Testing for Welfare Comparisons when Populations Differ in Size


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1 Cahier de recherche/workig Paper 039 Testig for Welfare Comparisos whe Populatios Differ i Size JeaYves Duclos Agès Zabsoré Septembre/September 200 Duclos: Départemet d écoomique, PEP ad CIRPÉE, Uiversité Laval, Caada Zabsoré: CIRPÉE, Uiversité Laval, Caada This work was carried out with support from SSHRC, FQRSC ad the Poverty ad Ecoomic Policy PEP Research Network, which is fiaced by the Govermet of Caada through the Iteratioal Developmet Research Cetre ad the Caadia Iteratioal Developmet Agecy, ad by the Australia Agecy for Iteratioal Developmet. We are grateful to Araar Abdelkrim, Sami Bibi, Joh Cockbur, Paul akdissi ad participats at the 200 SCSE ad CEA aual cofereces for useful commets ad advice.
2 Abstract: Assessmets of social welfare do ot usually take ito accout populatio sizes. This ca lead to serious social evaluatio flaws, particularly i cotexts i which policies ca affect demographic growth. We develop i this paper a littlekow though ethically attractive approach to correctig the flaws of traditioal welfare aalysis, a approach that is populatiosize sesitive ad that is based o criticallevel geeralized utilitariaism CLGU. Traditioal CLGU is exteded by cosiderig arbitrary orders of welfare domiace ad rages of poverty lies ad values for the critical level of how much a life must be miimally worth to cotribute to social welfare. Simulatio experimets briefly explore the ormative relatioship betwee populatio sizes ad critical levels. We apply the methods to household level data to rak Caada s social welfare across 976, 986, 996 ad 2006 ad to estimate ormatively ad statistically robust lower ad upper bouds of critical levels over which these rakigs ca be made. The results show domiace of recet years over earlier oes, except whe comparig 986 ad 996. I geeral, therefore, we coclude that Caada s social welfare has icreased over the last 35 years i spite or because of a substatial icrease i populatio size. Keywords: CLGU, welfare domiace, FGT domiace, estimatio of critical levels, welfare i Caada JEL Classificatio: C2, D3, D63, I30
3 Itroductio Is the value of a society icreasig with its populatio size? How ca that questio be dealt with i a ormatively robust framework? What sort of statistical procedures ca assess this empirically? What does the evidece actually suggest? To address these questios is the mai objective of this paper. Poverty ad welfare comparisos are routiely made uder the implicit assumptios that populatio sizes do ot matter, or equivaletly that populatio sizes are the same. Techically, this is implicitly or explicitly doe by callig o the socalled populatio replicatio ivariace axiom. The populatio replicatio ivariace axiom says that a icome distributio ad its kfold replicatio, with k beig ay positive iteger, should be deemed equivalet from a social welfare perspective. Welfare ad iequality comparisos ca the be performed i per capita terms. However, as Blackorby, Bossert, ad Doaldso 2005 ad others have argued, populatio size should probably matter whe assessig social welfare. We may ot be idifferet, for istace, to whether some icome or GDP statistics are expressed i per capita or i total terms. Whe total icome chages i a society, we may wish to kow whether this is due to chages i populatio size or chages i per capita icome; whe per capita icome chages, we may also wish to kow whether this is associated with a chage i populatio size. Geerally speakig, our assessmet of the welfare value of a chage i the distributio of icomes may deped o how populatio size also chages. I addressig these issues which we believe to be importat oes our work adopts as a coceptual framework for social welfare comparisos the criticallevel geeralized utilitariaism CLGU priciple of Blackorby ad Doaldso 984. CLGU essetially says that addig a perso to a existig populatio will icrease social welfare if ad oly if that perso s icome exceeds the value of a critical level. From a ormative perspective, the critical level ca be iterpreted as the miimum icome eeded for someoe to add value to humaity. The critical level has bee termed the value of livig by Broome 992b. Social welfare accordig to CLGU is the defied as the sum of the differeces betwee some trasformatio of idividual icomes ad the same trasformatio