Working Papers in Econometrics and Applied Statistics

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1 Workng Papers n Econometrcs and Appled Statstcs Estmatng Lorenz Curves Usng a Drchlet Dstrbuton Duangkamon Chotkapanch and Wllam E Grffths No. November 999 Workng Papers n Econometrcs and Appled Statstcs Department of Econometrcs Unversty of New England Armdale, NSW 235, Australa. ISSN ISBN

2 Estmatng Lorenz Curves Usng a Drchlet Dstrbuton Duangkamon Chotkapanch Curtn Unversty of Technology and Wllam E Grffths Unversty of New England Abstract The Lorenz curve relates the cumulatve proporton of ncome to the cumulatve proporton of populaton. When a partcular functonal form of the Lorenz curve s specfed t s typcally estmated by lnear or nonlnear least squares assumng that the error terms are ndependently and normally dstrbuted. Observatons on cumulatve proportons are clearly nether ndependent nor normally dstrbuted. Ths paper proposes and apples a new methodology whch recognzes the cumulatve proportonal nature of the Lorenz curve data by assumng that the proporton of ncome s dstrbuted as a Drchlet dstrbuton. Fve Lorenz-curve specfcatons were used to demonstrate the technque. Once a lkelhood functon and the posteror probablty densty functon for each specfcaton are derved we can use maxmum lkelhood or Bayesan estmaton to estmate the parameters. Maxmum lkelhood estmates and Bayesan posteror probablty densty functons for the Gn coeffcent are also obtaned for each Lorenz-curve specfcaton. Keywords: posteror dstrbuton; Metropols-Hastngs algorthm; Gn coeffcent;

3 3. Introducton The Lorenz curve s one of the most mportant tools upon whch the measurement of ncome nequalty s based. For a gven economy or regon, t relates the cumulatve proporton of ncome to the cumulatve proporton of populaton, after orderng the populaton accordng to ncreasng level of ncome. Two general approaches to Lorenz curve estmaton have been adopted. In the frst, a partcular assumpton about the statstcal dstrbuton of ncome s made, the parameters of ths ncome dstrbuton are estmated, and a Lorenz curve consstent wth the dstrbutonal assumpton, and consstent wth the parameter estmates for that dstrbuton, s obtaned. See, for example, McDonald 984 and McDonald and Xu 995. In the second approach, a partcular functonal form for the Lorenz curve s specfed and estmated drectly. It s ths second approach whch s the focus of ths paper. Early breakthroughs on Lorenz curve estmaton were those of Gastwrth 972 and Kakwan and Podder 973, 976. Kakwan and Podder recognzed the multnomal nature of grouped data and used a Lorenz curve specfcaton that, after transformaton, could be placed n an approxmate lnear model framework. Other specfcatons have typcally been estmated by lnear or nonlnear least squares wthout any regard for the fact that the assumpton of ndependent normally dstrbuted errors s unrealstc Kakwan 98, Basmann et al 99, Chotkapanch 993. Clearly, observatons on cumulatve proportons, or even ther logarthms f such a transformaton s convenent, wll be nether ndependent nor normally dstrbuted. Saraba et al 999 overcome ths problem

4 4 by suggestng a dstrbuton-free method of estmaton. Suppose that a Lorenz curve has n unknown parameters, and that M observatons on the cumulatve proportons are avalable. They fnd a set of parameter estmates for each of the K = M n subsets of n observatons. Snce each of the subsets yelds n equatons n n unknown parameters, a set of parameter estmates s obtaned by solvng these equatons. The medans of the sets of parameter estmates are recommended as the fnal set of estmates. No dstrbuton theory s avalable for ths procedure, but the authors do provde some bootstrap standard errors. An alternatve way to proceed, and the approach adopted n ths paper, s to choose a dstrbutonal assumpton that s consstent wth the proportonal nature of the data and to pursue maxmum lkelhood or Bayesan estmaton. Maxmum lkelhood estmators have well known statstcal propertes, and Bayesan estmaton provdes a framework for fnte sample nference wth several well recognzed advantages. See, for example, Porer 995. One multvarate dstrbuton whch has shares whch sum to one as ts vector of random varables s the Drchlet dstrbuton. By relatng the parameters of the Drchlet dstrbuton to Lorenz curve dfferences, we can allow for the cumulatve proportonal nature of the Lorenz curve data, and set up a lkelhood functon dependent on the unknown parameters of the Lorenz curve. A smlar approach was adopted by Woodland 979 for estmaton of share equatons that arse n demand and producton theory. Although our dscusson and examples relate to the use of grouped data, our methodology could also be appled to unt recorded data.

