Estimation of Gini coefficients using Lorenz curves

Transcription

1 Journal of Statstcal and Econometrc Methods, vol.1, no.2, 2012, ISSN: (prnt), (onlne) Scenpress Ltd, 2012 Estmaton of Gn coeffcents usng Lorenz curves Johan Fellman 1,2 Abstract Prmary ncome data yelds the most exact estmates of the Gn coeffcent. Usng Lorenz curves, the Gn coeffcent s defned as the rato of the area between the dagonal and the Lorenz curve and the area of the whole trangle under the dagonal. Varous attempts have been made to obtan accurate estmates. The trapezum rule s smple, but yelds a postve bas for the area under the Lorenz curve and, consequently, a negatve bas for the Gn coeffcent. Smpson s rule s better ftted to the Lorenz curve, but ths rule demands an even number of subntervals of the same length. Lagrange polynomals of second degree can be consdered as a generalsaton of Smpson s rule because they do not demand equdstant ponts. If the subntervals are of the same length, the Lagrange polynomal method s dentcal wth Smpson s rule. In ths study, we compare dfferent methods. When we apply Smpson s rule, we manly consder Lorenz curves wth decles. In addton, we use the trapezum rule, Lagrange polynomals and generalzatons of Golden s method (2008). No method s unformly optmal, but the trapezum rule s almost always nferor and Smpson s rule s superor. Golden s method s usually of medum qualty. Mathematcs Subject Classfcaton: 62P20, 91B15, 91B82 1 Swedsh School of Economcs and Busness Admnstraton, POB 479, FIN Helsnk, Fnland e-mal: fellman@hanken.f 2 Folkhälsan Insttute of Genetcs, Department of Genetc Epdemology, Helsnk, Fnland Artcle Info: Receved : May 4, Revsed : June 7, 2012 Publshed onlne : July 30, 2012

2 32 Estmaton of Gn coeffcents Keywords: Golden s method, Lagrange polynomals, Pareto dstrbuton, Smpson s rule, Trapezum rule 1 Introducton Prmary ncome data yelds the most accurate estmates of the Gn coeffcent. However, the estmaton must often be based on tables wth grouped data or on Lorenz curves. The Lorenz curves are usually defned for fve quntles or for 10 decles. If one uses the Lorenz curve, the Gn coeffcent s defned as the rato of the area between the dagonal and the Lorenz curve and the area of the whole trangle under the dagonal. For fve quntles, the trapezum rule s the most commonly used method. However, ths rule yelds postve bas for the estmate of the area under the Lorenz curve for every trapezum and, consequently, the rule causes negatve bas for the Gn coeffcent. Smpson s rule s better ftted to the Lorenz curve, but demands an even number of subntervals of the same length. Ths means, for example, that Lorenz curves wth 10 decles are sutable. One has three L values for each doubled subnterval. The area under ths part of the Lorenz curve s estmated so that the Lorenz curve s approxmated by a parabola obtanng the same L values. Smpson s rule obvously yelds exact results for quadratc curves but, n general, ths also holds for cubc curves. Lagrange polynomals of the second degree can be consdered as a generalsaton of Smpson s rule and do not demand subntervals of equal length, but the number of subntervals should stll be even. The polynomals obtaned have to be ntegrated n order to yeld approxmate areas and Gn coeffcents. If the subntervals are of the same length, the Lagrange polynomal method s dentcal wth Smpson s rule. Varous attempts have been made to produce more exact estmates. Gastwrth (1972) ntroduced nterval estmates of the Gn coeffcent n order to measure the accuracy of the estmates. Needleman s study (1978) starts from the trapezum estmate of the Gn coeffcent G L. He then ntroduces an mproved upper estmate G U. Hs fnal estmate follows the two-thrds rule that s G 2GU G L. 3 3 McDonald and Ransom (1981) consdered the Γ densty, appled Monte Carlo methods and ntroduced lower and upper bounds of the Gn estmates. Golden (2008) showed how a quck approxmaton of the Gn coeffcent can be calculated emprcally, usng numercal data n cumulatve ncome quntles. In ths study, we ntend to compare dfferent methods. When we apply Smpson s rule, we consder Lorenz curves wth decles. In addton, we use Lagrange polynomals and generalzatons of Golden s method.

