A Piecewise Method for Estimating the Lorenz Curve

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1 DEPATMENT OF ECONOMICS ISSN DISCUSSION PAPE 05/15 A Pecewse Method for Estatg the orez Curve ZuXag Wag 1 ad ussell Syth 2 Abstract: We roose a ecewse ethod to estate the orez curve for groued coe data. Our llustratve alcato shows that the ethod ca roduce ore lausble desty estates whe the coe dstrbuto data has ultle eaks ad the orez curve caot be odeled satsfactorly over the etre terval wth a sgle orez curve odel. 1 Deartet of Ecoocs, Wuha Uversty, Wuha , Cha 2 Deartet of Ecoocs, Moash Uversty, Clayto, Vc 3800 Australa; Telehoe: + (613) ; Fa: + (613) ; Eal: russell.syth@oash.edu 2015 ZuXag Wag ad ussell Syth All rghts reserved. No art of ths aer ay be reroduced ay for, or stored a retreval syste, wthout the ror wrtte ersso of the author oash.edu/ busess-ecoocs ABN CICOS Provder No C

2 1. Itroducto orez odels ca be used to odel groued coe data to obta both the orez curve ad frequeces of the uderlyg coe dstrbuto. May odels are avalable the lterature (eg. Basa et al. 1990; y & Slottje 1996; Ogwag & ao 2000; Saraba et al., 1999, 2001; ohde 2009). A drawback of these odels s that they do ot sultaeously rovde a good ft for both the orez curve values ad frequeces (Wag et al. 2011). We suggest a ecewse orez curve ethod to address ths drawback. Moreover, coe dstrbuto data wth ultle eaks les coe olarzato, whch has draw uch recet atteto (see, eg, Foster & Wolfso 2010). The ethod that we roose ca be used to odel such dstrbutos. Kakwa (1976) ad Cowell ad Mehta (1982) cosder coe data aroato by ecewse terolatg destes ad the obtag orez curves fro the destes. Our aroach s to do the ooste. Secfcally, we ecewse aroate orez curves ad the obta the destes fro the orez curves. 2. The ecewse ethod for estatg the orez curve Suose we have groued coe data o terval [ c, d] 1,,,, (1) 1 where s the roorto of coe uts whose coe s less tha, or equal to, ad ad are the coe share ad average coe resectvely for those coe uts o [ c, d]. Assue et l be the actual orez curve o [ c, d]. It follows: l( ), l( ), 1,2,,. (2) et the actual dstrbuto fucto for coes o [ c, d] be F () wth F ) ad F ), where ad are ostve tegers wth (, lyg c, ] ad [, d] resectvely. [ (, ad. Therefore, ad 1 are oulato shares o et ad be estated orez curves for coes o c, ] ad [ [, d] resectvely. Assue, ]. Defe the two-ece-orez-curve [

3 (TPC) estato for the groued coe data (1) as: F( ), (3) F( ), 1 1 where l ) ad F ). ( ( Theore: (3) s a orez curve. Furtherore, f ad are actual orez curves o c, ] ad [, d] resectvely, (3) equals l. [ Frst, f ad satsfy the defto of the orez curve, so does, because ( 0) 0, (1) 1, ( ) 0, 0. Secod, f s the actual orez curve for coes o c, ] [, the t ust satsfy l( ), (4) for ay F() wth. If s the actual orez curve for the coes o [, d], the t ust satsfy for ay F() wth 1 sde of (3) equals l for ay [0,1 ]. l( ) 1, (5). elatoshs (4) ad (5) ly that the rght-had The TPC estato rocedure (3) s useful whe there s a eak ether c, ] [ or [, d], or both, ad the orez curve o the etre terval caot be odeled satsfactorly wth a sgle orez curve odel. (3) ca be geeralzed to ultle cases. For eale, oe could estate a three-ece-orez-curve for the etre dstrbuto by further creatg a TPC for coes o [, d].

4 To aly the TPC, we use a orez odel to estate groued datasets for coes o c, ] ad [, d] resectvely, obtag estated orez curves ad [. Eterg the to (3), we obta the TPC o [ c, d]. Irresectve of the ethod aled locally to the two ortos, the covety of the resultg curve s reserved accordg to the theore. F() (3) s relaced by the estated dstrbuto fucto for all coes o [ c, d]. It ca be obtaed by solvg for ay 0 ad the related desty f () ca be estated by 1 f ( )., 1,, s used as a crtcal ot jog ad forg the TPC. Dfferet values of wll result dfferet F (), ad f (). ca be detered by solvg F( ) F( ). (6) Alteratvely, sce the estated desty s geerally dscotuous at ay selecto of, 1,,, we ca take to ze the ju desty at by solvg l 0 f ( ) f ( ). (7) 3. Illustratve alcato I ths secto we coare the ft of the TPC wth that of a sgle estated orez curve (EC) ftted to the etre dataset, aled to Swedsh coe dstrbuto data for 1977 rovded Cowell ad Mehta (1982). We use Swedsh coe data because t clearly cotas ultle eaks, as deostrated Fgure 1 below. The orez odel G (1 ) ( 1 ), (8) wth 1, 1 ad [0,1 ], s used to ft the etre dataset to obta the EC where

