Bounding Estimates of Wage Discrimination

Transcription

1 Equton Secton 1 Bounng Estmtes of Wge Dscrmnton by J. G. Hrschberg Deprtment of Economcs Unversty of Melbourne Prkvlle, 3052 n D. J. Slottje Deprtment of Economcs Southern Methost Unversty Dlls, TX August 2002 Abstrct: The Blner Oxc ecomposton metho for efnng scrmnton from the wge equtons of two groups hs h we egree of pplcton. However, the mplcton of ths mesure cn very rmtclly epenng on the efnton of the non-scrmntory wge chosen for comprson. Ths pper uses form of extreme bouns nlyss to efne the lmts on the mesure of scrmnton tht cn be obtne from these ecompostons. A smple pplcton s presente to emonstrte the use of the bootstrp to efne the strbutons of the scrmnton mesure. Key wors: Extreme Bouns Anlyss, Dscrmnton, Bootstrp JEL Coes: J7, C2

2 0. Introucton A rch lterture on the emprcl nlyss of lbor mrket scrmnton hs followe from the contrbutons of Blner (1973) n Oxc (1973). These reserchers were mong the frst to explore ths ssue econometrclly. It hs been unerstoo for some tme tht the chotomy n the verge wges of two groups (usully broken own by sex or rce n here referre to s the vntge n the svntge) s ue n prt to fferences n verge levels of prouctvty (or skll) (ther enowment) n s ue n prt to sprte tretment of the two groups once they enter the lbor mrket (the scrmnton). However, the ecomposton of the verge wge fferences nto these two fferent prts hs been foun to vry wth the metho use. In ths pper we propose metho for efnng the bouns on these mesures. Although recent contrbutons to the lterture hve nvestgte entry nto the lbor mrket n selectvty bs s tonl resons for the observton of lrge wge fferentls ths pper concentrtes on the vrton wthn the trtonl Blner-Oxc ecomposton whch for gener fferences hs recently been shown to be the most mportnt element n the ecomposton of wge fferentls (for exmple see Men 2000). Ths pper procees s follows. Frst, we revew the ecomposton n the methos tht hve been propose. Secon we efne the metho for bounng the non-scrmntory wge prmeters. Then we show how the mesures of scrmnton cn be boune. In the fourth secton we opertonlze the use of the bouns by provng pproxmtons to the symptotc vrnces of the scrmnton mesures. In Secton fve the bootstrp methos re efne for the estmton of the enstes of the bouns on the scrmnton mesures. Secton sx efnes smple pplcton usng t tht s wely vlble. 1

3 1. Decomposton of Wge Dfferences Becker (1971) efne mesure of scrmnton s the fference between the observe wge rto n the wge rto tht woul prevl n the bsence of scrmnton. Ths scrmnton coeffcent cn be expresse s (1) δ = W MP W MP MP MP where W s the verge vntge worker's wge n the mrket n W s the verge svntge worker's wge n the mrket. It s strghtforwr to see tht (2) MP MP W = W n the bsence of scrmnton n (2) follows from the usul cost mnmzton problem. Oxc (1973) ntrouce the formulton gven n (1). Followng Oxc (1973), Cotton (1988) note tht (1) cn be wrtten n logrthmc form (3) ln W ln W = ln MP ln MP + ln ( δ+ 1) where the frst term on the rght hn se (the fference n the logs of the mrgnl proucts) s ue to fferences n prouctvty of the two groups n the secon term on the rght hn se (ln(δ+1)) s ue to scrmnton. Oxc (1973) showe tht seprte lner moels of the log wge specfcton cn be estmte for svntge or 's ( ln( W ) X ) vntge or 's ( ln( W ) X ) = β n = β. The estmtes cn then be combne n the followng wy snce regresson lnes must pss through the vrbles' mens: (4) ln ( W ) ln ( W ) = X β X β 2

