Estimating Lorenz Curves Using a Dirichlet Distribution

Transcription

1 Estmatng orenz Curves Usng a Drchlet Dstrbuton Duangkamon Chotkapanch Department of Economcs, Curtn Unversty of Technology, Perth, WA 6845 chotkapanchd@cbs.curtn.edu.au Wllam E. Grffths Department of Economcs, Unversty of Melbourne, Vc 3, Australa b.grffths@unmelb.edu.au Abstract The orenz curve relates the cumulatve proporton of ncome to the cumulatve proporton of populaton. When a partcular functonal form of the orenz curve s specfed t s typcally estmated by lnear or nonlnear least squares, estmaton technques that have good propertes when 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 that recognzes the cumulatve proportonal nature of the orenz curve data by assumng that the ncome proportons are dstrbuted as a Drchlet dstrbuton. Fve orenz-curve specfcatons are used to demonstrate the technque. Maxmum lkelhood estmates under the Drchlet dstrbuton assumpton provde better-fttng orenz curves than nonlnear least squares and another estmaton technque that has appeared n the lterature. Keywords: Gn coeffcent maxmum lkelhood estmaton.

2 2. INTRODUCTION The orenz 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. A number of approaches to orenz curve estmaton have been adopted. In one approach, a partcular assumpton about the statstcal dstrbuton of ncome s made, the parameters of ths ncome dstrbuton are estmated, and a orenz 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. Ryu and Slottje 996 suggest another approach. They approxmate the orenz curve from any ncome dstrbuton by expandng the nverse dstrbuton functon n terms of a an exponental polynomal seres and b a sequence of Bernsten polynomal functons. When mcro-data are avalable, nonparameterc estmaton of the orenz curve and related nequalty measures s possble. See, for example, Beach and Davdson 983, Gastwrth and Gal 985, and Bshop et al 989. An alternatve approach, more suted to grouped data, s to specfy a partcular functonal form for the orenz curve and estmate t drectly. It s ths approach that s the focus of ths paper. Early breakthroughs on orenz 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 orenz curve specfcaton that, after transformaton, could be placed n an approxmate lnear model framework.

3 3 Other specfcatons have typcally been estmated by lnear or nonlnear least squares Kakwan 98, Basmann et al 99, Chotkapanch 993. Such exercses are useful for fttng orenz curves, but, because the covarance matrx estmates they provde are only relevant for ndependent normally dstrbuted errors, they do not provde a bass for nference about orenz curve parameters or any nequalty measures derved from them. 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 by suggestng a dstrbuton-free method of estmaton. Suppose that a orenz 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 subsets of n observatons. Snce each of the subsets yelds n 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 estmaton. A sutable dstrbuton s the Drchlet dstrbuton. It s a multvarate dstrbuton for a vector of random varables that are shares that sum to unty. By relatng the parameters of the Drchlet dstrbuton to orenz curve dfferences, we can accommodate the cumulatve proportonal nature of the orenz curve data, and set up a lkelhood functon dependent on the unknown parameters of the orenz curve. A smlar

4 4 approach was adopted by Woodland 979 for estmaton of share equatons that arse n demand and producton theory. To further motvate the choce of a Drchlet dstrbuton, note that, wth random samplng, the number of households n each of a number of ncome classes can be vewed as an observaton from the multnomal dstrbuton Agner and Goldberger 97, Kakwan and Podder 973. Furthermore, by usng a transformaton from cell numbers to cell proportons, the multnomal dstrbuton can be approxmated by a Drchlet dstrbuton Johnson 96, Johnson and Kotz 969, p.285. Thus, the Drchlet dstrbuton s a reasonable choce for share data, rrespectve of the orgnal ncome dstrbuton from whch the observatons were drawn. The choce of a Drchlet dstrbuton for ncome shares s much less arbtrary than choosng a specfc ncome dstrbuton. In addton, the number of recognzed multvarate dstrbutons that are drectly applcable to share data s very lmted. Apart from the Drchlet dstrbuton, only two other possbly-relevant generalzed beta dstrbutons are descrbed n Johnson and Kotz 972. These facts and the general lack of recognton of the share nature of the data n much of the lterature on orenz curve estmaton, make the Drchlet dstrbuton a useful alternatve to pursue. In Secton 2, we outlne the dstrbutonal assumptons and how they relate to orenz curve estmaton. The lkelhood functon for a set of unknown orenz curve parameters s derved. 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 orenz functons that we use n the emprcal work are presented. The results are gven