3 Johan Fellman 33 2 Methods There are several dfferent stuatons and, consequently, alternatve analyses of Gn coeffcents have to be performed. When Lorenz curves are consdered, the smplest stuaton s that they are defned for fve quntles or for 10 decles. In the frst case, the most commonly used method s the trapezum rule. For Smpson s rule, the number of subntervals should be even and the ntervals should have the same length. Consequently, the comparson of dfferent rules can be performed for Lorenz curves wth decles. Assume a Lorenz curve L ( p) wth decles. Let the observed values of the cumulatve Lorenz curve be p and L for 0, 1,..., 10. Note that p / 10, ( 0, 1,..., 10 ), that L 0 0 and that L Accordng to the trapezum rule, the estmated area under the Lorenz curve s 9 I L L p p (1) ½ 0 1 and the estmated Gn coeffcent, G T s 1 2I. Every trapezum yelds a postve bas to the estmated area, as can be seen n Fgure 1. Snce the bases obtaned add and no elmnaton of bases can be performed, the estmated Gn coeffcent always has a negatve bas. 1 Trapezum Fgure 1: A sketch showng the bas n the trapezum rule Compared to the trapezum rule, Smpson s rule gves more accurate approxmatons. As stressed above, Smpson s rule demands two restrctons: the number of subntervals has to be even and the subntervals have to be of equal length. In order to obtan Smpson s rule, the subntervals should be grouped two by two. Each doubled subnterval has three L values. The area under ths part of the Lorenz curve s estmated such that a parabola obtanng the same L values approxmates the Lorenz curve. Assumng 2n subntervals, the approxmate area

4 34 Estmaton of Gn coeffcents formula for a doubled nterval s I L 4L L 1 3n 0 1 2, the total sum s 4 1 I L2 4L21 L22 (2) 3n and G S 1 2I. Golden (2008) gave a detaled account of an alternatve method based on Lorenz curves wth quntles. He consdered p and L n percentages. The layout of the method s presented n Table 1. Frst he determned where the cumulatve ncome shortfall s greatest and defned Z as the largest quntle pont of the cumulatve ncome shortfall from perfect equalty dvded by 100. In order to obtan the largest cumulatve ncome shortfall he defned the transformed varable L L Ths transformaton, L L 1 20, ndcates a search for an nterval at whch L shfts from ncreases faster than p to slower ncreases. For low s, the transformed value L L. Later, there s a frst value such that L L. For ths value, one fnds an nterval for whch L s closely parallel wth the dagonal, the greatest shortfall s obtaned, and one defnes q ( 20 L ) / 100. The estmated Gn coeffcent n percentages, G G, s 50q( 3 q). When ths method was appled to 621 ncome observatons, G G Golden (2008) noted that hs approach performed better than the trapezum rule, also stressng that hs method could be appled to Lorenz curves wth decles. Followng Golden (1980), the data s gven n percentages. The transformed varable L 20 s gven n the text. Table 1: A layout of a Lorenz curve wth decles p L L 0 0 L 20 L40 L 60 L 80 L L L 0 0 L 20 L 40 L 60 L 80 L 100 We try to generalze Golden s method n the followng way. If the Lorenz curves are gven n decles, then Golden s transformaton should be L L 1 10 and f the p s are not equdstant, then one has to defne L L 1 p p1. Followng Golden s rule, these processes have to contnue untl L L. We then ntroduce q ( p L ) 100 and 50q( 3 q). / G G

