Variance Estimation for Measures of Income Inequality and Polarization ± The Estimating Equations Approach

Transcription

1 Journal of Of cal Stattc, Vol. 13, No. 1, 1997, pp. 41±58 Varance Etmaton for Meaure of Income Inequalty and Polarzaton ± The Etmatng Equaton Approach Mlorad S. KovacÆevc 1 and Davd A. Bnder 2 The etmatng equaton technque for varance etmaton demontrated on a varety of ncome nequalty and polarzaton meaure when data are obtaned n a complex urvey. Th method, baed on the Taylor lnearzaton, computatonally nonntenve and eay to mplement. Sx dfferent meaure are condered. An example baed on data from the Canadan Survey of Conumer Fnance gven. Key word: Coef cent of varaton; exponental meaure; Gn ndex; Lorenz curve ordnate; polarzaton curve ordnate; polarzaton ndex. 1. Introducton Etmate of meaure of ncome nequalty and polarzaton are often requred n tude of ncome dtrbuton. When ncome dtrbuton are compared from regon to regon or through tme one hould account for ther amplng varablty. Lack of nformaton on tandard error con ne the role of meaure of ncome nequalty to that of decrptve devce rather than nferental tool for formal tattcal nference. The common charactertc of thee meaure ther complexty. They are nonlnear functon of the obervaton. Some of them depend on the ordered obervaton or quantle. In addton, ncome data uually come from complex urvey (trat ed, multtage, cluter ample wth unequal probablte of electon). Conequently, the varance of thee meaure are not expreble by mple formulae. Snce they cannot be etmated by conventonal varance etmaton method, one ha to rely on approxmate varance etmaton technque. Here an approxmate tandard error etmaton technque preented. It baed on the theory of etmatng equaton (EE) a developed by Bnder (1991), and Bnder and Patak (1994). The problem of etmatng ncome nequalty meaure and ther tandard error ung etmatng equaton ha been addreed by Bnder (1992) and the extenon to urvey amplng wa made by Bnder and KovacÆevc (1995). In th artcle, the EE method appled to etmatng ome common meaure of ncome nequalty and ther tandard error that were not addreed prevouly by Bnder and KovacÆevc (1995). Thee 1 Mlorad S. KovacÆevc a Senor Methodologt wth the Socal Survey Method Dvon, Stattc Canada. 2 Davd A. Bnder Drector of the Bune Survey Method Dvon, Stattc Canada, R.H. Coat Buldng, Tunney' Pature, Ottawa, Ontaro, Canada K1A 0T6. Acknowledgment: We thank H.J. Mantel for h careful readng and ueful comment on prevou veron of th artcle. We are alo grateful to the aocate edtor and referee for uggeton mprovng the readablty of the artcle. q Stattc Sweden

2 42 Journal of Of cal Stattc meaure are the coef cent of varaton and the exponental meaure of nequalty. We alo preent the etmaton of the polarzaton curve and the polarzaton ndex a de ned n Foter and Wolfon (1992) and Wolfon (1994). Some bac EE theory revewed n Secton 2. The nte populaton veron of the meaure of ncome nequalty and polarzaton and ther etmate baed on a complex ample are ntroduced n Secton 3. Th ecton and the Appendx contan dervaton of the tandard error etmator for thee meaure. Fnally, we apply EE methodology to data on earnng from the Canadan Survey of Conumer Fnance from In the ummary ecton we gve a table wth expreon for the mot frequently ued ncome nequalty and polarzaton meaure and ther tandard error. In th artcle we conder only the form of the etmator of the amplng varance for thee complex meaure of ncome nequalty. A fuller undertandng of the properte of thee etmator requre the ue of mulaton tude. Reult from a large mulaton tudy undertaken at Stattc Canada (KovacÆevcÂ, Yung, and Pandher 1995) trongly con rm the advantage of the EE method over everal competng method for the varance etmaton of meaure baed on order tattc ± uch a quantle ± Lorenz curve ordnate and the polarzaton ndex. For the ame meaure, the tudy howed that the EE method and the boottrap method performed mlarly. We ummarze the relevant ndng of the tudy n Secton The Etmatng Equaton Method n Survey Samplng ± A Revew A general formulaton of the EE approach for large ample complex urvey gven n Bnder and Patak (1994). In th ecton we ummarze the man reult needed for the dervaton n Secton 3. Wherea Godambe and Thompon (1986) condered the optmalty of etmatng equaton, the approach we take follow that of Bnder and Patak (1994) who concentrate on the aymptotc properte for a gven et of EE'. In general, for n nte populaton wth a contnuou dtrbuton functon F y; v and dfferentable denty functon f y; v, an etmate of the parameter v may be obtaned from the maxmum lkelhood equaton v ˆS log f y ; v = v ˆ 0. The optmalty of th etmatng equaton for uperpopulaton model dcued by Godambe and Thompon (1986). In the cae of the nte populaton, parameter are de ned explctly a functon of the y value of all populaton unt,.e., v ˆ g y 1 ; ¼; y N. They may alo be de ned mplctly, for example, by maxmzng the lkelhood functon derved by conderng the nte populaton a a ample from an n nte parametrc populaton, or by mnmzng a certan lo functon. Example are the populaton mean m N ˆ SY =N for the rt, and the regreon coef cent for the econd type X y x 0 b 2! mn : In both cae, the parameter v can be regarded a the oluton, v N, to the equaton v ˆXN u y ; v ˆ0 1 The choce of an etmatng functon u y ; v for a partcular parameter may not be