5 5 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 to those obtaned by Saraba et al 999, and those obtaned usng nonlnear least squares. Snce orenz-curve estmaton s usually a frst step towards estmatng nequalty, maxmum lkelhood M and nonlnear least squares estmates for the Gn coeffcent are obtaned for each orenz-curve specfcaton. Fnally, we examne whch estmaton technque leads to the best fttng orenz curve. 2. MODES, ASSUMPTIONS AND ESTIMATION Suppose we have avalable observatons on cumulatve proportons of populaton, 2,, wth M = and correspondng cumulatve proportons of M ncome η, η2,, ηm wth η M = obtaned after orderng populaton unts accordng to ncreasng ncome. We wsh to use these observatons to estmate a parametrc verson of a orenz curve that we wrte as η = where s an n vector of unknown parameters. Clearly, one would not expect all data ponts to le exactly on the curve η =. It seems reasonable to assume, however, that condtonal on the populaton proportons, the ncome shares q = η η are random varables wth means E q E η E η = = Our proposal s to also assume q = q, q,, 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

6 Γ Γ Γ Γ = M M M M q q q q f " 2 where ',,, 2 M = are the parameters of the pdf and. Γ s the gamma functon. By relatng the to the orenz functon, we can fnd a pdf for q whch has the mean gven n equaton and whch s a functon of the orenz curve parameters. Workng n ths drecton, we set [ ] λ = 3 where λ s an addtonal unknown parameter. Ths defnton for gves the desred result because the mean of the Drchlet dstrbuton s gven by [ ] = λ λ = = M M q E 2 ] [ " = 4 snce = M and =. We can now wrte the pdf for q as = λ λ Γ λ = Γ θ M q q f ] [ ] [ 5 where ', ' λ = θ. The varances and covarances between the shares are gven by Johnson and Kotz, 972, p ] [ var λ + = q E q E q 6

7 7 E q E q j cov q, q j = 7 λ + Thus, the ncome shares are correlated, wth correlatons gven by / 2 E q E q j r j = 8 [ ][ ] E q E q j Snce the varances depend on E q, 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 orenz curve to the data. The maxmum lkelhood estmate for θ can be found by maxmzng the loglkelhood functon log[ f q θ] = log Γ λ + M = M = λ[ log Γ λ[ ] ] log q 9 3. DATA AND ORENZ 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 orenz curves. The

8 data on these two countres were chosen because of ther dfferences n the degree of nequalty n ncome dstrbutons. 8 A large number of functonal forms have been suggested n the lterature for modellng the orenz 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 orenz functons to whch we appled the two data sets are: k e k = k > k e, = [ ], < 2 γ 3, γ = [ ] γ, < 2 γ,, γ = [ ], γ, < 3 4 d 5 a, b, d = a b a >, < d, < b 4 The functon s the relatvely smple one-parameter functon suggested by Chotkapanch concdes wth the proposal of Ortega et al s a well-known form of orenz curve suggested by Rasche et al 98 and 4 s an extenson of 3 and 2 ntroduced by Saraba et al 999. Note that 4 nests both 2 and 3, wth 2 beng 4 wth γ = and 3 beng 4 wth =. Settng both γ = and = yelds the orenz curve = whch orgnates from the classcal Pareto dstrbuton. The functon 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 orenz curves. See, for example, Datt 998. Note that, when a = and d =, 5 s the same as 2 wth =.