5 Johan Fellman 35 In many emprcal stuatons, the ncome dstrbuton F (x) s gven n grouped tables. If the mean of or total ncomes n the groups are known, the cumulatve dstrbuton can be consdered as a Lorenz curve, but the subntervals are usually not of constant length. The trapezum rule holds, but t stll yelds a postve bas for the area and negatve bas for the Gn coeffcent. An obvously better alternatve s to approxmate the Lorenz curve wth Lagrange s nterpolaton (Berrut & Trefethen, 2004). We apply the Lagrange nterpolaton of second degree, whch s a generalzaton of Smpson s rule. However, we have to assume an even number of subntervals. Now the Lagrange polynomal s n1 ( p p2 1)( p p22) L( p) L2 0 ( p2 p2 1)( p2 p22) ( p p22)( p p2) ( p p2 1)( p p2) (3) L2 1 L22 ( p2 1 p22)( p2 1 p2) ( p22 p2 1)( p22 p2) Ths approxmate polynomal must be ntegrated n order to obtan an estmate of the area under the Lorenz curve. Ths attempt s a generalzaton of Smpson s rule for cases wth subntervals of varyng lengths. The comparson between dfferent estmaton methods s n general dffcult to perform. These dffcultes are manly caused by the fact that the true Gn coeffcent s unknown, but sometmes, where more detaled studes have already resulted n very accurate estmates, the comparsons are possble. Some authors (e.g., Gastwrth, 1972; Mehran, 1975; McDonald & Ransom, 1981; Rgo, 1985; Gorg & Palln, 1987) have ntroduced nterval estmates, but these are often rather broad and t s stll dffcult to dentfy the best method. Such comparson problems are elmnated f the numercal estmatons are appled to theoretcal dstrbutons. Needleman ((1978) stated that as the Lorenz curve s convex, the trapezum approxmaton s always greater than the actual area under the curve, so that the estmate based on ths approxmaton s always less than the actual value of the coeffcent. Furthermore, he noted that most authors usng the trapezum approxmaton ndcate that they are aware of the bas nvolved, but ether assume the error so small as to be nsgnfcant, or else use a large number of ntervals n the belef, usually justfed, that the bas wll then be neglgble. Needleman s own study started from the trapezum estmate of the Gn coeffcent G L. He then ntroduced an mproved upper estmate, G U. Hs fnal estmate follows the G 2GU two-thrds rule, that s G L. 3 3 McDonald and Ransom (1981) ntroduced lower and upper bounds of the Gn estmates. In order to estmate the bounds of the Gn coeffcent estmates,

6 36 Estmaton of Gn coeffcents y y 1 e they consdered the Γ densty, that s, g ( y) wth correspondng ( ) ( ½) G and / and appled Monte Carlo methods. ( 1) In order to perform comparsons between the estmated and theoretcal Gn coeffcents we analyze classes of theoretcal Lorenz curves wth varyng Gn coeffcents. In ths study we compare Gn estmates for the Pareto dstrbutons. We defne the Pareto dstrbuton as F ( x) 1 x, where x 1 and 1. 1 The frequency functon s f ( x) x, the mean s, the quantles are x p, the Lorenz curve L( p) 1 1 p and the Gn coeffcent 1 p 1 G. If we consder , then the Gn coeffcent satsfes the 2 1 nequaltes G Results Teppng data. Gastwrth (1972) presents nterval estmatons of the Gn coeffcent. The exact Gn estmate on Current Populaton Surveys (CPS) ncome data for 1968 was computed by Teppng, hs result beng Gastwrth s Table 2 shows Teppng s data grouped nto a 10 subgroup Lorenz curve. He compares hs Gn nterval estmates wth Teppng s fndng. Gastwrth (1972) consders a mnmum of restrctve condtons, obtanng the nterval G Mehran (1975) suggests an alternatve estmaton method, obtanng the nterval estmate G The groupng lmts n Table 2 are not equdstant and one cannot apply Smpson s rule. Applyng the trapezum rule yelds and the negatve bas s apparent. The Lagrange rule yelds and the modfcaton of the Golden s rule yelds the rather naccurate estmate Lorenzen data. Lorenzen (1980) presents nformaton about the total dstrbuton of ncome for households n Germany n 1973 n hs Tabelle 2. The Gn coeffcent calculated by Lorenzen s based on data pooled n hs Tabelle 3, whch yelded Usng Lorenzen s Tabelle 3, we perform a comparson of the estmates obtaned based on the trapezum rule and the Lagrange rule. The avalable emprcal data cannot yeld a comparson of the accuracy of the two methods. The estmated Gn coeffcent accordng to the trapezum rule shows negatve bases compared to Lorenzen s result, beng The Lagrange nterpolaton yelds the estmate and the modfed Golden method