5 , (0,1 ], 1 (1 ) e e 1 1, 0. Dvde the coe terval to two subtervals. The odel H (1 ), (9) wth 0, 0, 1 ad [0,1 ], s aled to both groued datasets for the sub-tervals. H s cosdered by Wag et al. (2011), s the orez curve for the classcal Pareto dstrbuto ad s roosed by Chotkaach (1993). We use the balaced ft aroach suggested by Wag et al. (2011), whch estates araeters by zg b 1 2 ) (1 b) Fˆ( ) 1 2 (, b [0,1 ], (10) where s the orez odel ad F ˆ ( ) s the solver of the equato, ad s the estated frequecy of coe uts at. s average coe. We oly use b 1 or b 0 below. The araeter estates are gve the Aed. The dashed curve Fgure 1 s the EC for the Swedsh data, whle the two sold curves together are the TPC for the data, both wth b 1. Fgure 2 dslays the couterart curves wth b 0. Note that wth 7 ad wth 9 are selected for both estates. The jog ots 8 detered by (7) are the sae for both b 1 ad b 0. The colus ttled EC Table 1 gve estated orez curve values whe odel (8) s used ad alteratvely b 1 or b 0 s aled. The colus ttled TPC gve the corresodg two-ece-orez-curve aroato whe odel (9) s used. We follow Saraba et al (1999) ad use the ea squared error (MSE), ea absolute error (MAE) ad the au absolute error (MAX) to easure goodess of ft. These are lsted the last three rows. The TPC s a better ft tha the EC wth b 1 or b 0. Betwee alteratve b values, the TPC erfors better whe b 1. Table 1 orez curve aroato

6 b 1 b 0 EC TPC EC TPC MSE MAE MAS Fgure 1. TPC ad EC usg balaced ft wth b 1

7 Fgure 2. TPC ad EC usg balaced ft wth b 0 Table 2 gves the frequecy estato fro the estated orez curves, whch the colu ttled f cotas frequeces of coe uts alteratve tervals. The estates fro the EC wth b 1 ad b 0 are dstgushable, lyg o roveet ca be obtaed through selectg b. The estates wth the TPC are better tha ther EC couterarts wth b 1 or b 0. The TPC desty wth b 1 Fgure 1 s vsually less lausble tha ts couterart wth b 0 Fgure 2. However, the forer roduces a very attractve orez curve estate. Net, we estate the G de ad the olarzato de roosed by Foster ad Wolfso (2010). We do so for both the etre oulato ad a ajorty larger tha 89 er cet of the oulato wth coe less tha h 70. The fdgs are reorted Table 3. The G estates for the etre oulato are larger tha 0.35 for the EC ad TPC wth b 1 or b 0, whch s slghtly larger tha the estates gve by Cowell ad Mehta (1982). The G estates are ot sestve across the EC ad TPC to the choce of b. The olarzato de for the etre oulato s alost the sae for the EC ad TPC for both values of b. However, the estates for the suboulato for the de are larger for the TPC, tha for the EC.

8 Table 2 Frequecy aroato fro estated orez curves b 1 b 0 f EC TPC EC TPC MSE MAE MAS Table 3 Iequalty ad olarzato estato wth Swedsh 1977 data Etre Poulato 89% at ower Icoe G Polarzato G Polarzato b 1 EC TPC b 0 EC EPC Cocluso We have roosed a ultle orez curve ethod to odel groued coe data. Usg Swedsh coe data fro 1977, we deostrate that the ethod overcoes the drawback that the sgle orez curve ethod caot roduce good

9 aroatos to both the orez curve ad to the frequeces at the sae te. We show that the TPC works artcularly well whe the coe data has ultle eaks.

10 efereces Basa,.., K. J. Hayes, D. J. Slottje, J. D. Johso, A geeral fuctoal for for aroatg the orez curve. Joural of Ecooetrcs 43, Chotkaach, D., A coarso of alteratve fuctoal fors for the orez curve. Ecoocs etters 3, Cowell F.A., F. Mehta, The estato ad terolato of equalty easures. evew of Ecooc Studes, 49, Foster J.E., M.C. Wolfso, Polarzato ad the decle of the ddle class: Caada ad the U.S.. Joural of Ecooc Iequalty 8, Kakwa, N., O the estato of coe equalty easures fro groued observatos. evew of Ecooc Studes 43, Ogwag, T., U.. Gouraga ao, Hybrd odels of the orez curve. Ecoocs etters 69, ohde, N A alteratve fuctoal for for estatg the orez curve. Ecoocs etters 105, y, H.K., D.J. Slottje, Two fleble fuctoal for aroaches for aroatg the orez curve. Joural of Ecooetrcs 72, Saraba, J., E. Castllo, D.J. Slottje, A ordered faly of orez curves. Joural of Ecooetrcs 91, Saraba, J., E. Castllo, D. J. Slottje, A eoetal faly of orez curves. Souther Ecooc Joural 67, Wag, Z., Y-K Ng,. Syth, A geeral ethod for creatg orez curves. evew of Icoe ad Wealth 57,

11 Aed: Paraeter estates of the orez curves eloyg Swedsh 1977 data 1 b b b 1 eft ght b 0 eft ght