Celestin Chameni Nembua University of YaoundéII, Cameroon. Abstract

Transcription

1 A ote o te decompositio of te coefficiet of variatio squared: comparig etropy ad Dagum's metods Celesti Camei Nembua Uiversity of YaoudéII, Cameroo Abstract Te aim of tis paper is to propose a ew decompositio of te coefficiet of variatio squared. Te approac is similar to te Dagum s metod we decomposig te Gii ide. We compare te ew metod to te former etropy decompositio of tis coefficiet. A empirical study is elaborated. Tis cocers a subgroup decompositio of te o food epediture cameroia ouseolds iequality. Citatio: Camei Nembua, Celesti, (006) "A ote o te decompositio of te coefficiet of variatio squared: comparig etropy ad Dagum's metods." Ecoomics ulleti, Vol. 4, No. 8 pp. 8 Submitted: November 0, 005. Accepted: February, 006. URL: ttp:// 05D60003A.pdf

2 - Itroductio Tis paper proposes a ew subgroup decompositio of te coefficiet of variatio squared. It is equal to twice te Hirscma-Herfidal ide wic belogs to te class of geeralized etropy idices obtaied wit te parameter or C. Te approac ere is similar to te oe used by Dagum (997a; 997b) we decomposig te Gii ide ad it is based o te iterpersoal epressio of te coefficiet of variatio squared; so, we sall cosider tis as Dagum s metod. Te compariso betwee tis metod ad te oe issued from te geeralised etropy idices decompositio (Cowell, 980) is operated. Te teoretical results are applied to aalyse te decompositio of ouseold o food epediture iequality i Cameroo. Te ECAMII-00(a ouseold survey carried out by Cameroo s Natioal Istitute of Statistics) data base is used ad te empirical study is carried out usig te free program provided o te web site ttp// for te etropy metod i oe ad, ad usig a computer program we ave desiged for applicatios of our metod i aoter ad. Te remaiig tet is subdivided ito four sectios i additio to te preset itroductio. I Sectio, we establis te iterpersoal epressio of te coefficiet of variatio squared (sectio.) ad we preset is Dagum subgroups decompositio. Sectio 3 is devoted to a rapid overview of te etropy decompositio of te coefficiet of variatio squared. As to sectio 4, te precedig results are applied to decompose te iequalities i o food epediture of Camerooia ouseolds. Te results obtaied ere lead to te compariso of te two decompositio metods ad we motivate our preferece to te Dagum decompositio by givig importat reasos for tat. Fially te paper is cocluded i sectio 5. - Te Dagum subgroup decompositio of coefficiet of variatio squared Lets cosider a populatio P wit icome uits,, 3 i were CV, Var ad are respectively te square of coefficiet of variatio, te variace ad te mea o P. We assume tat P is partitioed ito subpopulatios P,P,P 3,,P,,P wit respectively,, 3,,,, members;,,..., CV ad are respectively te icome uits, te square of coefficiet of variatio ad te mea o P.. Te iterpersoal epressio of te coefficiet of variatio squared y defiitio, Var i i CV i i i + i i i i + ² i + - i - i ( ) ( ) i ² i i i () () (3)

3 i i (4). Te epressio of te decompositio We set, for,, : f ad s. a) Decompositio ito two compoets Usig (4), we ave: CV i i (5) i i Te mea of te differeces of order betwee te subpopulatios P ad P is defied by: i i () () α α ( α α α α + ( + ) ( ) E X X ad it follows tat : (6) CV ) (7) Let s itroduce te ide of iequality betwee te subpopulatios P ad P : () G (8) + We ave i particular: G () CV ( ) G + i i CV (9) (0) CV + G ( + ) () f s CV + ( ) f s + f s G CV + () W CV Were CV W is te witi group compoet ad b) Decompositio ito tree compoets CV is te gross betwee group compoet. Te gross ecoomic wealt oted d, is defied betwee two subpopulatios P ad P suc tat >. d is te mea of te differece ( i ) for eac icome i of a member i P greater ta icome of a member i P : + y d df ( y) y df ( ) 0 0 Were i i > i ( E X X i ) () (3) i (4)