5 5 In Secton 2, we outlne the dstrbutonal assumptons and how they relate to Lorenz curve estmaton. The lkelhood functon and a general posteror probablty densty functon pdf for a set of unknown Lorenz curve parameters are derved. A Metropols-Hastngs algorthm that can be used to estmate margnal posteror pdfs for the parameters and ther moments s descrbed. To llustrate our suggested technques we use data on Sweden and Brazl consdered earler by Shorrocks 983 and revsted by Saraba et al 999. These data are descrbed n Secton 3; fve dfferent Lorenz functons that we use n the emprcal work are presented. The results are gven and dscussed n Secton 4. Several questons are nvestgated. To examne whether the results are senstve to the chosen estmaton technque we compare our estmates and ther standard errors and posteror standard devatons to those obtaned by Saraba et al 999, and those obtaned usng least squares after takng logarthms where relevant. Snce Lorenz-curve estmaton s usually a frst step towards estmatng nequalty, maxmum lkelhood ML estmates and Bayesan posteror pdfs for the Gn coeffcent are obtaned for each Lorenz-curve specfcaton. A comparson of the ML and Bayesan results gves an ndcaton of any dfferences between asymptotc and fnte sample nferences. Fnally, we examne whether functonal form preference s senstve to the chosen estmaton technque and form of nference. 2. Models, Assumptons and Estmaton Suppose we have avalable observatons on cumulatve proportons of populaton π, π2,, πm K wth π M = and correspondng cumulatve proportons of ncome η, η2, K, ηm wth η M = obtaned after orderng populaton unts

6 accordng to ncreasng ncome. We wsh to use these observatons to estmate a parametrc verson of a Lorenz curve that we wrte as η = L π; β where β s an n vector of unknown parameters. Clearly, one would not expect all data ponts to le exactly on the curve η = L π; β. It seems reasonable to assume, 6 however, that condtonal on the populaton proportons π, the ncome shares q = η η are random varables wth means E q = E η E η = L π; β L π ; β Our proposal s to also assume q = q, q, K, q ' follows a Drchlet 2 M dstrbuton whch s a dstrbuton consstent wth the share nature of the random vector q. The probablty densty functon pdf for the Drchlet dstrbuton s gven by f q α = Γ α Γ α + α + L+ α 2 M α α2 q q 2 Γ α2 KΓ αm Kq αm M 2 where α = α, α, K, α ' are the parameters of the pdf and Γ. s the gamma 2 M functon. By relatng the α to the Lorenz functon, we can fnd a pdf for q whch has the mean gven n equaton and whch s a functon of the Lorenz curve parameters. Workng n ths drecton, we set α = [ π ; β L π ; β ] λ L 3 where λ s an addtonal unknown parameter. Ths defnton for α gves the desred result because the mean of the Drchlet dstrbuton s gven by

7 7 [ ] = β π β π λ β π β π λ = + α + + α α α = M M L L L L q E 2 ] ; ; [ ; ; L ; ; β π β π = L L 4 snce ; = β π M L and ; = β π L. We can now wrte the pdf for q as = β π β π λ β π β π λ Γ λ = Γ θ M L L L L q q f ] ; ; [ ] ; ; [ 5 where ', ' λ β = θ. The varances and covarances between the shares are gven by ] [ var λ + = q E q E q 6, cov λ + = j j q E q E q q 7 Thus, the ncome shares are correlated, wth correlatons gven by 2 / ] ][ [ = j j j q E q E q E q E r 8 Snce the varances depend on q E, the shares are also heteroskedastc. The parameter λ acts as an nverse varance parameter. The larger the value of λ, the better the ft of the Lorenz curve to the data.