3 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 43 unque n general. However, the nference obtaned ung our approach wll not depend on the choce of etmatng functon, even when more than one et of etmatng functon ued to de ne the parameter of nteret. To etmate a nte populaton parameter when the etmatng functon u y ; v gven, uppoe that a random ample from a nte populaton avalable. The expreon v n (1) can be etmated by an extenon of the Horvtz-Thompon (HT) unbaed etmator, a dcued by Rao (1979), a à v ˆXN w u y ; v 2 where the weght w equal to 0 whenever the th unt not n the ample and S N w ˆ ÃN. If we ue the HT etmator, the weght are the nvere of the ncluon probablte ( w ˆ 1=p ; [ 0; Ó Or, for example, f we ue general regreon etmaton baed on an auxlary varable x X ÃX x >< ; [ w ˆ p ÃX 0 >: 0; Ó where X the known populaton total for the varable x, and ÃX and ÃX 0 are the HT etmate of the total for x and x 2, repectvely. The oluton v à of the equaton à v ˆ0 the EE etmate of the nte populaton parameter v N. The etmatng functon u y; v provde a Gau content etmate (Godambe and Kale 1991). That, f we oberve all value n the nte populaton then the etmate obtaned ung the etmatng equaton equal to the parameter. To etmate the varance of v à we proceed a follow. Equaton (2) can be rewrtten a 0 ˆ à v ˆXN à u y ; v u y à ; v N Š XN w u y ; v N XN u y ; à v u y ; v N Š w 1Š Th decompoton of the etmatng equaton analogou to the one gven n Bnder (1991). We denote the lat term n (3) by R. The remander R generally of order o jv à v N j, whch aymptotcally neglgble a v Ã! v N. Expandng the functon u y; v à around v N ung Young' form of Taylor' Theorem (Ser ng 1980, p. 45), we have 0 ˆ à v ˆXN à v à v N u y ; v v o jv à v N j XN w u y ; v N R v ˆ vn Ignorng the remander term, the dfference v à v N can be expreed a Ãv v N < XN w u y ; v N 3

4 44 Journal of Of cal Stattc where u y ; v N ˆ Jv 1 u y ; v N and J v ˆ XN u y ; v v v ˆ vn Once the expreon for u y; v N obtaned, etmaton of the mean quared error of v à become traghtforward. Snce v à v N can be approxmated by an etmator of the populaton total of u y ; v N ', we can ue the varance etmaton technque for the etmate of a total,.e.,! var v ˆvar à v à X v N < var w u y ; v N 4 Note that u y ; v N depend on an unknown parameter. When we ubttute t etmate nto u y; v N, we obtan u ˆ u y ; v, à and the value of S w u exactly zero. To approxmate the varance of v, à we mut treat the u ' accordng to the EE approach explaned n Bnder and Patak (1994) and replace v N by v à only n the nal expreon of the varance (4). Generally, the parameter v multdmenonal. Bnder (1991) and Bnder and Patak (1994) condered the cae where the rt component of the vector v the parameter of nteret v N and the other are nuance parameter l N.Inthcae v ˆ v; l and can be parttoned a 1 v; l ; 2 v; l Š 0 and etmated o that the followng equalty hold " à 1 v; à à # l 0 ˆ à 2 v; à l à 5 A decompoton mlar to (3) can be appled to (5). Hence 2 3 X N " à 1 v; à l à # u 1 y ; v; à l u à 1 y ; v N ; l Š à XN u 1 y ; v N ; l u à 1 y ; v N ; l N Š 0 ˆ à 2 v; à l à ˆ 6 X N u 2 y ; v; à l u à 2 y ; v N ; l Š à XN u 2 y ; v N ; l u à y ; v N ; l N Š where X N X N 2 R ˆ w u 1 y ; v N ; l N R 7 5 w u 2 y ; v N ; l N X N X N u 1 y ; v; à l u à 1 y ; v N ; l N Š w 1 u 2 y ; v; à l u à y ; v N ; l N Š w 1 and neglgble whenever R ˆ o j à v vj. 3