4 The formulton gven n (4) follows Neumrk's (1988) notton where X n X re vectors contnng the mens of the vrbles whch re presume to mpct prouctvty (n subsequently wges) n β n β re the estmte coeffcents. Emprcl work usng (4) hs been one usng two ecompostons. If X = X X n β=β β, then (4) becomes ether, (5) ln ( W ) ln ( W ) = X β + X β or where (5) n (6) re foun by ng ( X β X β ) to (5) n ng ( X X ) β β to (6). The Oxc moel ecomposes the frst term on the rght hn se of (5) nto the porton of the men log wge fferentl ue to fferences n verge prouctvty n the secon term s ue to fferent wge structures. The β's re gven ths nterpretton snce they reflect the returns tht nvuls wll get from ther personl chrcterstcs wth respect to wges. Unfortuntely, s Neumrk (1988) (mong others) hs ponte out, conserble vrton my exst n the estmte one gets of the wge fferentl ue to scrmnton f one uses (5) vs á vs (6). Neumrk (1988) presents nce exposton on where the screpncy les n usng (5) rther thn (6) or vce vers. If (5) s selecte s the moel to etect scrmnton, t s ssume the vntge worker's wge structure becomes the one tht woul exst n the bsence of scrmnton. In (6), the svntge worker's wge structure woul be the prevlng one. These cses re both strghtforwr to see snce wthout scrmnton (where the secon term woul spper n (5)), we woul ttrbute the men wge fference to fferences n chrcterstcs weghte by the vntge workers wge structure (β ). Neumrk (1988) me ths pont even clerer by generlzng Oxc's result to get broer ecomposton: (6) ln( W ) ln( W ) = X β + X β 3

5 (7) ( ln ( ) ln ( ) = X β + X ) ( β β + X β β ) W W where β s ssume to represent the wge structure tht woul prevl n the bsence of scrmnton. Neumrk (1988) shows tht (5) or (6) cn be generte s specl cses of (7) n thus emphszes the mport of wht one ssumes bout β n ttemptng to mesure scrmnton. Cotton (1988) performe smlr nlyss n rgue tht β shoul be constructe s weghte verge of vntge n svntge worker's wges weghte by the rto of the svntge to the vntge lbor force representton. Neumrk (1988) rghtly notes tht ths s n hoc specfcton n proposes fnng β bse on more theoretcl founton. Specfclly, Neumrk (1988) ssumes the employer erves utlty from profts n from the scrmnton-bse composton of the lbor force. The utlty functon s ssume to be homogenous of egree zero wth respect to the lbor nput. Ths mens tht f the numbers of the two groups of workers re chnge proportontely, utlty s unchnge. Neumrk nterprets ths to men tht employers only cre bout the reltve proportons of the two types of workers. Neumrk's moel ultmtely les to, (8) W N + W N MP j = N j + N j j j j j (where N s the number of vntge workers n N s the number of svntge workers) or tht the mrgnl prouct of the jth worker epens on the reltve proportons of the vrous types of lbor so tht snce W j = MP j n the bsence of scrmnton, the nonscrmnton wge cn be foun from (8). Neumrk (1988) fns the estmtor of the nonscrmnton wge structure (β ) by frst runnng regressons on the two sub-smples to get ftte log wge vlues n then fter combnng the ftte vlues of the log wges, by then runnng regresson on the whole smple. Those coeffcent estmtes wll then gve n estmte of β. One ffculty wth the mplementton of Neumrk s metho s tht the 4

6 smple use n estmton my not refect the number of employees prtculr employer hs hre n ech ctegory. It s qute common to pply these methos to t bse on smplng proceure tht s not nfluence by the employer s ctons. Neumrk's (1988) weghtng proceure s smlr to one use by Oxc n Rnsom's (O-R) (1988) whch ws use n the context of estmtng unon wge effects. Oxc n Rnsom (1991) lso propose weghtng mtrx whch ws specfe by (9) Ω = 1 N ( X X ) ( XX) where X s the observton mtrx for the poole (both clsses of workers) smple n X s the observton mtrx for the vntge smple. The nterpretton of Ω N s weghtng mtrx s rely seen by notng tht XX = X X + X X, where X s the observton mtrx for the svntge smple. where O-R showe tht (10) ( I ) β s the ornry lest squres estmtor from the poole smple (contnng both types of workers.) Thus, ths weghtng scheme ws foun by O-R to be the ornry lest squres estmtor from the combne groups s the wge structure tht woul exst n the bsence of scrmnton. They note tht ths estmte of the common wge structure s not n generl convex, lner combnton of the seprtely estmte vntge n svntge workers' wge structures n they get result smlr to tht of Neumrk. ( X X ) β =Ω β + Ω β N N N As O-R note, Cotton's (1988) weghtng s equvlent to O-R's when ( N )( XX ) =, f the frst n secon smple moments re entcl for ll workers. An becuse the smple men chrcterstcs for the vntge n svntge workers re the sme, ll of the fferences n wges re ue to scrmnton. 5