8 The maxmum lkelhood estmate for θ can be found by maxmzng the loglkelhood functon 8 log[ f q θ] = log Γ λ + M = M = λ[ L π ; β L π log Γ λ[ L π ; β L π ; β] ; β] log q 9 For Bayesan estmaton we use unform prors on the elements of β, over the feasble ranges for those parameters. Snce λ + s lke an nverse varance parameter, we use a unform pror for log λ +. Also, assumng a pror ndependence of β and λ, yelds the pror pdf I β f θ = f β, λ λ > λ + where I β s am ndcator functon equal to unty for feasble values of β and zero f β falls outsde the regon that defnes L π; β as a Lorenz curve. Applcaton of Bayes theorem nvolves multplyng together equatons 5 and to obtan the kernel of the posteror pdf for θ f θ q f θ f q θ For all the Lorenz-curve specfcatons that we estmate, the posteror pdf n s analytcally ntractable n the sense that we cannot carry out the necessary ntegraton to obtan margnal posteror pdfs for ndvdual parameters and the posteror moments of these parameters. These quanttes can be estmated, however, by usng a Metropols-Hastngs algorthm to draw observatons on θ from the posteror pdf f θ q. See, for example, Albert and Chb 996 and Geweke 999. We used the followng random-walk algorthm wth the

9 maxmum lkelhood covarance V θ used as a covarance matrx for the randomwalk generator functon. The steps for drawng the m + th observaton θ m + are: 9. Draw a canddate value * θ from a N θ m, cv dstrbuton where c s a θ scalar set such that 2. Compute * θ s accepted approxmately 4-5% of the tme. r = * f θ f θ m q q Note that ths rato can be computed wthout knowledge of the normalsng constant for f θ q. Also, f any of the elements of * θ fall outsde the feasble parameter regon, then f θ * q =. 3. Draw a value u for a unform random varable on the nterval,. 4. If u r, set * θ m + = θ. If u > r, set θ m + = θ m. 5. Return to step, wth m set to m +. Observatons generated n ths way can be placed n hstograms to estmate margnal posteror pdfs, and sample means and standard devatons can be used to estmate posteror means and standard devatons. 3. Data and Lorenz Curves To llustrate our suggested technques we use ncome dstrbuton data on natonal samples of ncome recpents for a year close to 97, for two countres: Sweden and Brazl. These data were used by Saraba et al 999. They were derved from Jan 975 and frst publshed n Shorrocks 983. The data are n the form of decle cumulatve ncome shares. Shorrocks used the data on these two countres as part of a group of twenty countres to examne the rankng of ncome dstrbutons gven dfferent socal states. Saraba et al 999 used the data to llustrate ther proposed method for the estmaton of Lorenz curves. The

10 data on these two countres were chosen because of ther dfferences n the degree of nequalty n ncome dstrbutons. A large number of functonal forms have been suggested n the lterature for modellng the Lorenz curve. For detals of the varous alternatves, see Saraba et al 999, and references theren. To keep our study manageable, we chose only 5, rangng from one smple functon wth only one unknown parameter, to two three-parameter functons whch are more flexble, but also harder to estmate precsely. The 5 dfferent Lorenz functons to whch we appled the two data sets are: kπ e L π; k = k > 2 k e L π; α, δ = π [ π ] α, < δ 3 2 α δ γ δ L π ; δ, γ = [ π ] γ, < δ 4 3 α δ γ L π ; α, δ, γ = π [ π ] α, γ, < δ 5 4 d 5 ; a, b, d = π aπ π b L π a >, < d, < b 6 The functon L s the relatvely smple one-parameter functon suggested by Chotkapanch 993; L 2 concdes wth the proposal of Ortega et al 99. L3 s a well-known form of Lorenz curve suggested by Rasche et al 98 and L 4 s an extenson of L 3 and L 2 ntroduced by Saraba et al 999. Note that L 4 nests both L 2 and L 3, wth L 2 beng L 4 wth γ = and L 3 beng L 4 wth α =. Settng both γ = and α = yelds the Lorenz curve L = π whch δ orgnates from the classcal Pareto dstrbuton. The functon L 5 s the beta functon proposed by Kakwan 98. It s consdered one of the best performers among a number of dfferent functonal forms for Lorenz curves. See, for example, Datt 998. Note that, when a = and d =, L 5 s the same as L2 wth α =.