9 Once a orenz 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 for the Gn coeffcents that can be derved from each of the orenz functons. In each case the Gn coeffcent s defned as G = 2 d 5 Alternatve expressons for G can be found for some of the orenz curves. However, wth the excepton of, they stll generally nvolve a numercal ntegral. We obtan M estmates by numercally evaluatng 5 n each case wth replaced by the M estmate ˆ RESUTS In addton to M estmaton usng the assumpton of a Drchlet dstrbuton, we also estmated each functon usng nonlnear least squares. Because nonlnear least squares has been popular n the lterature, t s useful to compare ts estmates and standard errors to those from M estmaton. However, conventonal nonlnear least squares N standard errors are computed assumng ndependent dentcally dstrbuted error terms, an assumpton that s unrealstc wth share data. Thus, for N standard errors we report those suggested by Newey and West 987. The estmates and standard errors obtaned by Saraba et al 999, for 2, 3 and 4 are also reported they provde further evdence on the senstvty of estmates to choce of estmaton technque. However, Saraba estmates for and 5 are not avalable nor are the standard errors for the Saraba-based Gn coeffcent estmates for all functons. Pont estmates and standard errors of the orenz curve parameters and the correspondng Gn coeffcents for Sweden are presented n Table. Wth the

10 excepton of the functon 4, the estmates of the orenz parameters and the Gn coeffcent are not senstve to the estmaton technque. Nonlnear least squares, M and Saraba lead to almost dentcal estmates. For 4 there s consderable varaton n the orenz parameter estmates, and the Saraba-estmated Gn coeffcent s notceably dfferent from the others. A somewhat remarkable outcome s that, wth the excepton of the Saraba et al estmate from 4, the pont estmates of the Gn coeffcent are relatvely nsenstve to estmaton technque and functonal form specfcaton. [Table near here] Although pont estmaton s robust wth respect to choce of estmaton technque and functonal form, assessment of the relablty of the estmates, va ther standard errors, s heavly dependent on estmaton technque. Choosng a maxmum lkelhood technque that s consstent wth the share nature of the data can have a bg mpact on the perceved precson of the estmates. In Table the standard errors for M are generally hgher than those for nonlnear least squares those reported by Saraba et al are hgher for some coeffcents and lower for others. The standard errors of the Gn coeffcent were calculated usng the asymptotc approxmaton var Gˆ = V 6 ' where V s the asymptotc covarance matrx for the M or N estmator for. Expressons derved usng 6 for each of the orenz curves are gven n the Appendx.

11 The remarks made about Sweden also hold for the estmates for Brazl gven n Table 2, wth some mnor exceptons. Once agan, there are vastly dfferent estmates for 4, confrmng consderable nstablty n the estmaton of ths functon. In contrast to Sweden, estmates of the parameter and correspondng Gn coeffcent are also senstve to choce of estmaton technque. The other functons reman nsenstve to choce of estmaton technque. Except for the Gn coeffcent estmates are nsenstve wth respect to both estmaton technque and choce of functonal form. Despte yeldng smlar pont estmates, the three estmaton technques yeld very dfferent standard errors. [Table 2 near here] We turn now to questons of goodness of ft, and choce between alternatve orenz functons. 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 and the M estmates. et qˆ denote the predcted ncome shares obtaned from an estmated model. Thel s 967 measure of nformaton naccuracy s defned as I = log 7 M q q = qˆ Estmated 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.

12 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 nonlnear and M estmates, and for the Saraba et al estmates for functons 2, 3 and 4. The outcomes are presented n Table 3. 2 [ Table 3 near here] For the Swedsh data, M estmaton provdes a better ft than nonlnear least squares for all functonal forms. It also provdes better fts than those from the technque suggested by Saraba et al for the functons they consdered. The dfferences are not great for, 2 and 3 they are most notceable for and. The large mprovement of M over nonlnear least squares n the case 4 5 of 5 s perhaps surprsng, gven the apparent smlarty of the two sets of orenz curve estmates. A closer examnaton of the two sets of predctons for ths case revealed that they were not as close as one mght suspect by comparng parameter estmates. Also, nonlnear least squares led to some relatvely large over predctons that were penalsed heavly by the nformaton crteron. Fnally, t s nterestng that a rankng of the relatve magntudes of the M standard errors for the Gn coeffcent corresponds exactly to a goodness-of-ft rankng of the M-estmated orenz functons.