7 Johan Fellman 37 In order to analyse the accuracy of the dfferent methods, we nclude some theoretcal studes n ths study. For the Pareto dstrbutons presented above, we consder , that s, G Ths G nterval corresponds to the most common Gn coeffcents. The results appear n Table 2 and Fgure 2. Note that Smpson s and Golden s rules yeld smlar accuracy, but the trapezum rule shows the largest errors for all levels of Gn coeffcents. Ths theoretcal study ndcates that Golden s rule s not unformly better than the trapezum rule. Table 2: The estmaton of the Gn coeffcent n per cent appled to the Lorenz curve for the Pareto dstrbutons. The estmated Gn coeffcents accordng to the trapezum rule are naccurate and show negatve bases. Smpson s and Golden s rules yeld smlar accuracy, but Golden s s best for large Gn values. Estmates Error G Trapezum Smpson Golden Trapezum Smpson Golden Errors n the Gn estmates for Pareto dstrbutons Trapezum error Smpson error Golden error Gn coeffcent (%) Fgure 2: Estmaton errors n the Gn coeffcents estmated by the trapezum, Smpson s, and Golden s rules.

8 38 Estmaton of Gn coeffcents Note that Smpson s and Golden s rules yeld smlar accuracy, but the trapezum rule shows the largest errors. 5 Dscusson Ths study ndcates that the based trapezum rule s almost always nferor and shows negatve bases. No method however s unformly optmal. Note that Smpson s and Golden s rules yeld smlar accuracy. Golden s method s usually of medum qualty, but ts accuracy fluctuates. ACKNOWLEDGEMENTS. Ths work was supported n part by a grant from the Magnus Ehrnrooth Foundaton. References [1] J. P. Berrut and L.N. Trefethen, Barycentrc Lagrange nterpolaton, SIAM Revew, 46(3), (2004), [2] J.L. Gastwrth, The estmaton of the Lorenz curve and Gn ndex, Rev. Economcs and Statstcs, 54, (1972), [3] G.M. Gorg and A. Palln, About a general method for the lower and upper dstrbuton-free bounds on Gn s concentraton rato from grouped data, Statstca, 47, (1987), [4] J. Golden, A smple geometrc approach to approxmatng the Gn coeffcent. J. Economc Educaton, 39(1), (2008), [5] G. Lorenzen, Was st en echtes Konzentratonsmaß? Allgemenes Statstsches Archv, 4, (1980), [6] J.B. McDonald and M. R. Ransom, An analyss of the bounds for the Gn coeffcent, Journal of Econometrcs, 17, (1981), [7] F. Mehran, Bounds on the Gn ndex based on observed ponts of the Lorenz curve, J Amer Statst. Assoc. JASA, 70, (1975), [8] L. Needleman, On the approxmaton of the Gn coeffcent of concentraton, The Manchester School, 46, (1978), [9] P. Rgo, Lower and upper dstrbuton free bounds for Gn s concentraton rato, Proceedngs Internatonal Statstcal Insttute, 45 th Sesson, Amsterdam, Contrbuted Papers, Book 2, (1985),