4 Followig Dagum, we set p d ( if ) >. p correspods to te trasvariatioal compoet. Te et ecoomic wealt betwee two subpopulatios P ad P suc tat >, is defied by te differece d p > 0 ad te relative ecoomic differece betwee two suc subpopulatios is give by : d p D (5) () It is clear tat, (), (), G ad D defie symmetric matrices ad it is well ow (Dagum 997b) tat D is a distace o te set of distributios X wic is ull if ad oly if tere is perfect overlappig betwee distributios ad 0 D. Accordig to () ad isertig (5) icv, CV f scv + ( f s + f s )( D + D ) G CV f scv + D ( f s + f s ) G CV CV W CVN + CVT CVW f + ( )( ) (6) D f s + f s G (7) + were (8) s CV CV ( f s + f s ) G N D iequality to te overall is te cotributio of te witi subgroup iequality to te overall CV. CV. is te et cotributio of te betwee subgroups CV T ( D )( f s + f s ) G measures te cotributio to te overallcv, of te iequality comig from te trasvariatio betwee te subgroup pairs. c) Cotributio of eac group to te gross betwee group compoet Te classical decompositio of te variace implies: CV s + CV (9) y equatig () ad (9) we obtai: ( ) + CV s f CV. (0) Wat permit to gauge te cotributio of P to CV : trasvariatio come from travariazioe wic is te term used by C.Gii i 96. 3

5 ( P ) + s ( f ) CV CV () Formula (0) reveals i particular tat te gross betwee groups compoet, ad cosequetly te total CV ide, are icreasig fuctios of te witi groups idices; wic meas tat tis decompositio satisfies te Sorrocs (994) subgroup cosistecy property. 3- Te etropy subgroup decompositio of coefficiet of variatio squared It is obtaied as particular case of geeralized etropy ratio wic is epress by: i i I ( + ) real. () i Te geeralised etropy ca be decomposed (Cowell 980) ito two compoets, te witi group compoet ad te betwee group compoet suc as : I I + (3) ( + ) I W + I Te coefficiet of variatio squared equal wit. We deduce from (3) tat: CV CV CV scv + s 4- Applicatio I + (4) I W + I (5) We study o food epediture ouseold iequality i Cameroo. Te ECAMII-00 database is used ad it icludes 099 ouseolds subdivided accordig to teir residetial areas: Urba (group, 4975) Semi-urba (group, 37) ad Rural (group 3, ). Te two decompositio metods itroduced above allow oe to ow if te iequalities are geerated by te epediture gaps witi te tree residetial areas or if iequalities come from te epediture gaps betwee te tree groups. Te computatio of te etropy decompositio is provided by te free program o te web site ttp// wile te results o te Dagum decompositio are obtaied by a program pacage tat we ave desiged for te circumstaces. Table illustrates tese results i givig te cotributio of eac compoet of te two decompositios to te global iequality. Te etropy metod sows tat te cotributio witi te subpopulatio represet 93.38% ad te differeces betwee te tree areas represet oly 6.6% of te global iequality; wereas te Dagum metod grats a little differece betwee te witi groups elemet (40.69%) ad te betwee group elemet (59.3%) wit, i cotrary, te predomiace of te betwee group compoet. Oly te Dagum metod ca provide te itesity of et betwee group compoet ad te itesity of trasvariatio wic is te part of te betwee groups disparities issued from te overlap betwee te tree distributios. Te results obtaied ere sows tat, a cosiderable part (59.03%) of te global iequality comes from overlappig betwee te distributios of te tree groups ad te trasvariatioal betwee group compoet represets 99.5% (tat is 4