5 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 45 Expandng the functon à v; à l à around v N ; l N we obtan " 0 < J #" # " # 1vJ 1l Ãv vn à 1 v N ; l N J 2v J 2l Ãl l N à 2 v N ; l N where J 1v ˆ 1 v; l v ; J 1l ˆ 1 v; l vˆvn ; lˆl N l ; J 2v ˆ 2 v; l vˆvn ; lˆl N v and J 2l ˆ 2 v; l l vˆvn ; lˆl N vˆvn ;lˆl N 6 Matrce J 1v, J 1l, J 2v, and J 2l are of order 1 1, 1 k, k 1, and k k, repectvely, where k the number of nuance parameter. Solvng equaton (6) wth repect to the dfference v à v N we obtan Ãv v N < J 1 1v J 1 1v J 1l J 2l J 2v J 1 1v J 1l 1 J 2v J 1 1v Š à 1 v N ; l N J 1 1v J 1l J 2l J 2v J 1 1v J 1l 1 à 2 v N ; l N In mot cae of practcal mportance we nd that 2 v; l doe not depend on v, o that J 2v ˆ 0. Alo, we aume that the rt dervatve of the functon u 1 y; v; l and u 2 y; v; l wth repect to v are ndependent of l. Takng thee aumpton nto account we reduce the above expreon to Ãv v N < à 1 v N ; l N J 1l J 1 à 2l 2 v N ; l N ŠJ 1 1v ˆ XN w u 1 y ; v N ; l N J 1l J 1 2l u 2 y ; v N ; l N ŠJ 1 1v ˆ XN w u y ; v N ; l N 7 ng th expreon we can etmate the varance of varou complex tattc a the varance of the etmated total. A mentoned earler, to approxmate the varance of v, à we replace v N, l N, and pobly N,byv, à l à and S w, repectvely, only n the nal expreon of the varance. It hould be noted that the dervaton of th expreon n Bnder and Patak (1994) wa baed drectly on con dence nterval contructon. Income data are uually collected on a trat ed, multtage ample wth many trata, where a few prmary amplng unt (cluter), n h $2, are ampled from each tratum wthout replacement. However, to mplfy calculaton for varance etmaton we aume that the cluter are elected wth replacement. Such an approxmaton lead to a conervatve etmate of the varance wth a mall relatve ba when the number of prmary amplng unt mall n each tratum. Let w hc be the weght attached to the th ultmate unt n the cth cluter of the hth tratum uch that the approprate etmator of the populaton total for ome charactertc, ay x, ÃT ˆ X w hc x hc

6 46 Journal of Of cal Stattc Then t varance can be etmated by 0 X X 12 w n var ÃT ˆX h X X hc x hc B c w n h 1 hc x hc A Accordngly, h var à v ˆvar à v v N < var c n h! X w hc u y hc ; v N ; l N 0 X X < X w n h X X hc u 12 hc B w n h h 1 hc u c hc A n c h where u hc ˆ u y hc ; à v; à l. 3. Income Inequalty and Polarzaton Meaure and Ther Standard Error We now conder everal meaure of nequalty and polarzaton. ually thee meaure are computed from grouped data. Here we provde ther etmate baed on a probablty ample from a nte populaton. Alo, we derve the etmate of ther tandard error. Snce our goal not to tudy thee meaure and ther properte n depth, we ummarze ome general noton about meaurng nequalty ung the elected meaure. Determnng whether the value of x are ``more equal'' than the value of y eentally concde wth meaurng the dperon of the correpondng dtrbuton. Dependng on whether we are ntereted n nequalty n a partcular egment or n the whole dtrbuton, the relevant meaure may be categorzed a extreme, average or ummary meaure. Extreme meaure focu on nequalty n the tal of the ncome dtrbuton. In th artcle we nclude one uch meaure, the exponental meaure (Wolfon 1986), whch lowncome entve n the ene that the man contrbuton to t value come from mall or negatve ncome. A maller value of the exponental meaure ndcate greater equalty among the poor. Although the coef cent of varaton belong formally to the category of average nequalty meaure, becaue of t entvty to hgh ncome t often ued to ae nequalty among hgh-ncome earner. Whle the coef cent of varaton meaure devaton from a central value (the mean ncome), the Gn ndex account for the devaton of all the value n the populaton among themelve. In th way the Gn ndex an average meaure of nequalty, and very robut to nequalte n the tal. The prmary ummary meaure the Lorenz curve. For a gven dtrbuton t plot the cumulatve percentage of the populaton (dplayed from the pooret to the rchet) agant t total ncome hare. The area between the Lorenz curve and the 45-degree lne known a the Lorenz area. The Gn ndex equal to twce the Lorenz area. A populaton wth the Lorenz curve cloer to the 45-degree lne ha a more equal dtrbuton of ncome. If all ncome are the ame, the Lorenz curve degenerate to the 45-degree lne. However, f Lorenz curve for two or more ncome dtrbuton nterect, only a partal rankng of

