GMM Estimation of Income Distributions from Grouped Data

Transcription

1 Deprtment of Economcs Workng Pper Seres GMM Estmton of Income Dstrbutons from Grouped Dt Gholmre Hjrgsht Wllm E. Grffths Joseph Brce D.S. Prsd Ro Dungkmon Chotkpnch Sep Reserch Pper Number 9 ISSN: ISBN: Deprtment of Economcs The Unversty of Melbourne Prkvlle VIC 3

2 GMM Estmton of Income Dstrbutons from Grouped Dt Gholmre Hjrgsht Deprtment of Economcs Unversty of Melbourne, Austrl Wllm E. Grffths Deprtment of Economcs Unversty of Melbourne, Austrl Joseph Brce Deprtment of Economcs Unversty of Queenslnd, Austrl D.S. Prsd Ro Deprtment of Economcs Unversty of Queenslnd, Austrl Dungkmon Chotkpnch Deprtment of Econometrcs nd Busness Sttstcs Monsh Unversty, Melbourne, Austrl Dec, Correspondng uthor: Wllm Grffths Deprtment of Economcs Unversty of Melbourne Vc 3 Austrl Phone: Fx: Eml: wegrf@unmelb.edu.u

3 SUMMARY We develop GMM procedure for estmtng ncome dstrbutons from grouped dt wth unknown group bounds. The pproch enbles us to obtn stndrd errors for the estmted prmeters nd functons of the prmeters, such s nequlty nd poverty mesures, nd to test the vldty of n ssumed dstrbuton usng J-test. Usng eght countres/regons for the yer 5, we show how the methodology cn be ppled to estmte the prmeters of the generled bet dstrbuton of the second knd, nd ts specl-cse dstrbutons, the bet-, Sngh-Mddl, Dgum, generled gmm nd lognorml dstrbutons. Ths work extends erler work (Chotkpnch et l., 7, ) tht dd not specfy forml GMM frmework, dd not provde methodology for obtnng stndrd errors, nd consdered only the bet- dstrbuton. Keywords: GMM; generled bet dstrbuton; grouped dt; nequlty nd poverty. JEL clssfcton numbers: C3, C6, D3

4 3. INTRODUCTION The estmton of ncome dstrbutons hs plyed n mportnt role n the mesurement of nequlty nd poverty nd, more generlly, n welfre comprsons over tme nd spce. Access to wht s vst lterture on the modelng of ncome dstrbutons, the chrcterstcs of dfferent specfctons, nd vrous estmton methods, s convenently cheved through volume by Kleber nd Kot (3), the collecton of ppers n Chotkpnch (8), nd ppers by Bndourn et l. (3) nd McDonld nd Xu (995). For crryng out lrge scle nvestgtons tht nvolve mny countres, dfferent tme perods, nd the estmton of regonl nd globl ncome dstrbutons (see, for exmple, Mlnovc () nd Chotkpnch et l. ()), complton of the necessry countryspecfc ncome dstrbuton dt s mjor reserch problem. The dt re generlly drwn from household expendture nd ncome surveys tht re conducted once n fve yers n most countres. Becuse complton of dt nd dt dssemnton from these surveys re resource ntensve, much of the rw dt re not redly vlble for reserchers. More regulrly dssemnted dt tke the form of summry sttstcs tht nclude men ncome, mesures of nequlty lke the Gn coeffcent, nd grouped dt n the form of ncome nd populton shres. Two sources of such dt re the World Bnk nd the World Insttute for Development Economcs Reserch. The focus of ths pper s estmton of country-level ncome dstrbutons from lmted dt of ths form. Specfclly, our objectve s to develop nd pply generled method of moments (GMM) estmtor for ncome dstrbutons, usng dt tht re n the form of populton shres nd group men ncomes, wth unknown group lmts. When dt re vlble s populton nd ncome shres, group men ncomes for gven country cn be computed from the redly vlble dt on the country s men per cpt ncome. To cheve comprblty over countres nd tme, men ncomes tht nd

5 4 hve been djusted for purchsng power prty re vlble from the World Bnk nd the Penn World Tbles. Some of us hve tckled ths problem before. Chotkpnch et l. (7) (herefter CGR) suggest GMM estmtor for the bet- dstrbuton, pply t to smple of 8 countres n two tme perods, nd llustrte how the estmted dstrbutons cn be combned to derve regonl dstrbuton, fnd Loren curves, nd mesure nequlty. In more extensve study, Chotkpnch et l. () use the sme technque to estmte the globl ncome dstrbuton s mxture of GMM-estmted bet- dstrbutons for 9 countres n 993 nd. The GMM technque s nturl cnddte becuse t cn be mplemented wth ggregted dt rther requrng ndvdul ncome observtons. Chotkpnch et l. (7, ) descrbe the mn fetures of ther pproch, nd show tht t works well, but ther method ws defcent n severl respects. They dd not set up ther estmtor wthn forml GMM frmework, they used n rbtrrly specfed weght mtrx, nd, becuse of the lck of n symptotc covrnce mtrx for the estmtor, they dd not provde ny stndrd errors. These defcences re remeded n ths pper. We defne forml set of moment condtons nd construct n optml weght mtrx, ledng to n symptotclly effcent estmtor. Dervng the optml weght mtrx for the estmtor mkes t possble to estmte the symptotc covrnce mtrx of the estmtor whch n turn provdes mesures of relblty n the form of stndrd errors for the estmted prmeters nd functons of the prmeters used regulrly n the re of ncome dstrbutons, such s nequlty nd poverty mesures. We lso extend the CGR frmework to one tht cn ccommodte ny ncome dstrbuton, not just the bet- dstrbuton. In our emprcl work we focus on the generled bet dstrbuton of the second knd (GB dstrbuton) nd ts populr specl

6 5 cses, the bet-, Sngh-Mddl, Dgum, generled gmm nd lognorml dstrbutons. We show how the dequcy of n ncome dstrbuton cn be ssessed usng () the J-test for the vldty of excess moment restrctons, nd () comprson of predcted nd observed ncome shres. We lso llustrte how estmtes of the prmeters cn be used to plot ncome dstrbutons nd ther confdence bounds, nd compute nequlty nd poverty mesures. It s useful to note how our current nd pst work dffers from relted prllel work by Wu nd Perloff (5, 7) who lso consder GMM estmton of ncome dstrbutons from grouped dt. Wu nd Perloff use mxmum entropy densty to pproxmte the underlyng ncome dstrbuton nd use smulton to estmte the optml weght mtrx tht s used n two-step estmtor. In ths pper, we show how the optml weght mtrx cn be expressed n terms of the moments nd moment dstrbuton functons of ny ncome dstrbuton; then, n our emprcl work, we estmte the GB s generl nd flexble clss of ncome dstrbutons. Knowng the prmetrc specfcton mens we re ble to specfy the optml weght mtrx s functon of the unknown prmeters nd obtn optml GMM estmtes n one step. Our pst work (Chotkpnch et l. (7, )) used the more restrctve bet- dstrbuton nd sub-optml weght mtrx. Other dstngushng fetures of our current work re our emphss on estmton of group bounds, nd the symptotc covrnce mtrx tht cn be used to fnd stndrd errors for ll estmted prmeters (ncludng the bounds) nd functons of those prmeters. The pper s orgned s follows. In Secton the GMM methodology s descrbed n generl for estmtng the prmeters of ny ncome dstrbuton. In Secton 3 we provde the expressons tht re needed for GMM estmton of the GB, bet-, Sngh-Mddl, Dgum, generled gmm nd lognorml dstrbutons. These expressons nclude the moments, dstrbuton functons nd frst-moment dstrbuton functons. We refer to n ppendx where dervtves for computng the GMM symptotc covrnce mtrx cn be found. Expressons