13 The nformaton naccuraces for the Brazlan data lead to the same conclusons wth two small modfcatons. Nonlnear least squares and M estmaton of 5 had the same ft. Nonlnear least squares provded a better ft than M for. 3 To provde nformaton about choce of functonal form we examned whether lkelhood rato tests suggested nested versons of 4 and 5 would be adequate. The avalablty of these tests s one of the advantages of the maxmum lkelhood methodology that we have proposed. Table 4 contans 2 χ values for lkelhood rato tests for varous hypotheses. These results suggest that 3 s an acceptable restrcted verson of 4 for both Sweden and Brazl. Also, 2 s an acceptable restrcted verson of 4 for Sweden, but not for Brazl. Fnally, a restrcted verson of 2, obtaned by settng =, s clearly rejected relatve to the bestfttng 5. [Table 4 near here.] 5. CONCUSIONS AND SUMMARY One way of estmatng a orenz curve s to assume a partcular dstrbuton for ncome, estmate the parameters of that dstrbuton, and derve the correspondng orenz curve. Another way s to assume a partcular orenz curve, and estmate ts parameters. For ths second approach we have suggested a dstrbutonal assumpton and a correspondng estmaton technque whch s consstent wth the proportonal nature of orenz-curve data, can be used to approxmate share

14 data from any ncome dstrbuton, and can be employed wth any orenz-curve specfcaton. 4 Our model and estmaton technque was 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 orenz-curve specfcatons. Our fndngs do not necessarly carry over to other data sets and other functons. Wth ths fact kept n mnd, we reached the followng conclusons. Pont estmaton of the Gn coeffcent was generally nsenstve to choce of dstrbutonal assumpton, estmaton technque and orenz-curve specfcaton. There were two exceptons to ths concluson. One was for the functon appled to the Brazlan data, usng the Drchlet dstrbuton. The second excepton was the estmate from 4 wth the Swedsh data and the estmaton technque of Saraba et al. The dscrepancy obtaned n ths case appears to be a consequence of estmaton nstablty assocated wth ths functon. 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 choce of estmaton technque. Wth respect to estmaton technque, we found that M estmaton, under our proposal to use the Drchlet dstrbuton, provded the best ft. Useful future work would be a Monte Carlo study to assess whether the standard errors produced by each estmaton technque are an accurate reflecton of fnte-sample varablty of the estmates.

15 5 APPENDIX: EXPRESSIONS FOR VARIANCES OF THE GINI COEFFICIENT ˆ k 2 ˆ2 2 For : ˆ 2 e e k var G + = var kˆ kˆ 2 kˆ e For 2 : G = 2 [ ] d 2 var Gˆ = var ˆ cov ˆ, ˆ cov ˆ, ˆ varˆ G where = 2 log [ ] d and = 2 log d For 3 : G = 2 [ ] d γ var Gˆ = γ var ˆ covˆ, γˆ covˆ, γˆ varˆ γ G γ γ where = 2 γ [ ] log d γ and = 2 [ ] log[ ] d γ

16 6 For 4 : G = 2 [ ] d var Gˆ = var ˆ cov ˆ, ˆ γ cov ˆ, γˆ γ cov ˆ, ˆ varˆ covˆ, γˆ cov ˆ, γˆ covˆ, γˆ varˆ γ G γ γ where = 2 log [ ] d γ = 2 [ ] log[ ] d γ γ = 2 γ [ ] log ] d For 5 : G = 2 [ a ] d var Gˆ = a d d b var aˆ cov aˆ, dˆ b cov aˆ, bˆ d b where = 2 d a cov aˆ, dˆ var dˆ cov dˆ, bˆ cov aˆ, bˆ G a cov dˆ, bˆ var bˆ G d b d b = 2 a log d d = 2 a log d d b b