7 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 47 the dtrbuton poble. In uch cae, the ue of dfferent meaure may rank the dtrbuton dfferently. A mall et of nequalty meaure that are entve to dfferent magntude of ncome could help nterpretaton of uch ``con ctng'' ndng. Snce the early 1980 there have been dcuon about the dappearng mddle cla phenomenon whch eentally dfferent from the noton of nequalty n ncome (or wealth) dtrbuton and therefore need a dfferent quant caton. The uual term for the hrnkng of the mddle cla polarzaton, ndcatng that the mddle movng toward the tal. Here we preent two ummary polarzaton meaure, the polarzaton curve and the polarzaton ndex, a de ned by Foter and Wolfon (1992). In th ecton we preent x meaure of ncome nequalty and polarzaton along wth ther complex ample etmator and the u varate needed for varance etmaton va the EE method. To mplfy notaton we drop the ubcrpt N from the nte populaton parameter The coef cent of varaton The coef cent of varaton (quared) de ned a CV 2 ˆ V=m 2 where V the varance and m the mean ncome of the populaton. A a meaure of ncome nequalty t belong to the famly of meaure that are hgh-ncome entve and relatvely robut to the low ncome part of the populaton. It eay to calculate and ha a famlar nterpretaton. It dadvantage, however, a hgh amplng varablty. The CV 2 can be obtaned a a oluton to the ytem of equaton 8 1 CV 2 ; m ˆX y =m 1 2 CV 2 Šˆ0 >< >: 2 CV 2 ; m ˆX y m ˆ0 Note that the parameter of nteret CV 2 and the nuance parameter m. Then the etmate of the coef cent of varaton can be obtaned a a oluton to the followng etmatng equaton 8 à 1 CV 2 ; m ˆX w y =m 1 2 CV 2 Šˆ0 >< >: à 2 CV 2 ; m ˆX w y m ˆ0 and take on the famlar form CV à 2 ˆ 1 X w y =Ãm 1 2 ÃN where Ãm ˆ S w y = ÃN. In order to etmate the varance of CV à 2, the u varate are needed. Ther dervaton eentally a three-tep procedure: () Frt we determne the dervatve J 1 ; CV 2 ˆ N; J 1;m ˆ 2NCV 2 =m; and J 2; m ˆ N

8 48 Journal of Of cal Stattc () Next, we ubttute thee dervatve nto equaton (7) and obtan u y ; CV 2 ; m ˆ y =m 1 2 2y =m 1 CV 2 Š=N () Fnally, var à CV 2 ˆvar S w u y ; CV 2 ; m. In the cae of multtage amplng the etmated varance take the form var CV à ˆX 2 n h X u hc Åu n h 1 h 2 h c where u hc ˆ S w hc u hc; u hc ˆ u y hc ; CV à 2 ; Ãm and Åu h ˆ S c u hc =n k. The remander term R n the cae of CV à 2 2 X y =Ãm 1 2 CV à 2 y =m 1 2 CV 2 3 Š w 1 R ˆ 6 X y Ãm y m Š w 1 The rt component, after ome mpl caton, become aymptotcally equvalent to the product o jcv à 2 CV 2 j : o jãm mj : o j ÃN Nj a CV à 2! CV 2, Ãm! m and ÃN! N. Smlarly, the econd component equvalent to o jãm mj : o j ÃN Nj. For the coef cent of varaton (unquared) a dfferent expreon for u obtaned u ˆ 1 2 y =Ãm 1 2 = à CV 2y =Ãm 1 à CVŠ= ÃN 3.2. The exponental meaure The exponental meaure de ned a the populaton mean of the exponentally tranformed ncome (ee Wolfon 1986) EX ˆ 1 X exp y N =m Th meaure entve to low ncome value but take a reaonable nte value when ncome n the neghbourhood of zero. Alo, t well de ned for negatve ncome. It can be obtaned a a oluton to the equaton 8 1 EX; m ˆX fexp y =m EXg ˆ0 >< 8 >: 2 m ˆX y m ˆ0 The dervaton of u varate mlar to the prevou cae of the coef cent of varaton and gven n detal n the Appendx. Here we preent jut the nal form u y ; à EX; Ãm ˆ exp y =Ãm à EX y Ãm ÃJ 1;m = ÃNŠ= ÃN where ÃJ 1;m ˆ 1=Ãm 2 S w y exp y =Ãm