7 6 for nequlty nd poverty mesures re lso provded. Secton 4 contns descrpton of the dt used to llustrte the theoretcl frmework nd the blty of the GMM technque to recover ncome denstes. We selected eght countres/res for the yer 5: Chn rurl, Chn urbn, Ind rurl, Ind urbn, Pkstn, Russ, Polnd nd Brl. The results presented n Secton 5 nclude prmeter estmtes nd ther stndrd errors, plots of ncome denstes nd ther confdence bounds, goodness-of-ft ssessment, nd nequlty nd poverty mesures. Concludng remrks re provded n Secton 6.. THE GMM ESTIMATOR Suppose tht we hve smple of T observtons y, y,, yt tht re ssumed to be rndomly drwn from prmetrc ncome dstrbuton f( y; ), nd tht these observtons hve been grouped nto N ncome clsses (, ),(, ),,( N, N ), wth nd N. Let the men clss ncomes for the N clsses be gven by y y,,, yn ; nd let the proportons of observtons n ech clss be gven by c, c,, cn. Gven vlble dt on the y nd the c, but not the, our problem s to estmte, long wth the unknown clss lmts,,, N. To tckle ths problem usng GMM estmton we crete set of smple moment condtons T H θ h θ () yt, T t such tht E Hθ, where,,,, θ. The GMM estmtor ˆθ s defned s N θˆ rg mn H θ ' W H θ () θ where W s weght mtrx. In wht follows we consder frst the moment condtons nd then the weght mtrx.

8 7. The Moment Condtons To set up the moment condtons correspondng to the smple proportons c, we note tht the correspondng populton proportons, tht we denote by where g θ k f( y; ) dy ; y k θ, re gven by g y f y dy,,, N (3) Eg y s n ndctor functon such tht g y f y otherwse The smple moment correspondng to Eg y k T T gyt c T t T θ s where T s the number of observtons n group. Thus, for the proporton of observtons n ech group, we hve N moment condtons T g yt E g y c k T t θ,,, N (4) N N The moment condton for N s omtted becuse the result k θ c mkes one of the N condtons redundnt. To obtn moment condtons tht use the nformton n the clss men ncomes y, t s convenent to consder T cy yg( y) yg( y) T T t t t t TT t T t

9 8 N Snce cy y, where y s smple men ncome, we cn vew c y s tht prt of smple men ncome tht comes from -th ncome clss. Its correspondng populton quntty s θ yf( y; ) dy ; yg y f y dy,,, N (5) y Eyg Thus, our second set of moment condtons re T yg t yt E yg y cy T t,,, N (6) Collectng ll the terms we cn wrte h y, θ t θ g yt k g y k yg t yg θ N t N y θ t y θ t N t N (7) nd θ c k T cn kn θ Hθ yt; h θ T t cy θ cnyn Nθ (8) If K s the dmenson of (the number of unknown prmeters n the ncome densty), then there re N moment condtons nd N K unknown prmeters.

10 9 For computtonl purposes, t s typclly more convenent to express k θ nd θ n terms of dstrbuton functons. If E( y) yf( y; ) dy s the men of y, F( y; ) s ts dstrbuton functon, nd y F ( y; ) t f ( t; ) dt (9) s ts frst moment dstrbuton functon, then, from (3), (5) nd (9), nd k θ F( ; ) F( ; ) () F( ; ) ( ; ) θ F () In Secton 3 we gve explct expressons for, F ( ; ) nd F ( ; ) for the GB dstrbuton, nd ts specl cses, the bet-, Sngh-Mddl, Dgum, generled gmm nd lognorml dstrbutons. Insertng these expressons nto the moment condtons n (7) nd (8), nd ncludng expressons for the elements of the weghtng mtrx tht we consder n the next secton, mkes the GMM estmtor opertonl. The moment condtons n (6) dffer slghtly from those used by CGR. They used y ( ) k ( ) nsted of cy ( ). Our new formulton s preferred becuse Ecy, nd, lthough y ( ) ( ) plm ( ) k ( ), E( y ) E T T y tg t y t T T g k t y t Use of cy ( ) lso mkes dervton of n optml weght mtrx more strghtforwrd.

11 . The Weghtng Mtrx The smplest weghtng mtrx s tht where W I. However, snce the lst N moment condtons nvolvng the clss mens re of much hgher order of mgntude thn the frst N, whch nvolve proportons, settng W I gves n undesrbly lrge reltve weght to the lst N condtons. Under these crcumstnces, the lst N condtons tend to domnte the estmton procedure nd, s noted by CGR, ths cn led to perverse outcomes where ˆ ˆ for some. To ensure both sets of moment condtons were on smlr scle, for ther moment condtons CGR used dgonl weghtng mtrx, wth dgonl elements ( c, c,, c N, y, y,, y N ). The motvton behnd ths weght mtrx ws tht t led to n estmtor tht mnmed the sum of squres of percentge errors n the moment condtons. It s not n optml weght mtrx, however. Its dgonl elements re not equl to the nverses of the vrnces of the moment condtons, nd t gnores correltons between moment condtons. Dervng n optml weght mtrx s crucl for dervng n symptotclly effcent estmtor nd fclttes dervton of stndrd errors for the prmeter estmtes. where The optml weght mtrx s gven by: W s trdtonlly estmted from T W plm yt, yt, h θ h θ T t () yt yt ˆ ˆ W h θ h θ (3) T ˆ,, T t wth ˆθ beng frst-step estmtor obtned by mnmng H( ) WH( ) for pre-specfed W. The estmtor Ŵ depends on both the smple dt nd estmtes of the prmeters ˆθ. It turns out not to be sutble for our problem becuse t contns terms of the form

12 whch re not vlble from the grouped dt. However, nsted of usng (3), T yg( ) t t y t we cn tke the probblty lmt n () nd obtn result tht depends only on the unknown prmeters, not on the smple dt. To present ths result, whch s derved n Appendx A., we defne the followng N-dmensonl vectors k (,,, N ) k k k μ (,,, N ) μ () () () (),,, N where () θ y f( y; ) dy ygy fy; dy y E y g (4) The rgument hs been dropped from these vectors to ese notton. The N dmensonl vectors obtned by deletng the lst element n ech of the bove vectors wll be denoted by k, μ nd N N () μ N, respectvely. Also, for ny vector x, we use the notton Dx ( ) to denote dgonl mtrx wth the elements of x on the dgonl. Then, from Appendx A., W Dk ( N) knkn Dμ ( N) N knμ D( μ N ) () μkn D( μn) μμ N Dk ( N) Dμ ( N) N k N D( μ N ) k N μ () D( μn ) μ N (5) where N s n ( N ) dmensonl vector of eros.

13 For mnmng the GMM objectve functon, we requre W, not vlues (or estmtes) of θ, W cn be redly found by numerclly nvertng W. For gven W. However, t s useful to provde n nlytcl expresson for W to mprove computtonl effcency nd to gve nsghts nto the mnmton process. In Appendx A., we show how n nlytcl expresson for W cn be derved from (5). The result s gven n (6); we use the notton v ( v, v,, vn ) where v k, nd ι N s n ( N ) -dmensonl vector () of ones. Also, wth slght buse of mtrx notton, we use Dxs to denote dgonl mtrx wth dgonl x s, x s,, xn sn, for ny two vectors x nd s. () () v v D μ v ι ι D μ v ι W= DμN vn Dkv N vnιn N N N N N N N N N N N (6) An dvntge of expressng W n ths form s tht t contns lrge number of eros n ll sub-mtrx blocks except the upper left. The totl number of eros s N N. It my lso help expln why, n our emprcl work, the CGR estmtor, whch uses dgonl W, led to estmtes comprble to those from optml GMM. The extr qunttes needed to compute W (tht were not lso needed to compute the moment condtons) re the (). It s convenent to wrte () () F( ; F( ; (7) where F ( ; y f( y; ) dy () s the second moment dstrbuton functon, nd () Ey ( ) s the second moment for y. Expressons for ( ; ) F nd (), for number of dstrbutons, re provded n Secton 3.

14 3.3 Emprcl Implementton of GMM Estmton We consder three GMM estmtors. The frst, θ ˆ CGR, uses weght mtrx W CGR whch s the weght mtrx used by CGR, modfed to sut our specfcton of the moment condtons. It s dgonl mtrx wth ts frst N dgonl elements gven by c nd ts lst N dgonl elements gven by cy. Wth ths weght mtrx the GMM estmtor mnmes the sum of squres of the percentge errors n the moment condtons. The second estmtor s twostge estmtor tht uses θ ˆ CGR to compute n estmte of the optml weght mtrx, W θ ˆ CGR. In our emprcl work we used n tertve verson of ths estmtor, updtng W fter ech terton, nd tertng untl convergence. The thrd estmtor s one-stge estmtor obtned by mnmng the complete objectve functon wth respect to θ where both the moment condtons nd W re functons of θ. Hnsen et l. (996) refer to the onestge estmtor s the contnuous-updtng estmtor. Although the tertve nd contnuously updtng estmtors hve the sme symptotc dstrbuton, they re typclly dfferent n fnte smples (Hll, 5, p.3). The three estmtors re gven by θˆ rg mn H θ ' W H θ (8) CGR STAGE θ θ CGR ' ˆ θˆ rg mn H θ W θ H θ (9) STAGE θ CGR θ ˆ rg mn H θ ' W θ H θ ().4 Asymptotc Covrnce for the GMM Estmtor To specfy the symptotc covrnce mtrces for the estmtors, we frst defne the mtrx of prtl dervtves of the moment equtons wth respect to the prmeters s

15 4 G NNK H θ () Then, n estmtor for the symptotc covrnce mtrx of the one nd two-stge estmtors s gven by (see, for exmple, Cmeron nd Trved, 5 p. 76) ˆ ˆ ˆ ˆ ( ) ( ) ( ) vr θ = G θ W θ G θ () T An estmtor for the symptotc covrnce mtrx of θ ˆ CGR s gven by ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ CGR CGR CGR CGR CGR vr θ = G W G G W W W G G W G (3) T where Ĝ nd ˆ W re equl to G( θ ˆ ) nd CGR W ( θ ), respectvely. ˆ CGR In our emprcl work we focused on stndrd errors for the one nd two-stge estmtors nd functons of them. We successfully used both nlytcl nd numercl dervtves to clculte the elements of the G mtrx. Some expressons for obtnng nlytcl dervtves for vrety of dstrbutons re gven n Appendx B. Ths ppendx s best consulted fter we consder specfc ncome dstrbutons n Secton 3..5 The J Sttstc Under the null hypothess tht the moment condtons re correct plmh θ =, the J sttstc ˆ ˆ ˆ d ' N J T H θ W θ H θ (4) In trdtonl GMM estmton ths test sttstc s used to ssess whether excess moment condtons re vld. In our cse, snce we ssume prtculr form of prmetrc ncome dstrbuton, nd use ts propertes to construct the moment condtons nd weght mtrx, the J sttstc cn be used to test the vldty of the ssumed ncome dstrbuton. K

16 5 3. INCOME DISTRIBUTIONS A lrge number of probblty densty functons hs been suggested n the lterture for modellng ncome dstrbutons. See, for exmple, McDonld nd Rnsom (979), McDonld (984), McDonld nd Xu (995), Creedy nd Mrtn (997), Bndourn et l. (3) nd Kleber nd Kot (3). One of the flexble dstrbutons ntroduced by McDonld (984) s the four prmeter generled bet dstrbuton of the second knd (GB). Ths dstrbuton hs nlytcl propertes tht mke t well suted to the nlyss of ncome dstrbutons (Prker, 999), nd, s we wll see, t provdes very good ft to the observed dt (see lso, Bordley et l. (996) nd Bndourn et l (3)). In ths secton we descrbe the GB dstrbuton nd ts chrcterstcs needed for GMM estmton. We lso present results needed for GMM estmton of the bet, Dgum, Sngh-Mddl, generled gmm nd lognorml dstrbutons tht cn be obtned s specl cses of the GB dstrbuton. 3. The Generled Bet Income Dstrbuton of the Second Knd The GB dstrbuton whose prmeters re ( b,, pq, ) hs probblty densty functon (pdf) p y f( y;, b, p, q) p y b Bp, q b pq y (5) where y s ncome, b, p, q, nd ( p) ( q) p q B( pq, ) t ( t) dt ( pq) s the bet functon. The GB dstrbuton s generlton of the stndrd bet dstrbuton defned on the (,) ntervl. If U s stndrd bet rndom vrble wth prmeters ( pq, ), then Y b U ( U) GB(, b, p, q). The nverse of ths trnsformton s

17 6 Yb U (6) Yb Thus, the cumultve dstrbuton functon (cdf) of the GB dstrbuton s gven by u p q ( ;,,, ) ( ) u (, ) F y b p q t t dt B p q B( p, q) (7) where u y b y b nd the functon B ( pq, ) s the cdf for the stndrd bet u dstrbuton. Expressng the GB cdf n ths form s convenent becuse B ( pq, ) s redly computed by computer softwre. The qunttes requred to compute the moment condtons nd the weghtng mtrx u for GMM estmton re the frst nd second moments () nd, the dstrbuton functon F( y; ), nd the frst nd second moment dstrbuton functons F ( y; ) nd F ( y; ). The moments of order k exst only f p k q, nd, f moment exsts, t s gven by (Kleber nd Kot, 3, p.88) k ( k ) bb pk, qk (8) B p, q Also, the k-th moment dstrbuton functon y F ( y; t k f( t; ) dt k ( ) k cn be wrtten s (Kleber nd Kot, 3, p.9) k k k k Fk( y; ) Fy;, b, p, q Bu p, q (9) Ths expresson s computtonlly convenent one, becuse t llows us to compute the frst nd second moment dstrbuton functons usng stndrd bet dstrbuton functon wth dfferent prmeters.

18 7 Income dstrbutons re often estmted to ssess nequlty nd poverty. To llustrte, we consder two nequlty mesures, the Gn nd Thel coeffcents, nd two poverty mesures, the hedcount rto nd the Foster-Greer-Thorbecke mesure wth n nequlty verson prmeter of ( FGT( ) (Foster, Greer nd Thorbecke, 984). Generl forms for the Gn nd Thel coeffcents, nd ther expressons n terms of the prmeters of the GB dstrbuton, re (McDonld, 984; McDonld nd Rnsom, 8) G yf( y; ) f( y; ) dy B p q,,, 3F, pq, p ; p, pq; B p q B p q p 3F, pq, p ; p, pq; p y y T ln f( y; ) dy p q lnb (3) (3) where 3 F s generled hypergeometrc functon nd () t dlog () t dt s the dgmm functon. The hypergeometrc functon cn be computed by Mtlb, but we found t ws more effcent nd relble to numerclly ntegrte G y F( y; ) f( y; ) dy. For gven poverty lne x, the hedcount rto s the proporton of populton wth ncomes less thn x, nd so, for the GB dstrbuton, t s smply gven by H Fx (;) B( pq, ) where x u xb u (3) x b The mesure FGT ( consders not just the proporton of poor, but lso how fr the poor re x below the poverty lne. It s defned s

19 8 x x y FGTx( f( y) dy x (33) Followng Kkwn (999), t s convenent to express FGT ( n terms of the men nd vrnce of the poor ( x x x nd x ), the hedcount rto x H x, nd the ncome gp rto g x x. Defntons of these qunttes, nd expressons for them n terms of the prmeters of the GB dstrbuton nd some of ts specl cses re gven n Tble I. 3. Some Commonly Used Dstrbutons A number of populr dstrbutons re specl cses of the GB dstrbuton. Borrowng from McDonld (984), we dsply those dstrbutons n Fgure I. In our work we estmted the GB, bet-, Sngh-Mddl, Dgum, generled gmm (GG) nd lognorml dstrbutons. In the remnder of ths secton we provde or refer to the qunttes, F( y; ) nd F ( y; ) (), F( y; ),, tht re requred for GMM estmton of these specl-cse dstrbutons. For the nequlty nd poverty mesures, we confned estmton to the GB, bet-, Sngh-Mddl, nd Dgum dstrbutons. Expressons for the nequlty nd poverty mesures n terms of the prmeters of these dstrbutons re gven n Tble I. GB q q p GG Bet Sngh-Mddl Dgum p q q Lognorml Gmm Webull Fgure I- Reltonshp of GB wth other dstrbutons (dopted from McDonld (984))

20 9 The requred frst nd second moments nd dstrbuton functons for the bet-, Sngh-Mddl, nd Dgum dstrbutons re redly obtned from the more generl GB expressons n (8) nd (9) by settng, p nd q, respectvely. Smplfctons for the mens of these dstrbutons re gven n Tble I. For the bet- dstrbuton t s lso worth notng tht the second moment reduces to () bpp ( ) ( q)( q) There re lso useful smplfctons for the cdf s for the Sngh-Mddl nd Dgum dstrbutons. For the Sngh-Mddl dstrbuton we hve nd, for the Dgum dstrbuton, y F( y;, b,, q) b y F( y;, b, p,) b 3.3 Generled Gmm Dstrbuton The generled gmm pdf s gven by q p p y f( y;, p, ) y exp p ( p) (34) It s obtned s specl cse of the GB dstrbuton n (6) by settng b q nd tkng the lmt s q (McDonld 984). The stndrd gmm pdf f u u b ( p) b p ( ) u exp p (35) cn be obtned from (34) usng the trnsformton u y, nd redefnng b s b. Thus, vlues for the cdf of the generled gmm dstrbuton cn be computed from

21 F( y; p,, ) G( pb, ) wth u u y nd b where ub (, ) p t u ( ) G p b t e dt p s the cdf of the stndrd gmm dstrbuton wth prmeters p nd b. The moments nd moment dstrbuton functons for the generled gmm re gven by (McDonld, 984; Butler nd McDonld, 989) k ( k ) pk ( p) (36) F ( y;, p, ) F y;, pk, G p k, b (37) k These expressons complete wht s needed for computng the moment condtons nd the weght mtrx. u 3.4 Lognorml Dstrbuton The lognorml pdf f (ln y ) y;, exp y cn be obtned s specl cse of the generled gmm dstrbuton by settng (38) nd p( ), nd tkng the lmt s (McDonld, 984). Its cdf s ln( y) Fy;, (39) where s the stndrd norml cdf. Its moments re k ( k ) exp k nd vlues for ts moment dstrbuton functons cn be computed from ln( y) k Fk y;, Fy; k, (4)

22 Detls cn be found n Atchson nd Brown (957) or Kelber nd Kot (3). 4. DESCRIPTION OF DATA AND SOURCES To llustrte the methodology descrbed n Sectons nd 3, we use ncome dstrbuton dt from the PovclNet webste developed by the World Bnk poverty reserch group. Ths dtbse s set up for the purpose of poverty ssessment for ndvdul countres, regons nd globlly. The dt re provded n grouped form nd cn be downloded from They re vlble for developng countres for number of yers rngng from 98 to 5. The ltest verson of the dt ws updted n August 8 to ncorporte 5 purchsng power prty estmted by the World Bnk Interntonl Comprson Progrm. To use resonbly dverse cross secton of countres to test the performnce of the estmtor, we chose s exmples Brl, Chn, Ind, Pkstn, Russ nd Polnd for the yer 5. Seprte dt re vlble for rurl nd urbn regons n Ind nd Chn, mkng totl 8 dfferent dt sets. We wll refer to ech dt set s comng from regon, where regon cn be country, or rurl or urbn Chn or Ind. For most of the chosen regons, populton shres c nd the correspondng ncome shres g were vlble for groups. Exceptons were Ind rurl nd urbn whch ech hd groups, nd Chn rurl whch hd 7 groups. In lne wth Ind rurl nd urbn, we ggregted the dt from the other regons nto groups. Hvng groups for ll regons hs the dvntge of unformty for estmton, nd t provdes n opportunty for checkng the blty of the estmted model to predct ncome shres for groups other thn those used for estmton, procedure tht we consder n Secton 5. The populton proportons n ech regon were not dentcl, but n most cses they were pproxmtely.5 for the frst nd lst two groups nd. for the remnng groups.

23 Also vlble from the World Bnk webste s ech regon s men ncome y, found from surveys nd then converted usng 5 purchsng-power-prty exchnge rte. To use the methodology descrbed n Sectons nd 3, we need the dt on clss men ncomes y. They re obtned s y g y c. For computng stndrd errors nd the J sttstc, we lso need the smple ses T for ech of the surveys. Unfortuntely, lthough the webste provdes dt on the populton se of ech regon, t does not hve comprehensve dt on the smple ses T for ech of the surveys. For our clcultons we use T,. Ths s conservtve vlue snce most of surveys hve smple ses whch re much lrger. If stndrd errors or J-sttstcs for other smple ses re of nterest, they cn be obtned from our results by multplyng by the pproprte sclng fctor. 5. EMPIRICAL ANALYSIS Our presentton nd dscusson of the results begns n Secton 5. where we consder the estmted ncome dstrbutons for the eght regons. Goodness-of-ft s ssessed n Secton 5., usng J-sttstcs nd comprson of predcted nd observed ncome shres. Levels of nequlty nd poverty obtned from the dfferent dstrbutons re reported n Secton Country-Specfc Income Dstrbutons Tble II contns the estmted clss lmts nd prmeters of the GB dstrbuton obtned usng the GMM estmton procedure outlned n Sectons nd 3. For ech regon, we report three dfferent sets of estmtes those from the CGR, two-stge, nd one-stge estmtons. Stndrd errors for both the two-stge nd one-stge estmtes re lso reported. There re no drmtc dfferences between the estmtes from the three dfferent estmtors. The two-stge nd one-stge estmtes re lmost dentcl, prtculrly for the clss lmts, nd the CGR estmtes re only slghtly dfferent. Smlrly, the stndrd errors obtned usng two-stge estmton re very close to those obtned from one-stge estmton. The mgntudes of the

24 3 stndrd errors for the clss lmts re very smll suggestng we re estmtng ther vlues precsely. However, stndrd errors for some of the estmtes b ˆ, ˆ, pq ˆ, ˆ re reltvely lrge, mplyng wde confdence ntervls round the correspondng prmeters. In most regons, seprte hypothess tests for H : p (Sngh-Mddl), H : q (Dgum), nd H : (bet-) would not be rejected. The stuton my chnge, of course, f we use lrger smple se, but one of the 3-prmeter dstrbutons my be n dequte representton for some cses. More lght s shed on ths ssue when we exmne goodness of ft. To sve spce we hve not reported estmtes nd stndrd errors for dstrbutons other thn the GB; they re vlble from the uthors on request. Estmtes of the nd ther stndrd errors were smlr for ll dstrbutons. Stndrd errors for the estmted prmeters of the bet-, Sngh-Mddl nd Dgum dstrbutons (whch hve prmeters n common wth the GB) were much smller thn those for the GB, reflectng the drop from 4 to 3 prmeters. In ll cses we computed stndrd errors usng both numercl dervtves nd nlytcl dervtves, nd where both sets were computble, they produced dentcl results. There were few cses where the computton of nlytcl dervtves fled bet- estmtes for Pkstn nd Ind (rurl nd urbn). These flures corresponded to solutons where p ws very lrge nd b ws very smll; estmton ws unstble, wth dfferent strtng vlues ledng to dfferent locl mnm. Anlytcl stndrd errors could not be found becuse the hypergeometrc functon n Mtlb broke down. Numercl stndrd errors could stll be found, however. Ths problem dd not rse wth the GB nd other dstrbutons. Fgure II contns grphs of the estmted GB pdfs for Chn rurl nd urbn, Ind rurl nd urbn, Brl nd Polnd, long wth 95% confdence bounds for these dstrbutons. To fnd the confdence bounds, stndrd errors were computed for the estmted pdfs t

25 4 number of ncome levels usng the covrnce mtrx of the prmeter estmtes, the delt rule, nd numercl dervtves. The nrrowness of the confdence bounds suggests we re ccurtely estmtng the pdfs, despte reltvely lrge confdence ntervls for some of the prmeters. A comprson of the urbn nd rurl pdfs for Ind nd Chn shows clerly the lrger ncomes of the urbn popultons. In Ind, t s nterestng tht the rurl nd urbn modes re smlr, but the urbn pdf hs much ftter tl. Usng populton weghted mxture of the rurl nd urbn components, n Fgure III we hve grphed the pdf nd cdf for ll of Chn, longsde those of the rurl nd urbn subpopultons. If more countres re consdered, smlr mxtures cn be obtned for lrger regons such s contnents or the whole world. See, for exmple, Chotkpnch et l. (). A potentl estmton problem for ll dstrbutons other thn the generled gmm nd lognorml, s the non-exstence of the second moment. For the exstence of the k-th moment, the GB dstrbuton requres q k. Ths condton s the sme for the moments of the Sngh-Mddl dstrbuton, t becomes q k for the bet- dstrbuton nd k for the Dgum dstrbuton. Snce the optml weghtng mtrx requres the exstence of second order moments, f the CGR estmtes volte one of these nequltes, we cnnot proceed wth two-step estmton of the offendng dstrbuton. Also, our experence suggests one-step estmton breks down. (CGR estmton s stll fesble.) We encountered ths problem wth Brl, country wth reltvely hgh nequlty, for estmtons wth the GB, Sngh- Mddl nd Dgum dstrbutons, but not the bet- dstrbuton. In the results reported n Tble II we overcme the problem by mnmng the objectve functon subject to the constrnt q. Ths soluton my not be entrely stsfctory. The underlyng ncome dstrbuton my ndeed not hve second moments, the stndrd errors for the boundry