17 7 REFERENCES Agner, D.J. and Goldberger, A.S. 97, Estmaton of Pareto s aw from Grouped Observatons, Journal of Amercan Statstcal Assocaton, 65, Basmann, R.., Hayes, K.J., Slottje, D.J. and Johnson J.D. 99, A General Functonal Form for Approxmatng the orenz Curve, Journal of Econometrcs, 43, Beach, C.M. and Davdson, R. 983, Dstrbuton-Free Statstcal Inference wth orenz Curves and Income Shares, Revew of Economc Studes, 5, Bshop, J.A., Chakrabort, S. and Thstle, P.D. 989, Asymptotcally Dstrbuton-Free Statstcal Inference for Generalzed orenz Curves, Revew of Economcs and Statstcs, 7, Chotkapanch, D. 993, A Comparson of Alternatve Functonal Forms for the orenz Curve, Economcs etters, 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.. 972, The Estmaton of the orenz Curve and Gn Index, Revew of Economcs and Statstcs, 54, Gastwrth, J.. and Gal, M.H. 985, Smple Asymptotcally Dstrbuton-Free Methods Comparng orenz Curves and Gn Indces Obtaned from Complete Data, n R.. Basmann and G.F. Rhodes, Jr., edtors, Advances n Econometrcs, 4, JAI Press, Greenwch, CT. Jan, S. 975, Sze Dstrbuton of Income, World Bank, Washngton. Johnson, N.. 96, An Approxmaton to the Multnomal Dstrbuton Some Propertes and Applcatons, Bometrka, 47, Johnson, N.. and Kotz, S. 969, Dscrete Dstrbutons, John Wley & Sons, New York. Johnson, N.. and Kotz, S. 972, Dstrbutons n Statstcs: Contnuous Multvarate Dstrbutons, John Wley & Sons, New York. Kakwan, N.C. 98, On a Class of Poverty Measures, Econometrca, 48, Kakwan, N.C. and Podder, N. 973, On Estmaton of orenz Curves from Grouped Observatons Internatonal Economc Revew, 4, Kakwan, N.C. and Podder, N. 976, Effcent Estmaton of the orenz Curve and Assocated Inequalty Measures from Grouped Observatons, Econometrca, 44,

18 McDonald, J. B. 984, Some Generalzed Functons for the Sze Dstrbuton of Income, Econometrca, 52, McDonald J.B. and Xu, Y.J. 995, A Generalzaton of the Beta Dstrbuton wth Applcatons, Journal of Econometrcs, 66, Newey, W. and West, K. 987, A Smple Postve Sem-Defnte, Heteroskedastcty and Autocorrelaton Consstent Covarance Matrx, Econometrca, 55, Ortega, P., Fernandez, M.A., odoux, M. and Garca, A. 99, A New Funatonal Form for Estmatng orenz Curves, Revew of Income and Wealth, 37, Rasche, R.H., Gaffney, J., Koo, A. and Obst, N. 98, Functonal Forms for Estmatng the orenz Curve, Econometrca, 48, Ryu, H.K. and Slottje, D.J. 996, Two Flexble Functonal Form Approaches for Approxmatng the orenz Curve, Journal of Econometrcs, 72, Saraba, J-M, Castllo, E. and Slottje, D.J. 999, An Ordered Famly of orenz 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,,

19 Table Estmates and Standard Errors for orenz Parameters and Gn Coeffcents Sweden N M Saraba γ Gn N M Saraba N M Saraba k Gn N M a d b Gn 5 N M

20 Table 2 Estmates and Standard Errors for orenz Parameters and Gn Coeffcents Brazl N M Saraba γ Gn N M Saraba N M Saraba k Gn N M a d b Gn 5 N.95.3 M

21 2 Table 3 Informaton Inaccuracy Measure Sweden Brazl M N Saraba M N Saraba Table 4 The kelhood Rato Test Sweden Brazl Crtcal Value 4 VS VS VS 2 wth =