultsappear: 23.7 DecomposingFGTindices. “WhatisthecontributionofdifferenttypesofearnerstototalpovertyinBurkinaFaso?” 1. Openbkf94I.dtaanddecomposetheaveragepovertygap a. withvariableofinterestexppc; b. withsizevariablesettosize; c. attheofficialpovertylineof41099FrancsCFA; d. andusingthegroupvariablegse(Socio‐economicgroups). 2. Dotheaboveexercisewithoutstandarderrorsandwiththenumberofdecimalssetto4. 115 Answers Q.1 Steps: Type use"C:\data\bkf94I.dta",clear Toopentherelevantdialogbox,type dbdfgtg Choosevariablesandparametersasin Figure42:DecomposingFGTindicesbygroups AfterclickingSUBMIT,thefollowinginformationisprovided: 116 Q.2 UsingtheRESULTSpanel,changethenumberofdecimalsandunselecttheoptionDISPLAY STANDARD ERRORS. AfterclickingSUBMIT,thefollowinginformationisobtained: 117 23.8 EstimatingLorenzandconcentrationcurves. “HowmuchdotaxesandtransfersaffectinequalityinCanada?” Byusingthecan6.dtafile, 1. Draw the Lorenz curves for gross income X and net income N. How can you see the redistributionofincome? 2. Draw Lorenz curves for gross income X and concentration curves for each of the three transfers B1, B2 and B3 and the tax T. What can you say about the progressivity of these elementsofthetaxandtransfersystem? “WhatistheextentofinequalityamongBurkinaFasoruralandurbanhouseholdsin1994?” Byusingthebkf94I.dtafile, 3. DrawLorenzcurvesforruralandurbanhouseholds a. withvariableofinterestexppc; b. withsizevariablesettosize; c. andusingthegroupvariablezone(asresidentialarea). Q.1 Steps: Type use"C:\data\can6.dta",clear Toopentherelevantdialogbox,type dbclorenz Choosevariablesandparametersasin 118 Figure43:Lorenzandconcentrationcurves AfterclickingSUBMIT,thefollowingappears: 119 Figure44:Lorenzcurves Q.2 Steps: Choosevariablesandparametersasin 120 Figure45:Drawingconcentrationcurves AfterclickingonSUBMIT,thefollowingappears: 121 Figure46:Lorenzandconcentrationcurves Q.3 Steps: Type use"C:\data\bkf94I.dta",clear Choosevariablesandparametersasin 122 Figure47:DrawingLorenzcurves Figure48:Lorenzcurves 123 23.9 EstimatingGiniandconcentrationcurves “ByhowmuchdotaxesandtransfersaffectinequalityinCanada?” Usingthecan6.dtafile, 1. EstimatetheGiniindicesforgrossincomeXandnetincomeN. 2. Estimate the concentration indices for variables T and N when the ranking variable is grossincomeX. “ByhowmuchhasinequalitychangedinBurkinaFasobetween1994and1998?” Usingthebkf94I.dtafile, 3. EstimatethedifferenceinBurkinaFaso’sGiniindexbetween1998and1994 a. withvariableofinterestexpeqzfor1998andexpeqfor1994; b. withsizevariablesettosize. Q.1 Steps: Type use"C:\data\can6.dta",clear Toopentherelevantdialogbox,type dbigini Choosevariablesandparametersasin 124 Figure49:EstimatingGiniandconcentrationindices AfterclickingSUBMIT,thefollowingresultsareobtained: Q.2 Steps: Choosevariablesandparametersasin 125 Figure50:Estimatingconcentrationindices AfterclickingSUBMIT,thefollowingresultsareobtained: Q.3 Steps: Toopentherelevantdialogbox,type dbdigini Choosevariablesandparametersasin 126 Figure51:EstimatingdifferencesinGiniandconcentrationindices AfterclickingSUBMIT,thefollowinginformationisobtained: 127 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.1 Steps: Type use"C:\data\can6.dta",clear Toopentherelevantdialogbox,type dbcdensity Choosevariablesandparametersasin Figure52:Drawingdensities 128 AfterclickingSUBMIT,thefollowingappears: Figure53:Densitycurves .00003 0 .00001 .00002 f(y) .00004 .00005 Density Curves 0 12000 24000 36000 48000 60000 y X Q.2 Steps: Toopentherelevantdialogbox,type dbc_quantile Choosevariablesandparametersasin 129 N Figure54:Drawingquantilecurves AfterclickingSUBMIT,thefollowingappears: Figure55:Quantilecurves 0 10000 Q(p) 20000 30000 Quantile Curves 0 .2 .4 Percentiles (p) X 130 .6 .8 N Q.3 Steps: Toopentherelevantdialogbox,type dbcnpe Choosevariablesandparametersasin Figure56:Drawingnon‐parametricregressioncurves AfterclickingSUBMIT,thefollowingappears: 131 Figure57:Non‐parametricregressioncurves 20000 Non parametric regression 10000 0 5000 E(Y|X) 15000 (Linear Locally Estimation Approach | Bandwidth = 3699.26 ) 0 12000 24000 36000 48000 60000 X values t b Q.4 Steps: Choosevariablesandparametersasin 132 Figure58:Drawingderivativesofnon‐parametricregressioncurves AfterclickingSUBMIT,thefollowingappears: Figure59:Derivativesofnon‐parametricregressioncurves Non parametric derivative regression 0 -.5 -1 dE[Y|X]/dX .5 1 (Linear Locally Estimation Approach | Bandwidth = 3699.26 ) 0 12000 24000 36000 48000 60000 X values t 133 b 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.1 Steps: Type use"C:\data\can6.dta",clear Toopentherelevantdialogbox,type dbsjdensity Choosevariablesandparametersasin Figure60:Plottingjointdensityfunction AfterclickingSUBMIT,thefollowinggraphisplottedinteractivelywithGnuPlot4.2: 134 Joint Density Function f(x,y) 3e-009 2.5e-009 2e-009 1.5e-009 1e-009 5e-010 0 0 10000 20000 0 10000 30000 Dimension 1 40000 20000 30000 Dimension 2 40000 50000 50000 6000060000 Q.2 Steps: Toopentherelevantdialogbox,type dbsjdistrub Choosevariablesandparametersasin 135 Figure61:Plottingjointdistributionfunction AfterclickingSUBMIT,thefollowinggraphisplottedinteractivelywithGnuPlot4.2: Joint Distribution Function F(x,y) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 60000 50000 40000 30000 20000 Dimension 2 10000 0 0 10000 40000 30000 20000 Dimension 1 50000 60000 136 23.12 Testingthebi‐dimensionalpovertydominance Using 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 the confidenceintervalofthatdifferencewhen Range alpha_j Var.ofinterest Dimension1 haz:height‐for‐age ‐3.0/6.0 0 Dimension2 sprob:survival 0.7/1.0 0 probability 2. Testforbi‐dimensionalpovertyusingtheinformationabove. Answer: Q.1 Steps: Toopentherelevantdialogbox,type dbdombdpov Choosevariablesandparametersasin 137 Figure62:Testingforbi‐dimensionalpovertydominance AfterclickingSUBMIT,thefollowinggraphisplottedinteractivelywithGnuPlot4.2: Bi-dimensional poverty dominance 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 6 Difference Lower-bounded Upper-bounded 5 4 3 2 1 Dimension 1 0 -1 -2 -3 1 0.8 0.840.82 0.880.86 0.9 0.940.92 Dimension 2 0.980.96 0.78 Q.2 138 To make a simple test of multidimensional dominance, one should check if the lower‐bounded confidenceintervalsurfaceisalwaysabovezeroforallcombinationsofrelevantpovertylines–or conversely. o Forthis,clickonthepanel“Confidenceinterval”andselecttheoptionlower‐bounded. o ClickagainonthebuttonSubmit. AfterclickingSUBMIT,thefollowinggraphisplottedinteractivelywithGnuPlot4.2: Bi-dimensional poverty dominance Lower-bounded 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -3 -2 -1 0 Dimension 1 1 2 3 4 0.96 0.9 0.92 0.94 0.88 5 0.86 0.84 6 0.78 0.8 0.82 Dimension 2 0.98 1 139 23.13 Testingforpro‐poornessofgrowthinMexico The three sub‐samples used in these exercises are sub‐samples of 2000 observations drawn randomlyfromthethreeENIGHMexicanhouseholdsurveysfor1992,1998and2004.Eachofthese threesub‐samplescontainsthefollowingvariables: strata Thestratum psu Theprimarysamplingunit weight Samplingweight inc Income hhsz 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.1 Steps: Toopentherelevantdialogbox,type dbcpropoorp 140 Choosevariablesandparametersasin(selecttheupper‐boundedoptionfortheconfidence interval): Figure63:Testingthepro‐poorgrowth(primalapproach) AfterclickingSUBMIT,thefollowinggraphappears 141 Relative propoor curve -.15 -.1 -.05 0 .05 (Order : s=1 | Dif. = P_2( (m2/m1)z, a=s-1) - P_1(z,a=s-1)) 0 600 Difference Null horizontal line 1200 1800 Poverty line (z) 2400 3000 Upper bound of 95% confidence interval Q.2 Steps: Toopentherelevantdialogbox,type dbcpropoord Choose variables and parameters as in (with the lower‐bounded option for the confidence interval): Figure64:Testingthepro‐poorgrowth(dualapproach)‐A 142 AfterclickingSUBMIT,thefollowinggraphappears Absolute propoor curves -.4 -.2 0 .2 .4 (Order : s=1 | Dif. = Q_2(p) /Q_1(p) - mu_2/mu_1 ) 0 .184 .368 .552 Percentiles (p) Difference Null horizontal line .736 .92 Lower bound of 95% confidence interval Q.2 Steps: 143 Toopentherelevantdialogbox,type dbcpropoord Choose variables and parameters as in (with the lower‐bounded option for the confidence interval): Figure65:Testingthepro‐poorgrowth(dualapproach)–B AfterclickingSUBMIT,thefollowinggraphappears 144 Absolute propoor curves 0 2 4 6 (Order : s=2 | Dif. = (GL_2(p) - GL_1(p) ) / GL_2(p) ) 0 .184 .368 .552 Percentiles (p) Difference Null horizontal line Q.4 Steps: Toopentherelevantdialogbox,type dbipropoor Choosevariablesandparametersas. .736 .92 Lower bound of 95% confidence interval 145 AfterclickingSUBMIT,thefollowingresultsappear: 23.14 BenefitincidenceanalysisofpublicspendingoneducationinPeru (1994). 1. Usingtheperedu94I.dtafile,estimateparticipationandcoverageratesoftwotypesofpublic spendingoneducationwhen: ‐ Thestandardoflivingisexppc ‐ The number of household members that benefit from education is fr_prim for the primarysectorandfr_secforthesecondaryone. ‐ The number of eligible household members is el_prim for the primary sector and el_secforthesecondaryone. ‐ Socialgroupsarequintiles. 146 Answer: Typedbbianinthewindowscommandandsetvariablesandoptionsasfollows: Figure66:Benefitincidenceanalysis AfterclickingonSubmit,thefollowingappears: 147 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: 2. 148 Figure67:BenefitIncidenceAnalysis(unitcostapproach) AfterclickingonSubmit,thefollowingappears: 149 150