11 Once a Lorenz curve has been estmated, one s usually nterested n varous nequalty measures that are related to t. As an example, we compute maxmum lkelhood estmates and posteror pdfs for the Gn coeffcents that can be derved from each of the Lorenz functons. In each case the Gn coeffcent s defned as G = 2 L π; β dπ 7 Alternatve expressons for G can be found for some of the Lorenz curves. However, wth the excepton of L, they stll generally nvolve a numercal ntegral. We obtan ML and Bayesan estmates by numercally evaluatng 7 n each case. For ML estmaton, numercal ntegraton s performed wth β replaced by the ML estmate βˆ. For Bayesan estmaton, the ntegral s evaluated for each draw of β from the posteror pdf of β. 4. Results In addton to ML and Bayesan estmaton usng the assumpton of a Drchlet dstrbuton, we also estmated each functon usng nonlnear least squares. Nonlnear least squares s optmal under the assumpton that the η are ndependent normally dstrbuted random varables wth mean L, β and constant varance. Although ths assumpton s not realstc for data whch are cumulatve proportons, nonlnear least squares s a popular estmaton technque, and so the senstvty of parameter estmates to the choce of technque s useful nformaton. π Pont estmates of the Lorenz curve parameters and the correspondng Gn coeffcents for Sweden and Brazl are presented n Tables and 2, respectvely. The Bayesan pont estmates are the posteror means estmated from 75,

12 draws usng the random-walk Metropols algorthm, after dscardng the frst, draws as a burn n. The estmates obtaned by Saraba et al 999, usng ther proposed technque, are also gven for L 2, L 3 and L 4. [Table near here] Table provdes the estmates for Sweden. For L, L2, L3 and L 5 the estmates of the Lorenz parameters and the Gn coeffcents are not senstve to the estmaton technques. For L 4 dfferent estmaton technques gve very dfferent Lorenz parameter estmates. Despte these dfferences, the estmates for the Gn coeffcent are very smlar across all functonal forms and estmaton technques. An excepton s the one obtaned from L 4 usng Saraba s method. Reasons for the atypcal outcomes from L 4 are addressed later. [Table 2 near here] The remarks made about Sweden also hold for the estmates for Brazl gven n Table 2. One dfference s the Gn coeffcent estmates obtaned from ML and Bayes, when usng L. They are.5 and.52, when all other estmates are approxmately.63. When we dscuss goodness of ft, we dscover that ths dfference can be attrbutable to a poor ft. Tables and 2 also reveal the dfference n nequalty n Sweden and Brazl, wth Sweden exhbtng the lower level of nequalty. 2 Standard errors for the ML and nonlnear least squares estmates, and posteror standard devatons for the parameters from Bayesan estmaton, are presented n Tables 3 and 4 for Sweden and Brazl, respectvely. The posteror standard devatons are estmated from the 75, Metropols draws, and correspondng

13 3 values of the Gn coeffcent. The standard errors for the Gn coeffcent for ML and nonlnear least squares were calculated usng the asymptotc approxmaton G G var Gˆ = Vβ 8 β' β where V β s the asymptotc covarance matrx for the ML or nonlnear least squares estmator for β. Expressons derved usng 8 for each of the Lorenz curves are gven n the Appendx. [Tables 3 and 4 near here] From Tables 3 and 4, we make the followng observatons:. Wth the excepton of L 4, to whch specal attenton s devoted later, the Bayesan posteror standard devatons are larger than the ML standard errors. Snce the ML standard errors are large-sample approxmatons, whereas the posteror standard devatons reflect fnte sample uncertanty, ths comparson reveals the extent to whch msleadng nferences can be made from a large-sample approxmaton. To llustrate ths pont further, we plotted the estmated posteror pdfs for α n the functon L 2 for Sweden Fgure, the Gn coeffcent from L 4 for Sweden Fgure 2, and the Gn coeffcent from L 5 for Brazl Fgure 3. Normal pdfs, centred at the ML estmates, and wth standard devatons equal to the ML standard errors, were also drawn on these fgures. When vewed through Bayesan eyes, these are the pdfs typcally used to make large sample nferences. In all three fgures, the Bayesan pdfs have fatter tals, suggestng that ML estmaton understates the uncertanty about these quanttes.