7 To summrze the lterture on the estblshment of hypothetcl el (wth no vntge or svntge gven) wge structure (β ) we summrze the fnngs n Tble 1 n whch we hve entfe the vrous efntons of Ω s propose n prevous reserch. We now propose fferent metho for etermnng the extent to whch the efnton of β mtters on the resultng efnton of scrmnton. 2. Bounng β Lemer s 1978 monogrph proposes metho for the etermnton of the frglty of regresson result. Ths s one by subjectng regresson moels to n nlyss tht etermnes the extreme bouns (EB) of prmeter estmtes bse on the ssumpton of pror strbuton for selecte prmeters. In the usul pplcton ths s nterprete s mens for the comprson of ll possble regresson moel specfctons n whch vrous subsets of regressors re consere for omsson from the regresson. The most wely cte exmple of ths form of nlyss cn be foun n Lemer s 1983 pper enttle Let s tke the con out of econometrcs. Subsequently number of ppers hve ppere tht hve crtcze the EB pproch to moel specfcton nlyss most notbly McAleer Pgn n Volker (1985) s focusng on very nrrow type of specfcton choces n for the tenency for these nlyss to reject too mny moels to be of much use. However, resurgence of pplctons n mofctons of Lemer s EB nlyss hve ppere n Levne n Renelt (1992), Gwne (1995), n Temple (2000) mong number of others. In ths pper we o not use the EB nlyss per sy n tht we o not nvestgte the mplctons of regresson specfcton chnges. However, we use one of the funmentl results on whch EB nlyss s bse whch llows us to efne boun ll the possble prmeter estmtes tht my be use for the nonscrmntory wge structure. Then we solve n optmzton problem tht llows us to efne two nonscrmntory wge structures. One tht wll mxmze the mesure of scrmnton n the other tht wll mnmze the mesure of scrmnton. 6

8 Chmberln n Lemer (1976) (C-L) conser the cse of vector β tht cn be efne s mtrx weghte verge of two vectors 1 (11) β = ( H ) ( + H H β+ H b β) where the weghtng mtrces H n H re postve efnte symmetrc. In the pplctons they conser these two sets of prmeters re entfe n terms of Byesn estmtor where one group woul be entfe s the t n the other s the pror wth the resultng el or non-scrmntory set of prmeters s the posteror n the H s re the corresponng precson mtrxes (or nverse covrnce mtrxes). Algebrclly there s no stncton between the pror n the t though n prctce Byesn methos re often pple where etle t strbutons re efne but prors re non-nformtve. In the cse of the ecompostons efne by ΩO, ΩR n Ω C s efne n Tble 1, we cn set H =Ω n H = Ι Ω. In the cse of the Neumrk ecomposton H = X X n H = X X n the resultng (posteror) men vector of prmeters s equvlent to the Byesn nterpretton of the OLS estmtor when there s n ton of t. Thus X woul be e to X to form totl smple from whch the estmte woul be obtne. (12) β = + ( I - ) Ω β Ω β Where the mtrx Ω s postve efnte symmetrc mtrx. Consequently, wge ecompostons prove n pplcton of methos evelope for the conserton of these lner Byesn moels. From Theorem 2 C-L prove tht the mtrx weghte verge (β ) must le wthn the ellpso efne by ( β c) ( β c) < ¼ β β c = β +β 2 the H H. Where ( ) rthmetc verge of the prmeter vectors n H s smple precson mtrx unque up to sclr multple. Ths proves constrnt on the extreme vlues of β s: 7

9 (13) ( β c ) H( β c ) = ¼ β H β Whch mples tht ny possble vlue of β efne by the fferent vlues of Ω must be contne wthn or on the surfce of ths ellpso. From the reltonshp n (7) we hve: ln ( W ) ln ( W ) = E + D where: (14) ( D = X ) ( β β + X β β ) D s the fference n the log wges tht s ttrbutble to the fferentl pyment scheule tht s often referre to s scrmnton. Where the term X ( β β ) compenston p to the vntge group n X ( β β ) compenston p to the svntge group. mesures the over mesures the uner (15) E = X β E s the fference tht s ue to the fferences n the worker s chrcterstcs/humn cptl whch s referre to s enowment. We cn solve for the vlue of β s the vlue tht ether mxmzes or mnmzes D. By mplcton, snce ln( W ) remns constnt, mnmzng D mxmzes E n mxmzng D s equvlent to mnmzng E. Thus we solve the followng optmzton problem: (16) ( E = X ) c H c β H β Mx/Mn β, st ( β ) ( β ) = ¼ Where we use the full smple cross proucts mtrx XX s the smple precson mtrx H or the pproprte nverse of the heteroscestc consstent covrnce mtrx. The constrne optmzton cn then be efne by Lgrngn of the form: (17) L = X β λ( ( β c) H( β c) ¼ β H β) 8