9 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton The Lorenz curve Earler we decrbed the Lorenz curve a a powerful decrptve and analytc tool for rankng ncome dtrbuton. It mply depct the cumulatve ncome agant the populaton hare. The formal nte populaton de nton of the Lorenz curve can be tated a L p ˆ 1 X y Nm Ify # y p g 0 # p # 1 where If:g denote an ndcator functon and y p the pth populaton ncome quantle. To ue the EE method, the Lorenz curve ordnate can be expreed a the oluton to the ytem of equaton 8 X Ify # y p g L p Šy ˆ 0 >< X >: Ify # y p g pš ˆ 0 for 0 # p # 1 The econd equaton de ne the nte populaton quantle. The reultng ample etmate ÃL p ˆ 1 X w y Ify # y ÃN à p g Ãm where y à p the pth ample quantle, y à p ˆ nffy [ j ÃF y $ pg, formally obtaned a a oluton to the equaton X w Ify # y p g pš ˆ0; or ÃF y p ˆp where ÃF y ˆSw Ify # yg= ÃN an etmate of the nte populaton cumulatve dtrbuton functon. For varance etmaton of the Lorenz curve ordnate we ue the value of u u ˆ 1 ÃN Ãm y à y p Ify # à y p g p à y p y ÃL p Š and formula (4). A detaled dervaton of the above expreon gven n Bnder and KovacÆevc (1995) The Gn ndex One of the mot popular meaure of ncome nequalty, the Gn ndex, de ned a the tandardzed Lorenz area,.e., the rato between the actual and the larget poble Lorenz area (whch 1/2). Hence, t take value n [0,1]. The nte populaton form gven a (ee Glaer 1962) G ˆ 1 X 2F N 1 y =m where F ˆ F y ˆ 1=N S j[ Ify j # y g the value of the nte populaton dtrbuton functon at y. The Gn ndex entve to ncome value n the mddle of the dtrbuton. It dadvantage that t not de ned for negatve ncome. It can be de ned a a

10 50 Journal of Of cal Stattc oluton to the equaton 1 G; ff g [ ; m ˆX 2F 1 y =m GŠ ˆ0 where the nuance parameter l ˆffF g [ ; mg the oluton to the ytem of equaton 8 ( ) X >< Ify j # y g F Š ˆ 0 j [ [ 9 X >: y m ˆ0 There are N unknown parameter (nce one of the F ' equal to 1). Wth N populaton value of y, we are able to olve the ytem. The etmate of the Gn ndex come a the oluton to the etmated rt equaton ÃG ˆ 1 X w 2 ÃF 1 y 10 ÃN Ãm where ÃF and Ãm are the oluton to the ytem of etmated equaton (9). The varance of the Gn ndex etmated by expreon (4) where the u varate are equal to u ˆ 2 ÃN Ãm ÃA y y ÃB y Ãm 2 ÃG 1 where ÃA y ˆ ÃF y ÃG 1 =2 and ÃB y ˆS w y Ify $ yg= ÃN. See the Appendx for detal The polarzaton curve In order to formalze the concept of polarzaton, Foter and Wolfon (1992) contructed a curve whch how, for any populaton percentle, how far t ncome from the medan. A larger value of the polarzaton curve ordnate mple a larger `pread' of the dtrbuton from the mddle, ndcatng a maller mddle cla. For a varable y wth a gven dtrbuton F y Foter and Wolfon (1992) de ned the polarzaton curve ordnate by B p ˆ p 0:5 F 1 q m m dq whch can alo be wrtten n a mpler form B p ˆ 0:5 p m m L p L 0:5 Š 0 # p # 1 12 where L p and m were prevouly de ned a the Lorenz curve ordnate and the mean, repectvely, and m the medan, m ˆ nffy [ jf y $ 0:5g. In term of the EE method, the polarzaton curve can be de ned a a oluton to the equaton h y 1 B p ; m; y p ˆX m Ify # y p g Ify # mg B p 0:5 p ˆ

11 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 51 where the nuance parameter y p and m are oluton to the ytem of equaton 8 X Ify # y p g pš ˆ0 0# p # 1 >< X >: Ify # mg 0:5Š ˆ0 The etmate of B p baed on a complex ample of the form gven by (12) wth parameter L p, m, and m replaced by ther etmate. An outlne of the dervaton for the polarzaton curve preented n the Appendx. Here we gve the nal form of the u varate u ˆ 1 ÃN Ãm Ãm y Ify # Ãmg y y à p Ify # y à p g y à p p Ãm 2 B p 0:5 à p 0:5 Ify # Ãmg ÃmÃf Ãm 14 Ãf Ãm where Ãf Ãm an etmate of the denty functon at the medan. Etmaton of the denty functon at etmated quantle wa dcued by Bnder and KovacÆevc (1995) The polarzaton ndex ng the analogy wth the Lorenz curve and the Gn ndex, Foter and Wolfon (1992) condered the area below the polarzaton curve a a ummary meaure of the polarzaton and named t the polarzaton ndex. A perfectly polarzed populaton dvded nto equal halve, each havng jut one of two poble value of ncome, the mnmum or the maxmum. If medan ncome n th cae de ned a the mddle pont between mnmum and maxmum (a lght departure from the uual de nton of the medan), the polarzaton curve then a horzontal lne wth ntercept 1/2 and the pont of dcontnuty at the 50th percentle, gvng the value of the polarzaton ndex of 1/2. If there no polarzaton, everyone ha the ame ncome. The polarzaton curve the [0,1] egment of the horzontal ax and the correpondng polarzaton ndex zero. Therefore, the polarzaton ndex take value between 0 and 1/2, and t tandardzed veron the prevou one multpled by 2. The tandardzed polarzaton ndex, a ntroduced n Foter and Wolfon (1992), P ˆ T 2 G m m where T ˆ m m L =m wth m, m L, m equal to the populaton mean ncome, the mean ncome for the populaton below the medan, and the mean ncome for the populaton above the medan ncome, repectvely. A before, G the Gn ndex and m denote the medan ncome. Snce m ˆ m m L =2, the polarzaton ndex can be wrtten a P ˆ 1 m m ml mg ˆ 1 X y Nm 1 2Ify # mg GŠ