6 almost te totality) of te gross betwee group compoet. I te oter ad, te Dagum metod permits to gauge te cotributio of eac pair of groups to te et ad te trasvariatioal betwee groups compoet. Table4 idicates tat te pair (group ; group3), (group; group) ad (group; group3) eplai respectively 6.83%, 33.4% ad 4.75% of te trasvariatioal compoet; wic ere idicates a sigificat overlappig of te amout of o food epediture distributios i group3 ad group, as well as relative lieess betwee a cosiderable fractio of tese distributios. Te pairs (group ; group3) ad (group; group3) cotribute otig to te et betwee group compoet, wat cofirms a perfect overlappig of teir correspodig distributios. Table reveals tat, usig te etropy metod, te group3 (Rural area) cotributes egatively (-0.86%) to te global iequality wile all te cotributios of te tree groups are positive i te Dagum metod. Tis peomeo of egative cotributio we usig etropy metod, are reiforced i table3 were te cotributio of group (-4.63%) ad group3 (-37.0%) to te betwee groups compoet are egative agai. Tis implies a ucomfortable situatio ad sows tat, te etropy decompositio of te coefficiet of variatio squared (or te Hirscma-Herfidal ide) sould be used wit a lot of precautios for, it betwee groups compoet ave egative terms wic may lead to osese iterpretatios. Altoug te two decompositios metod uveil togeter tat te group (Urba area) plays a cetral role i geeratig iequality (te cotributio of tis group to te overall iequality (cf. table) is iger i te two metods ad equal respectively to 95.68%, 89.37% i te etropy ad Dagum metod ), te differece of te results betwee tese two metods are importat. So it seems idispesable to direct te coice of te users of te decompositio of te coefficiet of variatio squared (or te Hirscma-Herfidal ide). We icite to privilege te Dagum decompositio for may reasos: ) Te Dagum metod is based o a iterpersoal epressio of te coefficiet of variatio squared ad tus itegrate te criteria of te iterpersoal utility compariso lie te Gii ide ( for more details, see Dagum, 980 or Mussard ad alii., 003 ). ) Te Dagum metod is built o a better betwee group specificatio; ot oly its betwee groups compoet is a effective iequalities betwee te subgroups, but also its trasvariatioal compoet costitutes a ericmet wic permit to gauge iequalities comig from overlappig betwee te icome distributios of various subgroups. Wile te etropy decompositio of te coefficiet of variatio squared as a betwee groups compoet wic is a simple differece i mea. Moreover, tis betwee groups cotributios are obtaied lie a residual ( I CV IW ) tat geerates egative terms wic may lead to osese iterpretatio as see above ad as Dagum as oticed i ) Te Dagum decompositio of te coefficiet of variatio squared satisfies te Sorrocs (994) subgroup cosistecy property. Tis meas tat cages (icrease or decrease) i oe of te witi group ide implies cages (i te same directio) i te overall iequality ide. 5- Coclusio We ave preseted aoter way to decompose te coefficiet of variatio squared similar to te oe used by Dagum we decomposig te Gii ide. Te compariso of te ew metod to te former etropy decompositio of te coefficiet of variatio squared as bee doe troug a empirical study cocerig te decompositio of te iequality i te o food epediture Camerooia ouseolds. Te results obtaied lead directly to te 5

7 preferece of te ew metod of decompositio. Sice our coice as bee motivated, we icite te users to give privilege to te ew metod. REFERENCES Cowell F.A. (980a) Geeralized etropy ad te Measuremet of distributioal Cage, Europea Ecoomics Review, vol 3: Cowell F.A. (980b) O te Structure of additive Iequality Measures, Review of Ecoomics Studies, vol 47, Dagum C. (997a) A ew Approac to te decompositio of te Gii Icome Iequality Ratio, Empirical Ecoomics, (4),p Dagum C. (988) Iequality Measures betwee Icome Distributio wit Applicatio, Ecoometrica Dagum C. (997b) Decompositio ad Iterpretatio of Gii ad te Geeralized Etropy Iequality Measures, Proceedigs of te America Statistical Associatio, usiess ad Ecoomic Statistics Sectio, 57 t Meetig, p Gii C.(96), II cocetto di trasvariazioe e le sue prime applicatiozioi, Giorale Musard S., Seyte F., Terraza M. (00b), Programme pour la decompositio de l idice de Gii de C.Dagum, site iteret ttp:// Musard S., Seyte F., Terraza M. (003), Decompositio of Gii ad te Geeralized etropy Iequality Measures, Ecoomics ulleti, vol.4 7,-6. Sorrocs A.F. (984), Iequality Decompositio by factors Compoets ad by Populatio Subgroup, Ecoometrica, vol.53, Table : Cotributio of eac elemet of te two decompositios to te overall iequality Decompositio Metod Witi group compoet etwee group compoet Net betwee group compoet Trasvariatioal compoet Absolute % Absolute % Absolute % Absolute % Etropy NA* NA* NA* NA* Dagum *NA: No available for tis metod Table : Cotributio of eac group to te global iequality Groups Etropy Metod Dagum Metod Absolute % Absolute % Group 3, ,68 3,86 89,37 Group 0, ,8 0,969 5,50 Group 3-0,099-0,86 0,7858 5,3 Global 3, ,00 3, ,00 6

8 Table 3: Cotributio of eac group to te witi ad betwee group compoet Etropy Metod Dagum Metod Groups Witi group compoet etwee group compoet Witi group compoet Gross etwee group compoet Absolute % Absolute % Absolute % Absolute % Group 3, ,4 0,365 4,65, ,00,759 84,8 Group 0,903 5,87-0,006-4,63 0,0374,6 0,5455 7,48 Group 3 0,05544,70-0, ,0 0,0957,38 0,590 7,70 Global 3, , ,478 00, Table4: Pair wise groups cotributio to te et ad trasvariatioal betwee groups compoet Net betwee groups compoet Trasvariatioal betwee groups compoet Group Group Group3 Group Group Group3 Group Group Group ,4-6,83 4,75 - Global N: Tis table is ot available for te etropy metod 7