14 2. The bootstrap standard errors computed by Saraba et al 999 are vastly dfferent from those provded by the other approaches. The dfference s suffcently great to cast doubt on ther valdty, partcularly when the dstrbuton theory for the Saraba et al technque s not avalable. 3. The standard errors for nonlnear least squares whch s optmal when the cumulatve ncome proportons are normally dstrbuted are also qute dfferent. Thus, although the pont estmates of the Lorenz parameters and the Gn coeffcent are qute nsenstve to the chosen estmaton technque, nterval estmates, and the assessment of estmaton precson, depend heavly on the dstrbutonal assumpton and related method of estmaton. 4. Overall, pont estmates of the Gn coeffcent are nsenstve to the Lorenz curve specfcaton. Those for L from ML and Bayes, usng the Brazlan data, are exceptons. There s, however, consderable varaton n the standard errors and posteror standard devatons. Thus, our knowledge or degree of uncertanty about the value of the Gn coeffcent does depend on the functonal form chosen for the Lorenz curve. Ths fact s clearly depcted by the posteror pdfs that are graphed n Fgures 4 and 5. Fgure 4 contans the posteror pdfs for Sweden s Gn coeffcent, obtaned usng L,L 4 and L 5. The 3-parameter Lorenz curves L 4 and L 5 suggest relatvely precse nformaton about the Gn coeffcent. The -parameter functon L exhbts consderable uncertanty. Fgure 5 contans the posteror pdfs for Brazl s Gn coeffcent, obtaned usng L 2, L 4 and L 5. Here, the story s smlar, except that the precson n estmaton mpled by L 5 s much greater than that mpled by L 2 and L 4. 4

15 5 We turn now to the queston of goodness of ft. Whch of the Lorenz functons best fts the data? As we wll see, the answer to ths queston has a bearng on precson of estmaton that we dscussed under the last pont 4. The problem of choosng between the alternatve functons can be addressed n a number of ways. For a straght goodness-of-ft comparson, we compare values of nformaton naccuracy Thel 967, 975. For testng nested functonal forms we use lkelhood rato tests for the ML estmates; from a Bayesan perspectve, we assess whether varous parametrc restrctons are true by examnng the posteror probablty n the regon near the restrctons. Let qˆ denote the predcted ncome shares obtaned from an estmated model. Thel s 967 measure of nformaton naccuracy s defned as I = M q q = qˆ log 9 Functons wth smaller values of I are better fts than those wth larger values. If the q are smlar to the qˆ, then knowng ther values provdes lttle nformaton relatve to knowledge of the predctons. The functon s a good ft. On the other hand, q qute dfferent from the qˆ convey consderable nformaton, leadng to a large value of I and a poor ft. The nformaton naccuracy measure was computed usng predctons from the ML estmates, and predctons from the Bayesan posteror means. The outcomes are presented n Table 5. In both countres, L 5 s the best ft, L 4 and L 3 are approxmately the same n terms of ft, and are preferred to L 2, whch, n turn, s preferred to L. There s vrtually no dfference n the measures obtaned from the

16 ML estmates and those obtaned from the Bayesan estmates. There s a dfference between Sweden and Brazl, however. For Brazl, the ft of the best functon L 5 s much better, and the ft of the worse functon L, s worse. Also, for Sweden, the functon L 2 s only margnally worse than L 3 and L 4. In the case of Brazl t s notceably nferor. 6 It s nterestng that the precson wth whch the Gn coeffcent s estmated s drectly related to how well the functon fts the data. The relatve magntudes of the posteror standard devatons for the Gn coeffcents Tables 3 and 4 reflect the relatve magntudes of the nformaton naccuracy measures. These relatvtes are also conveyed by the posteror pdfs n Fgures 4 and 5. The second way that we nvestgated choce of functonal form was by examnng whether nested versons of L 4 and L 5 would be adequate. Gven the results on goodness of ft, one would expect that at least L 3 would be an acceptable restrcted verson of L 4. Table 6 contans 2 χ values for lkelhood rato tests for varous hypotheses. These results confrm our conjecture about the relatonshp between L 3 and L 4 for both Sweden and Brazl. Also, L 2 s an acceptable restrcted verson of L 4 for Sweden, but not for Brazl, a concluson consstent wth goodness-of-ft results. Fnally, a restrcted verson of L 2, obtaned by settng α =, s clearly rejected relatve to the best-fttng L 5. The lkelhood rato test s a large-sample approxmate test whose propertes can be questonable n small samples, partcularly n our case, where there are only

17 observatons. An alternatve procedure, vald n fnte samples, s to examne the posteror probablty mass n the regon where the restrctons hold. Proceedng n ths drecton, we obtaned scatter plots of the Markov-Chan Monte-Carlo observatons for a and d n L 5. These scatter plots appear n Fgures 6 and 7, for Sweden and Brazl, respectvely. Settng a = and d = n L 5, and α = n L 2, gves the same restrcted verson of a Lorenz functon. Both plots show no probablty n the vcnty of a = and d =. For Brazl there s a concentraton of probablty around d =, but ths concentraton does not extend beyond a=.92, ndcatng no support for both restrctons. 7 The posteror pdfs for α from L 4 were plotted Fgures 8 and 9 to see f L 3 s an acceptable restrcted verson of L 4 from a Bayesan perspectve. For both Brazl and Sweden, these pdfs have modes near zero. The Swedsh one declnes very slowly t s almost unform from zero to.5, then sharply to.7. That for Brazl declnes almost lnearly from zero to.6. Both suggest α = s an acceptable value and hence there s nothng to gan by movng from the 2- parameter functon L 3 to the 3-parameter functon L 4. Fgures 8 and 9 also explan why, for L 4, the estmates of α were very senstve to estmaton technque Tables and 2. The ML estmate s approxmately equal to the mode of the pdf whch s near zero. The Bayesan estmate s the posteror mean whch s near the centre of the dstrbuton n each case. The above exercse was repeated for the parameter γ from L 4. See Fgures and. Interestngly, there was a symmetry between the pdfs for α and γ. For

18 Sweden, the pdf for γ was gradually ncreasng, but almost unform, from to.55. For Brazl t ncreased lnearly from to.35. After the ncreasng part of the functons, there was a sharp declne at the rght sde of the dstrbutons. The reason that a hypothess test suggested L 2 was an acceptable restrcted verson of L 4 for Sweden, but not for Brazl, s clear. There s substantal probablty mass at for the former, but not for the latter. 8 A remanng puzzle s: Why s the Gn coeffcent from L 4 estmated relatvely accurately, as reflected by the standard errors and standard devatons n Tables 3 and 4, and posteror pdfs n Fgures 4 and 5, when the parameters α and γ from L 4 are estmated wth lttle precson? We shed lght on ths queston by examnng scatter plots of the Markov Chan Monte Carlo observatons on α and γ. See Fgures 2 and 3. The cgar-shaped nature of these plots ndcates a very hgh correlaton between the parameters. Thus, although we cannot estmate the parameters accurately ndvdually, we can estmate combnatons of the parameters very accurately. It appears that the data does not dscrmnate between large γ wth small α and small γ wth large α, and that these combnatons have smlar mplcatons for the value of the Gn coeffcent. Also, we observe n the Swedsh case that, although the hypotheses α = and γ = are reasonable when consdered separately, the jont hypothess α =, γ = s clearly rejected.

19 9 Conclusons and Summary One way of estmatng a Lorenz curve s to assume a partcular dstrbuton for ncome, estmate the parameters of that dstrbuton, and derve the correspondng Lorenz curve. Another way s to assume a partcular Lorenz curve, and estmate ts parameters. For ths second approach we have suggested a dstrbutonal assumpton and correspondng estmaton technques whch are consstent wth the proportonal nature of Lorenz-curve data, can be employed wth any Lorenzcurve specfcaton and can be used wth grouped data or unt-record data. Our model and estmaton technques were appled to two data sets that have been the subject of past analyses, one for Sweden, a country wth relatvely low nequalty, and one for Brazl, a country wth relatvely hgh nequalty. Results were obtaned for 5 dfferent Lorenz-curve specfcatons. Our fndngs suggest that pont estmaton of the Gn coeffcent s generally nsenstve to choce of dstrbutonal assumpton, estmaton technque and Lorenz-curve specfcaton. There were two exceptons to ths concluson. One was for the functon L appled to the Brazlan data, usng the Drchlet dstrbuton. In ths case, the dfferent estmates were attrbutable to a poor ft. The second excepton was the estmate from L 4 wth the Swedsh data and the estmaton technque of Saraba et al. Ths dscrepancy s lkely to be a consequence of estmaton nstablty assocated wth the overparameterzed functon L 4. Although pont estmaton of the Gn coeffcent was robust, assessment of the precson of estmaton was not. It depended heavly on choce of functonal form and the dstrbutonal assumpton, and, to a lesser extent, on whether ML or