10 The frst orer ervtves of L wth respect to β n λ re gven s: (18) L = X 2 λh ( β c ) β (19) L = ( β c ) H( β c ) ¼ β H β λ We cn solve (18) for the optml vlue of β ( β ) by settng ths expresson equl to zero n we get: (20) β = c +ρ H X +, where ρ! = 2 λ! 1 1 then substtutng H for 1 c +ρ X β nto (19) whch s lso set to equl to zero we cn solve for ρ where we get two soluton vectors (21) ρ! = ±φ, where φ= ½ β H β 1 X H X Then two solutons for the optml β re foun to be: (22) 1 β = c + γφ X H where γ 1 = 1 n γ 2 = 1. The secon orer contons cn be estblshe by evlutng the mtrx of secon ervtves evlute t ech soluton s: (23) L( ) 2 ( λ) 2 1 β λ φ H 2φ X = γ β 2 φ X 0 Becuse the precson mtrx ( ) H s postve efnte mtrx n φ > 0, β 1 wll be the mxmum of E n the mnmum of D n β 2 wll be the mnmum vlue of E n the mxmum of D n we cn etermne the bouns on the possble vlues of the mesure of scrmnton. Note tht when β = β then β = β = β. 9

11 3. Bouns on the mesure of scrmnton (D). The extreme vlues of scrmnton mesure (D) whch we wll enote s β cn now be use to efne the extreme vlues of the D. From the efntons bove we hve tht (24) D = ln( W) X β or by substtuton ths cn be shown to be: D = ln( W) X β Thus (25) D = ln( W) Xc γ ½ β H β X H X 1 recll tht γ 1 = 1 n γ 2 = 1. Thus the fference between the lmtng vlues of the scrmnton mesure s gven by (26) D D = β H β X H X whch s weghte functon of fferences n the vector of prmeters ( β ) n ( X ). Thus the greter the fference n the prmeters or the greter the fference n the scrmnton mesures the lrger the spn of vlues one mght obtn from ny scrmnton mesure employe. The mesure D cn lso be shown to be rectly relte to the mesure of scrmnton efne n (1) s δ. From the reltonshp n (7) n (14) n (15) we hve: (27) ln W = E + D W If we re ntereste n removng the nfluence of the fferences n enowments, or equvlently mkng the ssumpton tht MP = MP we cn concentrte on the vlue of D. (28) ln W = D W or equvlently: 10

12 W = W (29) exp( D) s the rto of the verge wge for the vntge group to the svntge group. An we efne: (30) by equton (1). exp( ) ( 1 ) W = D W W Thus we hve tht: = +δ W (31) δ= exp( D) 1 Or tht δ s monotonc functon of D n the mxmzton of D wll conce wth the mxmum of δ n the mnmzton of D s lso the mnmum vlue of δ. Note tht when D <.3 the pproxmton tht δ D cn be use. We cn efne the estmte of δ usng ny prtculr efnton of β s: (32) δ = exp D 1 In orer to use the estmte vlues of D n β to mke nferences we nee to be ble to mke probblty sttements concernng ther estmtes. A frst step n mkng these nferences s the ervton of n estmte for ther vrnces. 4. The symptotc vrnce of D n β In compnon pper to ther 1994 pper Oxc n Rnsom (1998) present the methoology for the computton of the vrnces use n ther erler pper. The technque they employ s n pplcton of the wely use elt metho n whch frst orer Tylor seres expnson s use to lnerze D. In ths secton we lso pply the elt metho but we conser not only the estmte prmeters but n fference from Oxc n Rnsom we lso ssume tht the mens of the chrcterstcs of ech group re stochstc s well. Thus D s efne n terms of four rnom vectors ( β, β, X, n X) for whch we cn efne 11

13 estmtes of ther covrnces. By stckng these four vectors we efne vector of length 4k gven s θ whch s efne s: (33) θ = β β X X 14 k Where the covrnce of θ s efne s Ψ n we cn efne ths covrnce s: (34) