POSTERIOR DISTRIBUTIONS FOR THE GINI COEFFICIENT USING GROUPED DATA

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1 POSTERIOR DISTRIBUTIONS FOR THE GINI COEFFICIENT USING GROUPED DATA DUANGKAON CHOTIKAPANICH and WILLIA E. GRIFFITHS Curtn Unversty o Technology and Unversty o New England, Australa SUARY When avalable data comprse a number o sampled households n each o a number o ncome classes, the lkelhood uncton s a multnomal dstrbuton wth the ncome class populaton proportons as the unknown parameters. Two methods or gong rom ths lkelhood uncton to a posteror dstrbuton on the Gn coecent are nvestgated. In the rst method, the underlyng ncome dstrbuton s assumed to ollow a lognormal dstrbuton whch permts reparametersaton o the lkelhood uncton n terms o the lognormal's locaton and scale parameters. The Gn coecent s a nonlnear uncton o the scale parameter. The etropols algorthm s used to nd the posteror dstrbuton o the Gn coecent rom a sample o Bangkok households. The second method does not requre an assumpton about the nature o the ncome dstrbuton, but uses trangular pror dstrbutons on the locaton o mean ncome wthn each ncome class. By samplng rom these trangular dstrbutons, and the Drchlet posteror dstrbuton o the ncome class proportons, an alternatve posteror dstrbuton o the Gn coecent s calculated. Keywords: LOGNORAL DISTRIBUTION; DIRICHLET POSTERIOR.. BACKGROUND AND INTRODUCTION easurng nequalty n the dstrbuton o ncome s a maor concern o economsts. On a global level, montorng nequalty tells us whether wealth s becomng more concentrated, or whether there s decreasng world nequalty, whch s more n lne wth generally accepted socal values. On a natonal level, measurement o ncome dstrbuton s necessary or eectve admnstraton o government polces such as socal welare programs and taxaton. One powerul and popular method or measurng nequalty n the dstrbuton o ncome s the Gn coecent. It s dened as twce the area between a 45 degree lne and a Lorenz curve, where the Lorenz curve s a graph descrbng the share o total ncome η accrung to the poorest racton π o the populaton. See Fgure, where three Lorenz curves (dscussed later) are graphed. [Fgure near here] Lorenz curves and Gn coecents are obtaned as ollows. Let y denote ncome wth a dstrbuton characterzed by ts densty uncton ( y). The proporton o populaton earnng ncome up to a gven level r s gven by the dstrbuton uncton

2 r ( y) dy F( r) π = y = 0 The proporton o ncome earned by ths proporton o the populaton s gven by the rst moment dstrbuton uncton r η = y y = µ 0 ( y) dy H ( r) where µ s the mean level o ncome. The Lorenz curve s dened as the relatonshp between π and η. It s obtaned by substtutng the nverse o the uncton n () nto the π uncton n (). That s, η = η( π) = [ F ( )]. H The Gn coecent s gven by ( ) G = η π dπ (3) 0 I ncome dstrbuton normaton s avalable n the orm o grouped data or ncome classes, wth estmates o the proportons ( π, η ) avalable or each class, then a Lorenz curve based on lnear nterpolaton s commonly used. Ths Lorenz curve s not a π,. The two lower smooth curve but made up o lnear segments onng the ponts ( ) η curves n Fgure have been constructed n ths way. Followng ths procedure s equvalent to assumng that there s no nequalty wthn each ncome class. The Gn coecent rom a Lorenz curve made up o lnear segments s expressed as = + π ηπ+ = G = η (4) I the number o classes s small, assumng no nequalty wthn each class wll substantally underestmate the total nequalty. As the number o classes ncreases, G, dened n (4), approaches the true Gn coecent n (3). Both equatons (3) and (4) have been used n the lterature to estmate the Gn coecent. To use (3), t s necessary to estmate the parameters o an assumed orm o dstrbuton or ncome and to then nd the correspondng Lorenz curve and Gn coecent estmate. Alternatvely, sometmes the Lorenz curve s estmated drectly. To use (4), observatons on ( ) η π, are used drectly. There are exceptons, but most o the emprcal work on Gn coecent estmaton als to report any measure o uncertanty; typcally, only a pont estmate s provded. Furthermore, as ar as we are aware, there have been no studes where estmaton uncertanty has been descrbed by a posteror probablty densty uncton (pd). The obectve o ths paper s to start llng ths gap. We ocus on dervng posteror pd's where only grouped data (the number o sampled households n each o a number o ncome classes) are avalable. For estmaton usng equaton (3), we assume ncome ollows a log-normal dstrbuton. For estmaton usng equaton (4) we consder two scenaros, one where mean ncome wthn each class s known wth certanty, and another where pror dstrbutons are assgned to the mean () ()

3 3 ncomes or each class. A sample o 863 Bangkok households s used to llustrate the technques. Related prevous work s that by Kakwan and Podder (973, 976) who employed an approxmate lnear model ramework and grouped data to obtan a samplng theory estmate o the Gn coecent. A multnomal ramework or samplng theory estmaton o the parameters o a varety o ncome dstrbutons was used by Kloek and van Dk (978) and van Dk and Kloek (980), and or the Pareto dstrbuton by Agner and Goldberger (970). The methodology relevant or the assumpton o a log-normal ncome dstrbuton s descrbed n Secton. Secton 3 covers the case where no assumpton s made; the results are presented and dscussed n Secton 4. Although we ocus on the Gn coecent, our technques are readly extendable to alternatve measures o nequalty or other nonlnear unctons o ncome dstrbuton parameters. See, or example, Rongve and Beach (977). Furthermore, nequalty or concentraton measures have applcablty n ndustry economcs or measurng monopoly concentraton o output.. USING AN INCOE DISTRIBUTION ASSUPTION The scenaro we are consderng s one where avalable data take the orm o those n Table. That s, there are ncome classes wth endponts ( 0, z ),( z, z3), L,( z, ) ; household ncome or each household n the sample s not known exactly, but we do know the number o households n that all nto each o the ncome classes. Let p be the probablty that a household beng chosen randomly wll have ncome n class, and let n = ( n, n, K, n )' and P = ( p,p, K, p )'. The ont probablty densty uncton ( n P), whch gves the probablty that the number o ndvduals n group s n, =,,...,, ollows the multnomal dstrbuton. It can be expressed as n n ( n P) p p K p (5) n wth the restrctons n the sample. p = = and n = N, where N s the total number o households = [Table near here] Assumng the ncome dstrbuton can be represented by a partcular densty uncton means that the parameters p n equaton (5) can be expressed as unctons o a smaller number o parameters. In the case o a lognormal dstrbuton, where t s assumed log( Y ) ~ N( µ, σ ), Y beng the ncome o a randomly selected household, ( n P) can be wrtten as log z + µ log z µ Φ Φ (6) = σ σ ( n µ, σ) n

4 4 where Φ ( ) s the standard normal cumulatve dstrbuton uncton. Equaton (6) s the lkelhood uncton or ( µ, σ ) ; t can be maxmzed to provde maxmum lkelhood estmate or these parameters. Ths was the approach adopted by van Dk and Kloek (980) or a number o ncome dstrbutons. It was also utlzed by Zellner (97, p.36) who, buldng on the work o Agner and Goldberger (970), ound the posteror pd o the Pareto dstrbuton parameter. Usng equatons (), () and (3), t can be shown that the assumpton o a lognormal dstrbuton mples a Gn coecent gven by σ G = Φ Thus, estmaton o the scale parameter σ s a rst step towards estmaton o the Gn coecent. Usng the conventonal nonnormatve pror pd ( µ, σ) σ yelds the ollowng posteror pd or ( µ,σ) σ ( µ, σ n) ( n µ, σ) Ths posteror pd s not analytcally tractable. However, t s straghtorward to use a etropols-hastngs algorthm (see, or example, Geweke 998) to draw observatons on ( µ, σ) rom t. From the observatons on σ, we can compute correspondng observatons on G through equaton (7). These observatons can be used to estmate the posteror pd or G and ts moments. We used a random-walk etropols arkov Chan wth 85,000 observatons, and 0,000 o these dscarded as a "burn-n".. NO EXPLICIT ASSUPTION ABOUT THE INCOE DISTRIBUTION I no assumpton s made about the ncome dstrbuton, then the dscrete ormula or the Gn Coecent, wth the Lorenz curve represented by lnear segments, can be utlzed. π, can be wrtten as See equaton (4). The two ngredents n ths ormula ( ) η π = p = η = = = p p µ µ (7) (8). (9) where µ s the mean ncome rom wthn the th ncome class. Thus, we can nd posterors pd's or the p and the µ, we can draw observatons rom these pd's, compute correspondng draws or π and η rom equaton (9), and then compute correspondng values or G rom equaton (4). These values wll provde estmates o the posteror pd or G and ts moments. A possble nonnormatve pror or the p s ( P ) p p K p

5 5 See, or example, Gelman et al (995, p.399). Combnng ths pror pd wth the lkelhood uncton n equaton (5) yelds the posteror pd ( P n) = n p (0) Ths pd s a Drchlet dstrbuton rom whch we can draw observatons. The other set o parameters about whch we need normaton s the µ. There was no explct need or ths normaton when the assumpton o a lognormal dstrbuton was made because knowledge o the parameters o that dstrbuton mples knowledge o mean ncome wthn each class. Wthout an explct dstrbutonal assumpton, the only posteror normaton we have about a µ s that t les n the nterval ( z, z + ). It seems sensble to ntroduce addtonal pror normaton on where n each nterval the µ are lkely to le. An assumpton at one extreme s to assume all values between z and z + are equally lkely values or µ, or, n other words, assgn a unorm pror on µ between the two end ponts. At the other extreme, we could assume that each µ s equal to the mdpont o each class wth probablty one. Ths s an assumpton oten used n samplng theory studes. See, or example, Kakwan and Podder (973). As a benchmark, we derve the posteror pd or the Gn coecent under ths assumpton. A more realstc pror speccaton or the µ s one that assgns a low probablty around the end ponts o each nterval, and a hgh probablty around the centre o the nterval. Also, the mode o the pror need not be at the mdpont o each nterval. When an ncome class alls on the ncreasng part o the ncome dstrbuton pd, average ncome or that class s lkely to be closer to the upper class lmt than the lower class lmt. The converse s true or the declnng part o the ncome dstrbuton pd. Also, the steeper the ncome pd, the closer the average wll be to the class boundary. A class o densty unctons that can capture these characterstcs, s lexble enough to allow or changng modes over derent parts o the ncome dstrbuton, and s a smple one to specy and smulate rom, s the class o trangular dstrbutons. Ths class s dened by the dstrbuton uncton F ( µ ) = ( µ z ) ( z + z )( c z ) ( z + µ ) ( z z )( z c ) + + or z or c where c s the mode o the pror pd or the th ncome class. An example s depcted n Fgure. [Fgure near here] The values or c that we chose or our emprcal work are gven n Table. They are chosen on the assumpton that the mode o the ncome dstrbuton pd s at 4000 µ µ c z + ()

6 6 baht, as has been suggested n prevous work (Chotkapanch 994). It s also assumed, as s typcal wth most ncome dstrbutons, that the pd to the let o the mode s steeper than the pd to the rght o the mode, and the steepness declnes as ncome ncreases. For the last nterval, t was consdered that mean ncome could not be beyond baht and so we set z = 60,000. To obtan observatons rom the posteror pd or the Gn coecent, 30,000 ndependent draws were made rom the Drchlet dstrbuton ( P n) and rom the ndependent trangular dstrbutons ( µ ). 4. RESULTS There are three sets o results those that assume a lognormal ncome dstrbuton, those that assume mean ncome or each class s equal to the class mdpont wth probablty one, and those that use the trangular pror dstrbutons or the mean ncomes. The posteror pds or these three cases appear n Fgure 3 and ther posteror means and standard devatons n Table. [Fgure 3 near here] Table. Posteror oments or the Gn Coecent. ean Stand. Dev. Lognormal dponts Trangular We can make the ollowng observatons:. There s no overlap between the posteror pd that assumes the lognormal dstrbuton and that whch makes no dstrbutonal assumpton, but assumes the class means are equal to the class mdponts. Ths outcome suggests the assumptons or at least one o these two sets o results may be napproprate. To nvestgate ths queston urther we plotted the Lorenz curves correspondng to () the lognormal assumpton wth µ and σ set equal to ther posteror means, () the sample π, and the µ set equal to the class mdponts, and () the sample π, and the µ set equal to the pror modes c. These three Lorenz curves are plotted n Fgure. The choce o class mdpont or pror mode or µ seems to make lttle derence, but the uncton rom the lognormal assumpton s notceably derent. Investgatng urther, we ound that a ch-square goodness-o-t test on the approprateness o the lognormal assumpton led to ts reecton.. The spreads o the "lognormal pd" and the "mdpont pd" are smlar. Thus, assumng the mean ncome wthn each class s known wth certanty conveys a

7 7 smlar amount o normaton to an assumpton about the specc orm o the ncome dstrbuton. 3. Recognzng uncertanty n the locaton o the class means ntroduces consderable uncertanty n the value o the Gn coecent. The dramatc ncrease n the spread o the posteror pd derved rom the trangular prors on mean ncomes rases at least two ssues. Frst, t gves an dea o how senstve uncertanty measures can be to the degree o normaton about the mean ncomes. Secondly, t makes one wonder a more normatve pror than the trangular dstrbuton could be realstc. The trangular dstrbuton has relatvely at tals at the class endponts. Dstrbutons wth less probablty n these regons mght be more approprate. We have demonstrated how normaton about the Gn coecent, ncludng estmaton uncertanty, can be presented n terms o a posteror pd. Our results hghlght how derent assumptons nluence the senstvty o both a Gn coecent estmate, and the uncertanty assocated wth such an estmate. There s room or urther work on other pror pds or class mean ncome, and other ncome dstrbutons. REFERENCES Agner, D.J. and Goldberger, A.S. (970). Estmaton o Pareto s law rom grouped observatons. J. Amer. Statst. Assoc. 65, Chotkapanch, D. (994). Technques or easurng Income Inequalty: An Applcaton to Thaland. Hampshre: Avebury. Gelman, A., Carln, J.B., Stern, H.S. and Rubn, D.S. (995). Bayesan Data Analyss. London: Chapman and Hall. Geweke, J. (998). Usng smulaton methods or Bayesan econometrc models: nerence, development and communcaton. Econometrc Revews 7, orthcomng. Kakwan, N.C. and Podder, N. (973). On the estmaton o Lorenz curves rom grouped observatons. Int. Econ. Revew 4, Kakwan, N.C. and Podder, N. (976). Ecent estmaton o the Lorenz curve and assocated nequalty measures rom grouped observatons. Econometrca 44, Kloek, T. and van Dk, H.K. (978). Ecent estmaton o ncome dstrbuton parameters. J. o Econometrcs 8, Rongve, I. and Beach, C. (997). Estmaton and nerence or normatve nequalty ndces. Int. Econ. Revew 38, van Dk, H.K. and Kloek, T. (980), Inerental procedures n stable dstrbutons or class requency data on ncomes. Econometrca 48, Zellner, A. (97). An Introducton to Bayesan Inerence n Econometrcs. New York: John Wley.

8 8 Table. Avalable Data and Pror odes or Average Income. ncome class household z z + requency (n ) pror modes (c ) 0 -,000,000-4,000 4,000-6,000 6,000-8,000 8,000-0,000 0,000 -,000,000-4,000 4,000-6,000 6,000-8,000 8,000-0,000 0, ,800 3,700 4,450 6,500 8,550 0,600,650 4,750 6,750 8,800 35,000 Fgure Lorenz curves

9 9 ( µ ) z c z + µ Fgure. A Trangular Pror. Fgure 3 Posteror pd or Gn coecent