26 5 solutons tht result my not be vld, nd nequlty ppered to be underestmted reltve to vlues reported by the World Bnk. We lso found tht the generled gmm dstrbuton cn be dffcult to estmte. Sometmes estmton would brek down, prtculrly when there ws tendency for the estmte for to become smll. We suspect tht smll vlues of were mkng clculton of p troublesome. We tred dfferent prmetertons nd dfferent strtng vlues, nd n ll cses mnged to get convergence. However, we re not confdent tht ll our solutons correspond to globl mnm. 5. Goodness-of-Ft Anlyss In ths secton we ssess the dequcy of the vrous dstrbutons for modellng the observed populton nd ncome shres. Two crter re used: () the J test to test whether the moment condtons re vld for ech of the dstrbutons, nd () comprson of observed nd predcted ncome shres. Tble III presents the p-vlues for the J sttstcs clculted for ll dstrbutons consdered nd for ll exmple regons, obtned usng the -stge estmtes. Under the null hypothess tht the moment condtons re correct, the J sttstc hs dstrbuton wth degrees of freedom equl to the number of excess moment condtons. In the cse of the GB dstrbuton, we hve 3 moment condtons nd 5 prmeters gvng degrees of freedom of 8. For the 3-prmeter dstrbutons the degrees of freedom s 9, nd for the log norml t s. At 5% sgnfcnce level, the moment condtons for the GB dstrbuton re not rejected for 5 out of 8 of the regons. All the other dstrbutons re rejected for ll other countres wth the excepton of the bet- dstrbuton for Chn rurl nd Russ. These test outcomes re not s plesng s one mght hope. It would hve been more stsfctory f the GB dstrbuton ws ccepted for ll regons, nd some of the other dstrbutons were

27 6 ccepted for wder selecton of countres. However, cceptnce of prtculr dstrbuton s lkely to be dffcult wth such lrge smple se, nd, despte the test outcomes, the blty of the GB dstrbuton to predct ncome shres ws very good. Goodness-of-ft n terms of predctng ncome shres ws crred out by comprng the observed ncome shres g wth the predcted ncome shres derved from the estmted dstrbutons. The dstrbutons were estmted usng groups, obtned by ggregtng orgnl groups n ll regons except Ind rurl nd urbn nd Chn rurl. No ggregton ws crred out on the orgnl groups vlble for Ind rurl nd urbn, nd, n the cse of Chn rurl, 7 groups were ggregted to. To ssess goodness-of-ft, we exmned the blty of the models to predct the ncome shres n the orgnl groups ( n most cses) from the dstrbutons estmted from groups. The ncome shres were predcted n the followng wy. Begnnng wth the orgnl populton shres c, nd correspondng cumultve proportons c j, we found clss lmts (not necessrly the sme s the prevously-estmted clss lmts) by solvng the j equtons F ; ˆ. Then, predcted cumultve ncome shres ˆ were found from the frst moment dstrbuton functon ˆ ˆ F ;, gvng the predcted ncome shres g ˆ ˆ. ˆ Note tht when the number of groups used for estmton dffers from the number used for predctng the ncome shres, the clss lmts ( ) n the bove two equtons wll, by necessty, be dfferent from the estmted. When the sme number of groups s used for estmton nd predcton, we hve two lterntves for predctng the ncome shres. We cn use the bove two equtons s lredy descrbed, or we cn smply use ˆ ˆ ; ˆ F where

28 7 ˆ re the orgnl estmtes of the clss lmts. We used the former pproch n ll cses. Snce t uses less nformton from GMM mnmton, t s lkely to be more strngent test of predctve blty. We present comprson of the predcted nd ctul ncome shres (n percentge form) for the GB dstrbuton n Tble IV. Tble V contns the root-men-squred errors, ˆ N N g g, for ll dstrbutons. In Tble IV the observed nd predcted ncome shres re remrkbly smlr for ll regons, gvng strong support for the GB dstrbuton. Ths outcome s very encourgng gven tht the prmeters of the dstrbutons hve been estmted from lmted dt, the predctons re prtlly out-of-smple for most countres, nd the clss lmts mpled by the estmted prmeters, not the gvng the best ft, were used to compute the predcted ncome proportons. In Tble V, the GB dstrbuton performs the best or close to the best for ll regons except Ind rurl nd Brl. The generled gmm nd lognorml dstrbutons generlly performed poorly reltve to the other dstrbutons. The other specl cses of the GB dstrbuton dd well for some regons nd not so well for others. A possbly counterntutve result s tht the lognorml dstrbuton outperformed the generled gmm dstrbuton. Snce the lognorml cn be vewed s -prmeter specl cse of the 3-prmeter generled gmm, we would expect the generled gmm to do better. The problem wth the generled gmm seemed to le n predctng the shre of the lst group. If ths group ws omtted, the predctons from the lognorml were worse. We speculted erler tht, wth the generled gmm, we my not hve lwys reched globl mnmum. Tht could be the reson for poor predcton of the lst shre.

29 8 5.3 Inequlty nd Poverty In ths subsecton we llustrte how the prmeter estmtes cn be used to estmte nequlty nd poverty. The Gn nd Thel coeffcents were clculted usng the expressons gven n (3) nd (3) for the GB dstrbutons nd usng those n Tble I for the bet-, Sngh- Mddl nd Dgum dstrbutons. Stndrd errors were computed numerclly usng the delt rule nd the covrnce mtrx of the prmeter estmtes. Tble VI reports the estmted Gn nd Thel coeffcents nd ther correspondng stndrd errors. It s found tht GB nd bet- gve smlr results for the Gn nd Thel coeffcents whle Sngh-Mddl nd Dgum gve slghtly dfferent results. In terms of the stndrd errors, those from bet- seem to be the smllest. However, n ll cses the stndrd errors re reltvely smll compred to the estmted coeffcents. Inequlty s hghest n Brl followed by Ind urbn; t s lowest n Ind rurl. Tble VII reports poverty ncdence usng the hedcount rto (HCR) nd the FGT() mesure, both expressed s percentges, usng poverty lne of $.5 per dy, or, n the monthly unts used n estmton, $38. Vlues re clculted from the expressons n Tble I for the bet-, Sngh-Mddl, Dgum nd GB dstrbutons. The correspondng stndrd errors re lso reported. Poverty s gretest n rurl nd urbn Ind, followed by rurl Chn nd Pkstn, then Brl. There s much less poverty n urbn Chn, Russ nd Polnd. The estmtes cn be senstve to the chosen dstrbuton, prtculrly when we re n the tl of the dstrbuton where the level of poverty s low; see, for exmple, Russn nd Polnd. 6. SUMMARY AND CONCLUSIONS Estmton of ncome dstrbutons s crtcl for montorng nequlty nd poverty t both ntonl nd nterntonl levels. Studes whch ttempt to estmte the globl ncome dstrbuton tkng nto ccount both wthn-country nd between-country nequlty typclly utle dt provded n ggregted form by ether the World Insttute for Development

30 9 Economcs Reserch (WIDER) or the World Bnk. See, for exmple, Mlnovc () nd Chotkpnch et l (). Prevous work by Chotkpnch et l () used method-ofmoments estmtor to estmte bet- ncome dstrbutons from ths dt. In ths pper we hve extended ther work by provdng moment condtons nd the optml weght mtrx tht cn be used for GMM estmton of ny clss of ncome dstrbutons. Specfc expressons for the moment condtons nd the optml weght mtrx were provded for the more generl GB dstrbuton, ts obvous specl cses the bet-, Dgum nd Sngh-Mddl dstrbutons, nd ts less obvous specl cses, the generled gmm nd lognorml dstrbutons. We lso show how to get stndrd errors for the optml GMM estmtes. Once the prmeters hve been estmted, long wth the covrnce mtrx of the estmtor, they cn be used n vrety of wys. Vlues for the densty functon, dstrbuton functon nd Loren curve nd ther confdence bounds cn be found t number of ncome vlues nd then grphed. Dstrbutons for lrger regons cn be obtned s populton weghted mxtures of ndvdul countres. Inequlty nd poverty mesures nd ther stndrd errors cn lso be computed. We hve llustrted the methodology nd how number of economclly relevnt qunttes cn be estmted from t, usng dt on 6 selected countres tht ncluded 8 dfferent regons. We found tht the methodology cn be redly mplemented nd tht the GB dstrbuton generlly provdes good ft n terms of the vldty of ts moment condtons, nd the ccurcy of predcted ncome shres from the estmted dstrbutons. Acknowledgements Ths project hs been supported by Austrln Reserch Councl Grnt DP9463.

31 3 A. Fndng We requre APPENDIX A: OPTIMAL WEIGHTING MATRIX W s Probblty Lmt P plm,, plm T W h θ h θ NN NN yt yt T t ' Q NN MNN It s strghtforwrd to show tht the elements of P, Q nd M re p g y k g y k g y k c k ck T T ( t) ( t) ( t) T t T t T p g y k g y k ck c k kk T ( ) ( ) j j t j t j j j j t T q g y k y g y c y c kc y k ( ) ( ) t t t T t T q g y k y g y c kc y k T ( ) ( ) j j t t j t j j j j j t m y g y y g y c y T T t ( t ) t ( t ) T t T t T m y g y y g y c y c y T ( ) ( ) j j t t t j t j j j j j t Q Usng plmc k, plmcy, nd T () t t t plm T y g ( y ), we hve for the elements of the mtrx W, plm p k k p kk plm j j plmq k q k plm j j m () plm plm m j j

32 3 Usng these results nd the notton defned n the body of the pper, we cn wrte W Dk ( N) knkn Dμ ( N) N knμ D( μn ) () μkn D( μn) μμ N A. Dervng W from W Wth obvous defntons for A nd c, we wrte W s Dk ( N) Dμ ( N) N k D( μ ) μ N N W D( μn ) k N μ A cc () To nvert ths mtrx, we use the result Frst, we need to obtn A cc A A cca ca c A whch we prtton s A A A A A Usng results on the prttoned nverse of mtrx, we hve () D μ A DkNDμ N N Dμ Ths s dgonl mtrx of dmenson N wth dgonl elements N N () Thus, N N () () k k () () v A D μ v. Also, D μ Dk v () N N N N A D μ DkN DμN N () N N N

33 3 nd A A N N N Dk N Dμ N N () D μn v N N N N Then, Dk v () D μn vn D μn vn N k N A c DμN vn DkN vn N μn () N N N N nd ι N N () N N ca c v N N N k () () N N Thus, () N N N ιn ιn ι () N ιnιn ι N () N vn v N A cca N v ca c N N N N N ι N () () N ι () vn v NN N N Fnlly, W=A A cca ca c () N N () ιnιn ι N D μn vn D μn vn N vn v N DμN vn DkN vn N () N N N N N ι N () vn v NN () () N NN v N N N N N N vn D μ v ι ι D μ v ι DμN vn DkN vn N vn ι N kn vn N

34 33 APPENDIX B: DERIVATIVES OF MOMENT CONDITIONS To fnd the symptotc covrnce mtrx of the estmtors we need G, the mtrx of dervtves of the moment condtons wth respect to ll the prmeters. We cn clculte the elements of G usng numercl dervtves. However, t s lso possble to clculte them nlytclly. To do so we frst note tht ths mtrx hs the followng structure: k H G H k H H k where H s prttoned s, H H H wth k H denotng the moment condtons for the clss proportons nd H denotng the moment condtons for the clss mens. Also, we prtton θ s θ, where ncome dstrbuton. The elements n respectvely, nd s the vector of prmeters n the,,, N H k nd k θ F( ; ) F( ; ), nd θ F( ; ) ( ; ) F H for whch we requre dervtves re, In ths ppendx, we focus on the dervtves of F ( ; ) nd F ( ; ) wth respect to nd the elements n. Fndng the dervtves of nd combnng these dervtves wth those of F ( ; ) nd ( ; ) F to fnd the requred dervtves of θ s strghtforwrd, lthough tedous. The bsc tool used to fnd expressons for the dervtves of F ( ; ) nd F ( ; ) s the followng stndrd result from clculus: u( ) u( ) d f ( x, ) du du f ( x, ) dx dx f ( u, ) f ( u, ) d d d u( ) u( )

35 34 B. Dervtves for the GB Dstrbuton Let u b b b. Dervtves for B p k, q k u re provded. Settng k gves the requred expressons for F ( ; ); settng k wll gve the requred expressons for F ( ; ). Dervtves for the bet-, Sngh-Mddl nd Dgum dstrbutons, cn be obtned by settng, p nd q, respectvely. u p k B p k, q k pk qk ( ) u u dx (, ), qk u x x u B p q B pk qk pk qk pq b Bpk, qk b B pk, qk u j j, pk qk B p k q k u pk qk u u u b b B pk qk b b B pk qk, pq, Bu pk, qk k k k ( pq) p Bu p, q p u pk qk x ( x) ln( xdx ) B( pk, qk ) pk Bpk, qk 3 ( pq) ( pk ) ln u Bu pk, qk x pk F pk, pk, kqp ; k, pk; u

36 35 Bu pk, qk k k k ( pq) q Bu p, q q u pk qk x ( x) ln( xdx ) B( pk, qk ) qk Bpk, qk 3 ( pq) ( qk ) ln u Bu pk, qk ( x) qk F qkq, k, k pq ; k, qk; u Bu pk, qk k k k k k B,, u p q B u p q p q pk qk b (ln ln b) pq b Bpk, qk In the dervtves wth respect to p nd q, s the dervtve of the log of the gmm functon nd 3 F represents the generled hypergeometrc functon. B. Dervtves for the Generled Gmm Dstrbuton For the generled gmm dstrbuton, we need the dervtves of G p k, b nd k where u. They re pk G pk q x x b u b u u pk ( pk ) u, exp exp dx pk b pk b pk G pk, q u j pk exp u b pk b pk ( j) G pk, b x exp xb exp u b u pk pk u p k dx pk b b b p k b p k u for k

37 36 u pk Gu pk, b x exp x b ( pk ) k ln( x) dx ln( b) Gu p, b pk p b ( pk ) ( pk ) pk Gu pk, b exp u b ln k G ( pk, q) u pk b ( p k ) p The ntegrl n the dervtve wth respect to p cn be evluted numerclly. B.3 Dervtves for the Lognorml Dstrbuton For the lognorml dstrbuton, we need the dervtves of ln( ) k nd k. They re for k ln( ) k ln k j ln( ) k for j k k ln ln ln( ) k ln( ) k ln( ) k where (.) denotes the stndrd norml pdf. References Atchson J, Brown J.A.C The Lognorml Dstrbuton. Cmbrdge Unversty Press: Cmbrdge. Bndourn R, McDonld JB, Turley RS. 3. Income Dstrbutons: An Inter-temporl Comprson over Countres. Estdstc 55: 35-5.

38 37 Bordley RF, McDonld JB, Mntrl A Somethng New, Somethng Old: Prmetrc Models for the Se Dstrbuton of Income. Journl of Income Dstrbuton 6: 9-3. Butler RJ, McDonld JB Usng Incomplete Moments to Mesure Inequlty. Journl of Econometrcs 4: 9-9. Cmeron AC, Trved PK. 5. Mcroeconometrcs: Method nd Applcton, Cmbrdge Unversty Press: New York. Chotkpnch D. (ed) 8. Modelng Income Dstrbutons nd Loren Curves. Sprnger: New York. Chotkpnch D, Grffths WE, Ro DSP 7. Estmtng nd Combnng Ntonl Income Dstrbutons usng Lmted Dt. Journl of Busness nd Economc Sttstcs 5: Chotkpnch D, Grffths WE, Ro DSP, Vlenc V.. Globl Income Dstrbutons nd Inequlty, 993 nd : Incorportng Country-Level Inequlty Modelled wth Bet Dstrbutons. The Revew of Economcs nd Sttstcs forthcomng. Creedy J, Mrtn VL. (eds.) 997. Nonlner Economc Models: Cross-sectonl, Tme Seres nd Neurl Network Applctons. Cheltenhm: Edwrd Elgr. Foster J, Greer J, Thorbecke E A Clss of Decomposble Poverty Mesures. Econometrc 5: Hll, AR. 5. Generled Method of Moments. Oxford Unversty Press: Oxford Hnsen LP, Heton J. Yron A Fnte-Smple Propertes of Some Alterntve GMM Estmtors. Journl of Busness nd Economc Sttstcs 4(3): 6-8.

39 38 Kkwn N Inequlty, Welfre nd Poverty: Three Interrelted Phenomen. In Hndbook on Income Inequlty Mesurement, Slber J (ed), Kluwer: The Netherlnds; Kleber C, Kot S. 3. Sttstcl Se Dstrbutons n Economcs nd Acturl Scences. John Wley nd Sons: New York. McDonld JB Some Generled Functons of the Se Dstrbuton of Income, Econometrc 5: McDonld JB, Rnsom MR Functonl Forms, Estmton Technques nd the Dstrbuton of Income. Econometrc 6: McDonld JB, Mchel RR. 8. The Generled Bet Dstrbuton s Model for the Dstrbuton of Income: Estmton of Relted Mesures of Inequlty. In Modelng Income Dstrbutons nd Loren Curves, Chotkpnch D (ed) Sprnger: New York. McDonld JB, Xu YJ A Generlton of the Bet Dstrbuton wth Applctons. Journl of Econometrcs 66: Errt 69 (995), Mlnovc B.. True World Income Dstrbuton, 988 nd 993: Frst Clcultons bsed on Household Surveys Alone. The Economc Journl : 5-9. Prker SC The Generled Bet s Model for the Dstrbuton of Ernngs. Economcs Letters 6: 97-. Wu X, Perloff J. 5. Chn s Income Dstrbuton: The Revew of Economcs nd Sttstcs 87: Wu X, Perloff J. 7. GMM Estmton of Mxmum Entropy Dstrbuton wth Intervl Dt. Journl of Econometrcs 38:

40 39 The men The Gn Thel ndex Hed count rto Men ncome of the poor Vrnce of the ncome of the poor Tble I. Expressons for the nequlty nd poverty mesures Generl form GB Bet- ( ) Sngh-Mddl ( p ) Dgum ( q ) p q bp b yfydy b q b p p q q q p G yf( y) f( y) dy Equton (3) y y ln f H x x F( x) x where moment y dy T p q ln b x B p,q G pb p, q b T p q ln G qq pp G q q p p T q ln b T p ln b u, Hx Bu p, q Hx Bu p, q Hx Bu p, q xb x b where u xb xb x b x b where u where u x b x b H B p q where u yfydy Bu p, q x B (, ) Fx u p q bbp q () x x x () x s the second () u, x H x p q pq Bu p, q x B ( p, q) u () b p x Hx q B p, q u Bu, q x B p, q bb q () u, x H x u q q Bu p, x B p, q bb p () u, x H x u p p FGT ( x x x y f ( ydy ) x FGTx( H xgx gx x x where g x x x x s known s the ncome gp rto nd x s the poverty lne.

41 4 Tble II: Estmted coeffcents from GB dstrbutons Chn Rurl Chn Urbn CGR Two Stge SE One Stge SE CGR Two Stge SE One Stge SE b p q Ind Rurl Ind Urbn CGR Two Stge SE One Stge SE CGR Two Stge SE One Stge SE b p q

42 4 Tble II. Estmted coeffcents from GB dstrbutons (cont.) Pkstn Russ CGR Two Stge SE One Stge SE CGR Two Stge SE One Stge SE b p q Polnd CGR Two Stge SE One Stge SE CGR Two Stge SE One Stge SE b p q Brl

43 4 Tble III. p-vlues from J-sttstcs GB B SM Dgum GGmm LogN Chn Rurl Chn Urbn Ind Rurl Ind Urbn Pkstn Russ Polnd Brl......

44 43 Tble IV: Observed nd estmted percentge shres of ncome bsed on GB Chn Rurl Chn Urbn Ind Rurl Ind Urbn Estmted Observed Estmted Observed Estmted Observed Estmted Observed Pkstn Russ Polnd Brl Estmted Observed Estmted Observed Estmted Observed Estmted Observed