20 Bayesan nference was adopted. Wth respect to choce of functonal form, we found that L 5 provded the best ft, L 4 tends to be an unnecessary overparametersaton, and L can ft poorly. Wth respect to tools of analyss, we showed how Bayesan posteror pdfs can be an effectve means for conveyng knowledge about unknown parameters and nequalty measures, and how they can be used to assess the valdty of parametrc restrctons on Lorenz functons. 2

21 2 Reference Albert, J. H. and Chb, S. 996, Computaton n Bayesan Econometrcs: An Introducton to Markov Chan Monte Carlo, n R. C. Hll ed., Advances n Econometrcs Volumn A: Bayesan Computatonal Methods and Applcatons, JAI Press, Greenwch. Basmann, R.L., Hayes, K.J., Slottje, D.J. and Johnson J.D. 99, A General Functonal Form for Approxmatng the Lorenz Curve, Journal of Econometrcs, 43, Chotkapanch, D. 993, A Comparson of Alternatve Functonal Forms for the Lorenz Curve, Economcs Letters, 4, Datt, G. 998, Computatonal Tools for Poverty Measurement and Analyss, FCND Dscusson Paper No. 5, Internatonal Food Polcy Research Insttute, World Bank. Gastwrth, J.L. 972, The Estmaton of the Lorenz Curve and Gn Index, Revew of Economcs and Statstcs, 54, Geweke, J. 999, Usng Smulaton Methods for Bayesan Econometrc Models: Inference, Development and Communcaton, Econometrc Revews, 8, Jan, S. 975, Sze Dstrbuton of Income, World Bank, Washngton. Kakwan, N.C. 98, On a Class of Poverty Measures, Econometrca, 48, Kakwan, N.C. and N. Podder 973, On Estmaton of Lorenz Curves from Grouped Observatons Internatonal Economc Revew, 4, Kakwan, N.C. and N. Podder 976, Effcency Estmaton of the Lorenz Curve and Assocated Inequalty Measures from Grouped Observatons, Econometrca, 44, McDonald, J. B. 984, Some Generalzed Functons for the Sze Dstrbuton of Income, Econometrca, 52, McDonald J.B. and Y.J. Xu 995, A Generalzaton of the Beta Dstrbuton wth Applcatons, Journal of Econometrcs, 66, Ortega, P., Fernandez, M.A., Lodoux, M. and A. Garca 99, A New Funatonal Form for Estmatng Lorenz Curves, Revew of Income and Wealth, 37, Porer, R.A. 995, Intermedate Statstcs and Econometrcs: A Comparatve Approach, Cambrdge: MIT Press. Rasche, R.H.,Gaffney, J., Koo, A. and N. Obst 98, Functonal Forms for Estmatng the Lorenz Curve, Econometrca, 48, 6-62.

22 Saraba, J-M, Castllo, E. and D.J. Slottje 999, An Ordered Famly of Lorenz Curves, Journal of Econometrcs, 9, Shorrocks, A.F. 983, Rankng Income Dstrbutons, Economca, 5,3-7. Thel, H. 967, Economcs and Informaton Theory, Amsterdam, North Holland. Thel, H. 975, Theory and Measurement of Consumer Demand, Amsterdam, North Holland. Woodland, A.D. 979, Stochastc Specfcaton and the Estmaton of Share Equatons, Journal of Econometrcs,,