of the critical level. CLGU is a social evaluatio approach that is both ormatively attractive ad surprisigly little kow; it has also ot yet to our kowledge bee tested ad applied. There are, however, two major difficulties i implemetig CLGU. First, it is difficult i practice to agree o a oarbitrary value for the critical level. I a world of heterogeous prefereces ad opiios, it is ideed difficult to evisage a relatively wide cosesus o somethig as fudametally ucosesual as the value of livig. Secod, it is also difficult to agree o which trasformatio to apply to idividual icomes whe computig social welfare. We get aroud these difficulties i this paper by applyig stochastic domiace methods for makig populatio comparisos uder a CLGU framework. This avoids havig to specify a particular form for the trasformatio of idividual icomes. This also eables assessig the rages of critical levels over which ormatively robust CLGU comparisos ca be made. I a poverty 2
4 compariso cotext, it also makes it possible to derive the rages of poverty lies over which robust CLGU comparisos ca be obtaied. Although the paper s mai objective i this paper is to compare welfare through CLGU, the use of CLGU for social evaluatio purposes has importat implicatios for the desig of policy ad for the aalysis ad moitorig of huma developmet i geeral. Accordig to CLGU, the socially optimal populatio size maximizes the product of populatio size ad the differece betwee a sigleidividual socially represetative icome ad the critical level. This results i policy prescriptios that optimize the tradeoff betwee populatio size ad some measure of per capita wellbeig i excess of the critical level. For istace, the process of demographic trasitio through a reductio of both fertility ad mortality i which a large part of humaity has recetly egaged is ofte ratioalized as oe that maximizes per capita welfare uder resource costraits. It is ulikely for developed coutries that this process also maximizes social welfare i a CLGU perspective. As we will also see i our illustratio, Caada s CLGU has robustly icreased i the last 35 years despite a sigificat icrease i populatio size. For developed coutries, such a social evaluatio perspective ca thus provide a ratioale for promotig policies that ecourage fertility, such as the provisio of relatively geerous child beefits for families with may childre. Whether the curret demographic trasitio is cosistet with CLGU maximizatio i developig coutries depeds much o the value that is set for the critical level. A social plaer would favor a populatio icrease oly if the additioal persos ejoyed a level of icome at least equal to that level. This would be more difficult to achieve i less developed coutries, where average icome is lower relative to the critical level, so a smaller populatio might the be desirable. Optimal policies would the aim to icrease per capita icome ad raise social welfare by limitig demographic growth particularly of the poor people. This could ivolve compulsory measures of birth cotrol for the poor ad measures for icreasig the life years oly of the more affluet. The use of CLGU thus eables social evaluatios to be made whe the distributios ad policy outcomes to be compared ivolve varyig populatio sizes. These are certaily the most geerally ecoutered cases i theory ad i practice. This is also almost always the appropriate settig whe makig welfare comparisos across time. A few papers have recetly cosidered comparisos of populatios of uequal sizes without usig the replicatioivariace axiom. Oe of the most recet is Aboudi, Tho, ad Wallace 200, who geeralize the wellkow cocept of majorizatio ad suggest that a icome distributio should be deemed more equal tha aother oe if the first distributio ca be costructed from the secod distributio through liear trasformatios of icomes. Pogge 2007 proposes the use of the Pareto criterio to compare social welfare i icome distributios with differet umbers of idividuals. Cosiderig oly the most welloff persos i the larger populatio such that their umber be equal to the size of the smaller populatio, Pogge 2007 suggests that social welfare i the larger populatio should be greater tha i the smaller populatio if every perso i the larger populatio reduced to the size of the smaller oe ejoys a level of wellbeig greater tha that of every perso i the smaller popu 3
5 latio. Other relatively recet iterestig cotributios iclude Broome 992b, ukherjee 2008 ad Gravel, archat, ad Se Our paper differs from these earlier papers by focussig o how to rak distributios ad outcomes ormatively ad empirically usig CLGUbased domiace criteria. The paper s ormative settig is described i Sectio 2, where CLGU is itroduced ad motivated ad social welfare domiace relatios are defied. Sectio 2 also discusses how this relates to wellkow poverty domiace criteria. This domiace cotext exteds Blackorby ad Doaldso 984 s focus o CLGU idices. It also builds o the theoretical cotributio of Traoy ad Weymark 2009, who proposes a CLGU domiace criterio that is a extesio of geeralized Lorez domiace ad secodorder welfare domiace. Sectio 3 presets the statistical framework that is used for aalyzig domiace relatios, both i terms of estimatio ad iferece. It also develops the apparatus ecessary to estimate ormatively robust rages of critical levels. Sectio 4 provides the results of a few simulatio experimets that show how ad why populatio size may be of cocer ormatively ad statistically for social welfare rakigs. Sectio 5 applies the methods to comparable Caadia Surveys of Cosumer Fiaces SCF for 976 ad 986 ad Caadia Surveys of Labour ad Icome Dyamics SLID for 996 ad Caada s populatio size has icreased by almost 50% betwee 976 ad We assess whether social welfare has icreased or decreased over that period i Caada, allowig for variatios i populatio size ad icome distributios ad usig rages of poverty lies or cesorig poits ad values of critical levels. Usig asymptotic ad bootstrap tests, we fid that Caada s welfare has globally improved i the last 35 years despite the substatial icrease i populatio size ad the fact that ew lives do ot ecessarily icrease society s value i a CLGU framework. ore surprisigly perhaps, Caada s smaller populatio i 986 is evertheless socially better tha Caada s larger populatio i 996 for a relatively wide rage of critical levels ad despite a sigificat icrease i average ad total icome. Hece, ot oly ca average ad total utilitariaism preset sigificat ethical weakesses, but their social evaluatio rakigs ca differ importatly from those derived from criticallevel utilitariaism. Sectio 6 cocludes. 2 CLGU: a alterative approach to assessig social welfare 2. Average ad total utilitariaism The most popular methods to assess social welfare i the cotext of variable populatio sizes are based o average utilitariaism. Usig average utilitariaism as a social evaluatio criterio implicitly assumes that populatio sizes should ot matter. Oe cosequece of this is that a populatio with oly oe idividual will domiate ay other populatio of arbitrarily larger size as log as those larger populatios average utility is perhaps oly slightly 4
6 smaller tha the sigle perso s utility level see for istace Cowe 989, Broome 992a, Blackorby, Bossert, ad Doaldso 2005, ad Kabur ad ukherjee This social evaluatio framework would seem to be too biased agaist populatio size: it would say for istace that a society made of a sigle very rich perso Bill Gates for example would be preferable to ay other society of greater size but lower average utility. A alteratively popular social evaluatio criterio is total utilitariaism. Adoptig total utilitariaism leads, however, to Parfit 984 s repugat coclusio. Parfit 984 s repugat coclusio bemoas the implicatio that, with total utilitariaism, a sufficietly large populatio will ecessarily be cosidered better tha ay other smaller populatio, eve if the larger populatio has a very low average utility: For ay possible populatio of at least te billio people, all with a very high quality of life, there must be some much larger imagiable populatio whose existece, if other thigs are equal, would be better, eve though its members have lives that are barely worth livig. Parfit 984, p.388. Such a social evaluatio framework agai seems to be too strogly biased, this time agaist average utility. 2.2 Criticallevel geeralized utilitariaism Blackorby ad Doaldso 984 have proposed CLGU as a alterative to ad i order to address the flaws of average ad total utilitariaism. To see how CLGU is defied, cosider two populatios of differet sizes. The smaller populatio of size has a distributio of icomes or some other idicator of idividual welfare give by the vector u, ad the larger populatio of sizen has a distributio of icomes give by the vectorv, with < N. Let u := u,u 2,...,u, where u i beig the icome of idividuali, ad v := v,v 2,...,v N with v j beig the icome of idividualj. Let the level of social welfare i u ad v be give by ad W u;α = gu i gα i= W v;α = N gv j gα, 2 j= where g is some icreasig trasformatio of icomes ad α is a critical level. Note that social welfare i the two populatios remais uchaged whe a ew idividual with icome equal to α is added to the populatio. The smaller populatio exhibits greater social welfare tha the larger oe give this if ad oly ifw u;α W v;α. 5
7 CLGU thus aggregates the differeces betwee trasformatios of idividual icomes ad of a critical level. It ca therefore avoid some of average utilitariaism s problems, sice the additio of a ew perso will be socially profitable if that perso s icome is higher tha the critical level, although that icome may ot ecessarily be higher tha average icome. CLGU ca also avoid the repugat coclusio sice it is socially udesirable to add idividuals with icomes lower tha the critical level, regardless of how may there may be of them. Overall, CLGU provides a relatively appealig ad trasparet basis o which to make social evaluatios ad avoid the flaws associated to average ad total utilitariaism. Suppose ow that we may wish to focus o those icome values below some cesorig poit z. This is a typical procedure i poverty aalysis. Suppose that z is the maximum possible level for such a cesorig poit or maximum poverty lie i a poverty cotext. Also deote u α := u,α,...,α as u expaded to size of populatio v by addig N α elemets. For a poverty lie z, the wellkow FGT Foster, Greer, ad Thorbecke 984 poverty idices with parameter s order s i what follows for distributio v are defied as P s v z = N N z v j s Iv j z, 3 i=j where I is a idicator fuctio with value set to if the coditio is true ad to 0 if ot. Similarly, the FGT idices for the expaded populatiou α are defied as P s u α z = N z u i s I u i z i= N These expressios will be useful to test for CLGU domiace. 2.3 CLGU domiace z α s Iα z. 4 The welfare fuctios i ad 2 deped o g ad α. Oe could choose a specific fuctioal form for g ad a specific value for α, but that would be icoveiet i the sese that the welfare rakigs of u ad v could the be criticized as depedig o those choices. It is thus useful to cosider makig welfare rakigs that are valid over classes of fuctios g ad rages of critical levels α. To do this, let s =,2,..., stad for a order of welfare domiace. Cosider C s as the set of fuctios R R that are s times cotiuously 6
8 differetiable. Defie the classf s z,z of fuctios as F s z,z := g z C s z z, g z x = g z z for all x > z, g z x = g z z for all x < z, ad where k dk g z x 0 k =,...,s. dx k Also deote W s α,z,z as the set of CLGU social welfare fuctios with g z F s z,z ad critical levelα. For ay vector of icomev R N, N, this set is defied as: N } Wα,z s,z {W := W v;α = g z v i g z α whereg z Fz s,z adv RN. i= 6 The first ad third lies i 5 say that the cesorig poit z must be below some upper level z. The secod lie says that for social evaluatio purposes we ca set to z those icomes that are lower tha z this assumptio is mostly made for statistical tractability reasos, to which we come back later. The fourth lie o the derivatives ofg z imposes that the social welfare fuctios be Paretia fork =, be cocave ad thus icreasig with a trasfer from a richer to a poorer perso fork = 2, be trasfersesitive i the sese of Shorrocks 987 for k = 3, etc.. The greater the order s, the more sesitive is social welfare to the icome levels of the poorest. We ca the defie the partial CLGU domiace orderig sw α,z,z as u sw α,z,z v W u;α W v;α W Ws α,z,z. 7 The welfare orderig 7 cosiders u to be better tha v if ad oly if W u;α is greater thaw v;α for all of the fuctiosw that belog tow s α,z,z. Similarly, defie the partial FGT domiace orderig sp z,z as u α sp z,z v Ps u α z P s vz 0 for allz z z. 8 This FGT orderig 8 cosiders u to be better tha v if ad oly if the FGT curve P s u α z foru α is always below the FGT curvep s v z forvfor all values ofz z z. Duclos ad Zabsoré 2009 demostrate that the two partial orderigs are equivalet, for someα,z ad z : u sw α,z,z v u α sp z,z v. 9 This result is used as a foudatio for the statistical ad the empirical aalysis of the rest of the paper. The curret paper uses i fact a atural extesio of 9 by focussig o domiace over a rage of critical levelsα α,α ]: u sw α,z,z v, α α,α ] u α sp z,z v, α α,α ]
9 This provides us with a social orderig that is robust over a class s of fuctios g ad over rages z,z ] ad α,α ] of cesorig poits ad critical levels. 3 Statistical iferece This sectio develops methods to ifer statistically the above domiace relatios. For the purpose of statistical iferece, we assume that the populatio data have bee geerated by a data geeratig process DGP from which a fiite but usually large populatio is geerated. For some but ot for all of the results, we will eed to assume that this DGP is cotiuous, but this is differet from sayig that the populatios must be cotiuous or of ifiite size too. For purposes of iferece o the populatios, we will use data provided by a fiite typically relatively small sample of observatios draw from the populatios. We defie F ad G as the distributio fuctios of the DGP that geerate the populatio vectors u ad v respectively. 3. Testig domiace The equivalece betwee FGT domiace ad CLGU domiace coveietly allows focusig o FGT domiace. As above, letαdeote the critical level adα be the maximum possible value that we assume this critical level ca take. For ay poverty lie z, defie the FGT idex of order s s for the expaded populatiou α as P s F α z = z 0 z u s df α u, where F α z := N Fz Iα z is the distributio of the expaded populatio u N N α ad Fz is the distributio fuctio of u. The FGT idex of the populatio v is similarly defied as P s Gz = z 0 z v s dgv. 2 The task ow is to itroduce procedures to test for whether a populatio CLGUdomiates aother oe at order s, ad this, over itervals of cesorig poits ad critical levels. Two geeral approaches ca be followed for that purpose. The first is based o the followig formulatio of hypotheses: H s 0 : P s G z Ps F α z 0 for all z,α z,z ] α,α ], 3 H s : P s Gz P s F α z > 0 for some z,α z,z ] α,α ]. 4 8
10 This formulatio leads to what are geerally called uioitersectio tests. It amouts to defie a ull of domiace ad a alterative of odomiace. The ull above is that v domiates u, but that ca be reversed. It has bee used ad applied i several papers where a Wald statistic or a test statistic based o the supremum of the differece betwee the FGT idices is geerally used to test for domiace see for example Bishop, Formby, ad Thistle 992 ad Barrett ad Doald 2003 ad Lefrac, Pistolesi, ad Traoy Davidso ad Duclos 2006 discuss why this formulatio leads to decisive outcomes oly whe it rejects the ull of domiace ad accepts odomiace. This, however, fails to order the two populatios. I those cases i which it is desirable to order the populatios, it may be useful to use a secod approach ad reverse the roles of 3 ad 4 by positig the hypotheses as H s 0 : P s Gz P s F α z 0 for some z,α z,z ] α,α ], 5 H s : P s G z Ps F α z < 0 for all z,α z,z ] α,α ]. 6 This formulatio leads to itersectiouio tests, i which the ull is the hypothesis of odomiace ad the alterative is the hypothesis of domiace. This test has bee employed by Howes 993 ad Kaur, Prakasa Rao, ad Sigh 994. Both papers use a miimum value of the tstatistic. A alterative test is based o empirical likelihood ratio ELR statistics, first proposed by Owe 988 see also Owe 200 for a comprehesive accout of the EL techique ad its properties. Here, we follow the procedure of Davidso ad Duclos 2006, which ca also be foud i Bataa 2008, Che ad Duclos 2008 ad Davidso Ulike these papers, we must, however, pay special attetio to the value of the critical level ad to the sizes of the two populatios. Letmadbe the sizes of the samples draw from the populatiosuadvrespectively ad let w i u ad w j v be the samplig weights associated to the observatio of idividualii the sample of u ad idividualj i the sample of v respectively. Suppose also that u i, w i u ad vj, w j v are idepedetly ad idetically distributed iid across i ad j. For the purposes of asymptotic aalysis, defiew u i adw v j such that w u i = m w u i ad w v j = w v j. 7 These quatities ca be used ad iterpreted as estimates of the populatio sizes of u ad v respectively. They remai of the same order asmadted to ifiity. We ca the compute ˆP F s α z ad ˆP G sz, which are respectively the sample equivalets ofps F α z adpg s z. They are give by 9
11 ˆP s F α z = m i= m w u i z u i s i= w u i / / j= j= w v j w v j ] z α s 8 ad ˆP s G z = j= w v j z v j s / j= w v j, 9 wherez x s z xs Ix z for ay icome valuex. We use the above to compute a ELR statistic. Letp u i adpv j be the empirical probabilities associated to observatios i ad j respectively. The ELR statistic is similar to a ordiary LR statistic, ad is defied as twice the differece betwee the ucostraied maximum of a empirical loglikelihood fuctio ELF ad a costraied ELF maximum. Subject to the ull 5 that u domiates v at some give value of z ad α, the costraied ELF maximum ELF z,α is give by m ] ELF z,α = max logp u p u i logp v j 20 i,pv j subject to i= j= ad p u i =, p v j = 2 i= j= i= p u i w u i z u i s p v jwj v j= p u i wi u z α s i= j= p v jw v j z v j s. 22 The ucostraied maximum ELF is defied as 20 subject to 2. Notice that 22 ca also be rewritte as p u i wi u i= z ui s z αs ] p v jwj v j= z vj s ] z αs. 23 0
12 I the spirit of Davidso ad Duclos 2006, we compute the ELR statistic for all possible pairs of z,α z,z ] α,α ], so that we ca ispect the value of that statistic whe the ull hypothesis i 5 is verified at each of these pairs separately. The fial ELR test statistic is the give by where LR = mi LRz,α, 24 z,α z,z ] α,α ] LRz,α = 2ELF ELF z,α]. 25 Whe, i the samples, there is odomiace of u o v at some value of z ad α i z,z ] α,α ], the costrait 23 does ot matter ad the costraied ad ucostraied ELF values are the same. The resultig ucostraied empirical probabilities are give by p u i = m adpv j =. 26 I the case where there is domiace i the samples, the costrait 23 bids ad the probabilities obtaied from the resolutio of the problem are: ad p u i = m ρ ν w u i z ui s ] 27 z αs p v j = ρ ν wj v z vj s The costatsρad ν are the solutios to the followig equatios, i= p u i wu i p v j wv j j= z ui s z vj s ] z αs = z αs ]. 28 z αs p v j wv j z vj s ] z αs j= ] 29 = ν, with p u i ad p v j give i 27 ad 28. The solutios caot be foud aalytically, so a umerical method must be used. A alterative, though aalogous, statistic is the tstatistic of Kaur, Prakasa Rao, ad Sigh 994, which is the miimum oftz,α overz,z ] α,α ], where tz,α = ˆP G sz ˆP F s α z varˆps G z ˆP ] /2, 30 F s α z
13 ad varˆps G z ˆP F s α z is the estimate of the asymptotic variace of ˆP G sz ˆP F s α z for some pairz,α. Deote that miimizedtstatistic by t. Testig the ull of domiace makes sese oly whe there is domiace i the origial samples.we ca the proceed with asymptotic tests ad/or bootstrap tests with either LR or t statistics, although for bootstrap tests we must first obtai the empirical probabilities of the ELF approach. LetLR a adt a deote the statistics i the case of asymptotic tests ad letlr b adt b be the statistics for the bootstrap tests. For asymptotic tests ad for a test of levelβ, the decisio rule is to reject the ull of odomiace i favor of the alterative of domiace if t a exceeds the critical value associated toβ of the stadard ormal distributio. Note thatlr ad the square oftare asymptotically equivalet see Sectio 8.3 i the Appedix for more details. We ca therefore also use a decisio rule of rejectig the ull of odomiace i favor of the alterative of domiace if LR a exceeds the critical value associated to β of the chisquare distributio. The bootstrap testig procedure is formally set up as follows: Step : For two iitial samples draw from two populatios, compute LRz,α ad tz,α for every pair z,α i z,z ] α,α ] as described above. If there exists at least oe z,α for which ˆP s G z ˆP s F α z 0, the H s 0 caot be rejected; choose the a value equal to for thepvalue ad stop the process. If ot, cotiue to the ext step. Step 2: Search for the miima statistics, that is to say, fid LR as the miimum of LRz,α ad t as the miimum of tz,α over all pairs z,α. Suppose that LR is obtaied at z, α ad deote p u i ad p v j the resultig probabilities give by 27 ad 28 ad evaluated at z, α. Step 3: Use p u i ad p v j to geerate bootstrap samples of size m for u ad of size for v by resamplig the origial data with these probabilities. The bootstrap samples are thus draw with uequal probabilities p u i ad p v j. It ca result that, i some of the bootstrap samples, the estimated size of populatio u becomes larger tha that of populatio v. I such cases, the roles of F α ad G are subsequetly reversed, that is, we cosider F adg α. Step 4: As is usual, cosider 399 bootstrap replicatios, b =,...,399. For each replicatio, use the bootstrap data ad follow previous step 3. Compute the two statistics LR b ad t b for everyb 399 as i the origial data. Step 5: Compute thepvalue of the bootstrap statistics as the proportio oflr b that are greater thalr the ELR statistic obtaied with the origial data or as the proportio of t b that are greater tha t thetstatistic obtaied with the origial data. Step 6: Reject the ull of odomiace if the bootstrap pvalue is lower tha some specified omial levels. This will ot occur i samples where all observatios have the same weights. 2
14 3.2 Estimatig robust rages of critical levels To get aroud the problem of the absece of empirical/ethical cosesus o a appropriate rage of values for the critical level, we ca search for evidece o the rages of critical levels that ca order distributios see Blackorby, Bossert, ad Doaldso 996 ad Traoy ad Weymark 2009 for a discussio. For this, cosider agai two populatios u ad v of sizes ad N respectively. Suppose that we have two samples draw from u ad v ad assume for simplicity that they are idepedet ad that their momets of order 2s are fiite. Deote m ad the sizes of the two samples. For some fixed z ad z, defie α s ad α s respectively as follows: ad α s = max{α P s F α z P s G z for all z z z } 3 α s = mi{α P s F α z P s Gz for allz z z }. 32 I the light of how they are defied, we ca refer to α s as a upper boud of the critical level ad α s as a lower boud of the critical level. I order to have FGT domiace made robustly over rages of cesorig poits, we ca also defie critical values for the maximum cesorig poit as: ad z s = max{z P s F α z P s Gz for all z z z } 33 z s = max{z P s F α z P s G z for all z z z }, 34 where α is some fixed value of critical level. z s is the maximum cesorig poit for which v domiatesuad z s is the maximum cesorig poit for whichudomiatesv. Give the defiitios 3 ad 32 ad assumig thatα s adα s exist, it is useful to defie the followig assumptios. Forα s, suppose that { N Ps F z Ps G z for all z α s N Ps F z < Ps G z for somez α VDU s ǫ ad z z z s, where ǫ is some arbitrarily small positive value. For α s, cosider first the case of s = ad suppose that { N P F N P F z N N Iα z P G z for all z z z z N N > P G z for somez α ǫ, UDV where ǫ is agai some arbitrarily small positive value. Whe s 2, we modify the above assumptios slightly ad defieα s as: 3
15 { N Ps F z N N N Ps F zs N N zs α s s z α s s Ps G z for allz z z = P s G zs forα s < z s z UDV s with z x s = maxz x s,0]. Suppose that z s exists ad is the crossig poit betwee the FGT curves PF s α ad PG s. I most cases, we would expect zs to coicide with z see Sectio 8. i the Appedix for more details. Assumptios VDU s ad UDV s are useful for the estimatio of α s ad α s. I order to better uderstad their role, cosider the case of s =. Figures ad 2 graph cumulative distributios fuctios adjusted for differeces i populatio sizes. 2 It is supposed that the larger populatio v domiates the smaller populatio u for a rage 0,z ] of cesorig poits α 0,α ]. This is expressed by the fact that the cumulative distributiog ofvis uder the cumulative distributiof ofu adjusted by the ratio up toα N > 0. Figure shows the case where the critical level is equal to 0. I this case, the larger populatio clearly domiates the smaller oe. At the critical level value α, the two fuctios cross; v just domiates u whe the critical level is equal to α. However,v does ot domiateuwhe the critical level takes a valueα 0 > α. I Figure 3, u is assumed to domiate v. The domiace of u over v is preserved whe the critical level has a value at least equal to α. But this is ot true for ay critical level α 0 lower thaα. 3 Note that α ad α are the crossig poits of FGT curves. This suggests the applicatio of the procedure of Davidso ad Duclos 2000 for estimatio ad iferece of the populatio values ofα adα. Cosider the populatiosuadvwith sample sizes equal tomad respectively. Usig assumptioudv also see Figure 3 ad assumig cotiuity of the DGP at α, we have that N N P F α PG N α = Deotigψz = N P N F z N Iα z PG z, theψz 0 for allz z z adψα = 0. Recall that m ˆP F z = m i= wu i Iu i z, ˆP G z = m m i= wu i j= wv j Iv j z, 36 j= wv j where wi u ad wj v are give i the previous sectio. A atural estimator of α would be α such that ˆ ˆN ˆ ˆP F ˆN ˆα ˆN ˆP G ˆα = 0, 37 2 See Sectio 8. i the Appedix for the case ofs >. 3 The Appedix illustrates graphically two cases of domiace ofuoverv whes >. 4
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