12 52 Journal of Of cal Stattc or a the oluton to the equaton 1 1 P; m; G ˆX m y 1 2Ify # mg GŠ P ˆ 0 15 It etmated a ÃP ˆ 1 X w y 1 2Ify # Ãmg ÃGŠ ÃN Ãm where Ãm ˆ nffy [ j ÃF y $ 0:5g and ÃG gven by (10). The varance of the polarzaton ndex etmated by (4) wth u de ned a u ˆ 2 Ãm Ãm y Ify # Ãmg 1 G ÃA y 2 y ÃB y à 1 G Ãm à 2 2 y P à Ãm f à Ify # Ãmg 1 ÃP = ÃN Ãm 2 where ÃA y and ÃB y are de ned n (11), and Ãf Ãm an etmate of the denty functon at the medan. The detal of the dervaton of u for the polarzaton ndex are gven n the Appendx. 4. Illutraton The EE methodology wa appled to etmate the tandard error of etmate of everal ncome nequalty meaure ung a le on the earnng of all effectve labour force partcpant aged 18 to 64 n An effectve labour force partcpant an ndvdual wth annual labour ncome of at leat 5% of the average wage. Data were collected by the Canadan Survey of Conumer Fnance (SCF) n Aprl The SCF an annual upplement to the monthly Canadan Labour Force Survey, whch baed on a trat ed, multtage ample of houehold. Approxmately 40,000 houehold provded detaled ncome nformaton for ndvdual 15 year of age or older. The data et ued for th llutraton contaned 50,701 ndvdual, tuated n 4,201 cluter (PS'), allocated to 1,139 trata. Attached to each record an ndvdual urvey weght whch an adjuted ample weght; th allow u to compute the tandard error for etmate of nteret. The pont etmate of the ncome nequalty and polarzaton meaure, ther tandard error and the correpondng coef cent of varaton are preented n Table 1. An analy of the unweghted data reveal the heavy rght kewne and the extreme kurto of the data dtrbuton. Th may explan the large tandard error of ÃCV and ÃCV 2 whch are entve to large ncome value. Alo, the polarzaton ndex exhbt entvty to the data pread to the rght. On the other hand, mot of the varablty of the exponental meaure come from the low ncome value whch are concentrated n a relatvely mall range. The Gn ndex robut to extreme obervaton and depend prmarly on the varablty n the mddle of the dtrbuton. Th may explan the mall tandard error of the latter two meaure. The Lorenz curve ordnate were found to have maller coef cent of varaton than the polarzaton curve ordnate. Th dfference can be partly attrbuted to the contrbuton of the etmated denty functon at the medan ued n the

13 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 53 varance etmaton of the polarzaton curve ordnate and the polarzaton ndex, and partly to the entvty of thee meaure to large obervaton. A mentoned n the ntroducton, the man goal of th artcle to provde etmator for the amplng varance of dfferent meaure of ncome nequalty ung the EE method. However, further nght nto the properte of thee varance etmator can be obtaned Table 1. Canadan SCF 1991: Earnng of all effectve labour force partcpant Meaure Etmate Standard error and CV% Mean 26, Medan* 22, CV CV Gn ndex Exponental meaure Polarzaton ndex p Lorenz curve ordnate Polarzaton curve ordnate ÃL p Standard error and ÃB p Standard error and CV% CV% ± ± *The tandard error of the medan alo obtaned by the EE method (Bnder and KovacÆevc 1995).

14 54 Journal of Of cal Stattc through an emprcal comparon wth ome other etmator commonly ued. A mulaton tudy degned to compare everal reamplng method wth the EE method for varance etmaton of ncome nequalty meaure wa conducted at Stattc Canada. The reult are reported n KovacÆevcÂ, Yung, and Pandher (1995). The tudy focued on ncome nequalty meaure that are functon of the quantle, and dd not cover the coef cent of varaton, the exponental meaure, or the ordnate of the polarzaton curve. In the followng we ummarze ome of the relevant ndng n the mulaton tudy. Fve dfferent method for varance etmaton were compared: jackknfe ``delete-onecluter,'' the grouped balanced half-ample method, repeatedly grouped balanced halfample, the boottrap and the Taylor lnearzaton va the EE approach. The underlyng populaton wa the mcrodata le from the Canadan Survey of Conumer Fnance n Ten thouand ample were drawn from the mcropopulaton ung a cluter ample degn wth the electon probablte proportonal to ze. The accuracy and the precon of the condered method were evaluated by ther relatve bae and relatve tablty. For the Lorenz curve ordnate the EE method howed very mall negatve relatve ba, n the range between 0:4% for the quantle p ˆ 0:6 and 5:2% for p ˆ 0:95. For the ame p-value, the relatve ba of the jackknfe etmator wa 20.49% and 39.02%, repectvely. However, the boottrap etmator exhbted the mallet relatve ba at thee pont, 0.3% and 1:91%. Concernng tablty, the EE method along wth the boottrap performed the bet. Smlar reult were obtaned for the polarzaton ndex. The relatve ba of the EE etmator wa computed a 4.2%, wherea for other method t vared between 2.9% (for the boottrap) and 95.4% for the jackknfe. In term of tablty, the EE and the boottrap etmator performed mlarly and outperformed other method. For the varance etmaton of the etmate of the Gn ndex all method condered performed mlarly: all howed a mall negatve relatve ba, n the range of 0:7% and 2:2% and had tablty n the range 87.0% to 99.2%. The EE method had a relatve ba of 1:5% and a tablty of 87.0%: Reult of the tudy con rm the advantage of ung the EE method over mot reamplng method condered for varance etmaton of the Lorenz curve ordnate, the polarzaton and the Gn ndex. The excepton wa the boottrap method whch performed lghtly better. Although, the polarzaton curve wa not tuded emprcally, t mlarty to the Lorenz curve mple that the performance of the EE method hould be acceptably good. 5. Summary Varance etmaton of complex tattc uch a meaure of ncome nequalty can be done by the method of etmatng equaton. The advantage of th approach that t can be ued under a wde cla of amplng degn and doe not requre ntenve computaton, whch mot of the reamplng alternatve requre. In order to etmate meaure of ncome nequalty, one mut rt compute the u varate (gven n the econd column of Table 2) and then compute ther total value after multplyng by the correpondng weght. To etmate the varance of uch etmate one need to compute the u value (ummarzed n the thrd column of Table 2) and ubttute them nto (4).

15 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 55 Table 2. Summary of the lnearzed term for the pont etmaton (w ) and varance etmaton (u ) Meaure u u CV 2 y Ãm 2 = ÃN Ãm 2 y =Ãm 1 2 2y hc =Ãm 1 CV à Š = ÃN CV y =Ãm 1 2 = CV à 2y =Ãm 1 CVŠ à = 2 ÃN Gn ndex 2 ÃF y 1Šy = ÃN Ãm 2 ÃA y y ÃB y Ãm ÃG 1 =2Š = ÃN Ãm (*) Exponental 1= ÃN exp y =Ãm y Ãm ÃJ 1; Ãm = ÃN exp y =Ãm EXŠ à = ÃN (**) meaure 2 Polarzaton y 1 2Ify # Ãmg ÃGŠ = ÃN Ãm Ãm Ãm y Ify # mg 0:5 ndex A y y B y ÃG 1 Ãm o = 2 ÃGy = 2 Š P à Ãm Ãf Ify Ãm # Ãmg 0:5 ÃP = ÃN Lorenz curve y Ify # y à p g = ÃN Ãm y y à p Ify # y à p g py à p y ÃL p Š = ÃN Ãm Polarzaton f0:5 p y Ify # y à 1 p g Ify # ÃmgŠ = Ãmg = ÃN N à Ãm Ify # Ãmg y y à p Ify # y à p g y à p p Ãm=2 curve ÃB p 0:5 p 0:5 Ify # Ãmg Ãm Ãf Ãm Š = Ãf Ãm g (*) A y ˆ ÃF y à G 1 2 and B y ˆS w j y j Ify j $ yg= ÃN (**) ÃJ 1; Ãm ˆ Sw y exp y = Ãm = Ãm 2 The extenon of th method to comparon between doman and comparon over tme are traghtforward nce we have reformulated the problem to one of etmatng varance of lnear tattc n the Godambe (1955) cla, allowng the ue of tandard method appled to thee lnear tattc. Of coure all the complexte arng from havng overlappng unt over tme would have to be accounted for. 6. Appendx Detaled dervaton of the u varate 6.1. The exponental meaure The rt dervatve of 1 EX; m and 2 m, gven by (8), are J 1; EX ˆ N; J 1;m ˆ 1 X m 2 y exp y =m ; and J 2;m ˆ N After ubttutng nto (7) and ung the etmate ntead of parameter u ˆ 1 ÃN exp y =Ãm EX à y Ãm ÃJ 1; Ãm = ÃNŠ where ÃJ 1; Ãm ˆ 1 X Ãm 2 w y exp y =Ãm 6.2. The Gn ndex The correpondng rt dervatve are J 1G ˆ N, J 1l ˆ f2y =mg [=o ; G=mŠ 1 N, where 0 the label of the maxmum y, J 2G ˆ 0 N 1, and J 2l ˆ NI N N, where I the dentty matrx.

16 56 Journal of Of cal Stattc Subttutng nto equaton (7) and replacng parameter wth ther etmate, we obtan u ˆ 2 ÃA y y ÃB y Ãm ÃN Ãm 2 ÃG 1 where ÃA y ˆ ÃF y ÃG 1 =2 and ÃB y ˆS w y Ify $ yg= ÃN 6.3. Polarzaton ndex We preent the dervaton of the u varate for the polarzaton ndex n full detal becaue t nvolve approxmaton that are pec c for quantle and functon of them. The two approxmaton for the nte populaton quantle y p ' mportant for the ubequent development are Ãy p y p < 1 f y p p ÃF y p Š ˆ X 1 w ÃNf y p p Ify # y p gš A1 where f y p the value of the denty functon at y p. Th an extenon of the Bahadur repreentaton (Bahadur 1966) for a quantle to the nte populaton cae. Contnung, let m a ˆ 1=N S y Ify # ag. Note that the prevouly ntroduced m L, the mean of the lower half of the populaton, equal to 2m m. Alo, the Lorenz curve ordnate at p equal to m y p =m. The followng approxmaton hold for quantle m y à X p m y p ˆ1 y N Ify # y à p g Ify # y p g < y p f y p à y p y p < X w y p p Ify # y p g = ÃN; a à y p! y p A2 The approxmaton (A2) appear n a more general form n Bnder and KovacÆevc (1995). The etmate of the equaton (15) can be expreed a 0 ˆ à 1 ÃP; Ãm; ÃG ˆ X h y w Ãm 1 2Ify # Ãmg ÃG ÃP < N P ÃP X X h w h y Ãm 1 2Ify # Ãmg ÃG y m 1 2Ify # mg G y m 1 2Ify # mg G P Approxmatng the functon y = Ãm around the medan m by t rt order Taylor expanon y =m y Ãm m =m 2, ubttutng nto the expreon above, and mplfyng, we have the followng ÃP P< 1 m ÃmL m L m m Ãm m ÃG G m 2 m Ãm L m ÃG X h y w m 1 2Ify # mg G P =N A3

17 KovacÆevc and Bnder: Varance Etmaton for Meaure of Income Inequalty and Polarzaton 57 Replacng Ãm L wth Ãm L m L m L and ÃG wth ÃG G G n the thrd term of (A3) and then removng the mxed product from approxmaton (A3) a hgher order term, we arrve at the followng lnearzaton ÃP P< 1 m ÃmL m L m m ÃG G P Ãm m m X h y w m 1 2Ify # mg G P =N Fnally, we ue (A1), (A2), and (11) to approxmate the dfference Ãm m, Ãm L m L, and ÃG G, repectvely ÃP P < X w u y ; P; m; G ˆ X 2 w m m y P Ify 2f m # mg 1 2 A y y B y G 1 m G 2 2 y P =N Th expreon provde the nal form of the u varate a gven by (16) The polarzaton curve The polarzaton curve ordnate are gven by expreon (12). To etmate ther varance we proceed a n the cae of the polarzaton ndex, tartng wth the decompoton of the etmate of (13) 0 ˆ X h y w Ãm Ify # y à p g Ify # Ãmg ÃB p 0:5 p < N B p ÃB p Š X X h w h y Ãm Ify # y à p g Ify # Ãmg y m Ify # y p g Ify # mg y m Ify # y p g Ify # mg B p 0:5 p Approxmatng the functon y = Ãm around the medan m by t rt order Taylor expanon y =m y Ãm m =m 2, and ung mlar ubttuton a n the cae of the polarzaton ndex, we obtan the dfference ÃB p B p a ÃB p B p < X ˆ X w u y ; B p ; m; y p w 1 m y Nm Ify # mg y y p Ify # y p g y p p m 2 B p 0:5 p 0:5 Ify f m # mg mf m Fnally, to etmate the varance of ÃB p we ue formula (4) for the varance of total and u varate gven by (14).

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