23 23 Appendx: Expressons for varances of the Gn coeffcent. ˆ k 2 ˆ2 2 For L : ˆ 2 e e k var G + = var kˆ kˆ 2 kˆ e For L2 : G = 2 π [ π ] dπ α δ 2 G var Gˆ = α G δ var αˆ cov αˆ, δˆ G cov αˆ, δˆ α var δˆ G δ G α δ where = 2 π log π[ π ] dπ α G α δ and = 2 π π log π dπ δ For L3 : G = 2 [ π ] dπ δ γ G var Gˆ = δ G var δˆ γ cov δˆ, γˆ G cov δˆ, γˆ δ var γˆ G γ G δ γ δ where = 2 γ [ π ] π log π dπ δ G δ γ δ and = 2 [ π ] log[ π ] dπ γ

24 24 For L4 : G = 2 π [ π ] dπ G var Gˆ = α G δ α δ var αˆ G cov αˆ, δˆ γ cov αˆ, γˆ γ cov αˆ, δˆ var δˆ cov δˆ, γˆ G cov αˆ, γˆ α G cov δˆ, γˆ var γˆ δ G γ G α δ γ where = 2 π log π[ π ] dπ α G γ G δ α δ = 2 π [ π ] log[ π ] dπ α γ δ γ = 2 π γ [ π ] π log π] dπ δ δ For L5 : G = 2 [ π aπ π ] dπ G var Gˆ = a G d d b var aˆ G cov aˆ, dˆ b cov aˆ, bˆ G d b where = 2 π π dπ a cov aˆ, dˆ var dˆ cov dˆ, bˆ G cov aˆ, bˆ G a cov dˆ, bˆ var bˆ G d b G d G b = 2 aπ π log π dπ d = 2 aπ π log π dπ d b b

25 25 Table Estmates for Lorenz Parameters and Gn Coeffcents Sweden α δ γ Gn L 2 L 3 L 4 NL ML Bayes Saraba NL ML Bayes Saraba NL ML Bayes Saraba L k Gn NL ML Bayes L5 a d b Gn NL ML Bayes

26 26 Table 2 Estmates for Lorenz Parameters and Gn Coeffcents Brazl α δ γ Gn L 2 L 3 L 4 NL ML Bayes Saraba NL ML Bayes Saraba NL ML Bayes Saraba L k Gn NL ML Bayes L5 a d b Gn NL ML Bayes

27 27 Table 3 Standard Errors Devatons for Lorenz Parameters and Gn Coeffcents Sweden α δ γ Gn L 2 L 3 L 4 NL..37. ML Bayes Saraba.8.33 NL ML Bayes Saraba NL ML Bayes Saraba L k Gn NL ML Bayes L5 a d b Gn NL ML Bayes

28 28 Table 4 Standard Errors Devatons for Lorenz Parameters and Gn Coeffcents Brazl α δ γ Gn L 2 L 3 L 4 NL ML Bayes Saraba NL ML Bayes Saraba.73.4 NL ML Bayes Saraba..4.9 L k Gn NL ML Bayes L5 a d b Gn NL ML Bayes

29 29 Table 5 Informaton Inaccuracy Measure Sweden Brazl ML Bayes ML Bayes L L L L L Table 6 The Lkelhood Rato Test Sweden Brazl Crtcal Value L 4 VS L L 4 VS L L 5 VS L

30 ML 5. Bayes Fgure : Pdfs for α for L 2 and Sweden 2 ML Bayes Fgure 2: Pdfs for Gn coeffcent for L 4 and Sweden.

31 ML Bayes Fgure 3: Pdfs for Gn coeffcent for L 5 and Brazl 9 L5 8 L L Fgure 4: Posteror pdfs for the Gn coeffcent for Sweden.

32 L5 L4 5 L Fgure 5: Posteror pdfs for the Gn coeffcent for Brazl

33 d a Fgure 6: Jont scatter plot a, d for L 5, Sweden d a Fgure 7: Jont scatter plot a, d for L 5, Brazl

34 Fgure8: Posteror pdf for α for L 4, Sweden Fgure 9: Posteror pdf for α for L 4, Brazl

35 Fgure : Posteror pdf for γ for L 4, Sweden Fgure: Posteror pdf for γ for L 4, Brazl

36 gamma alpha Fgure 2: Jont scatter plot α, γ for L 4, Sweden gamma alpha Fgure 3: Jont scatter plot α, γ for L 4, Brazl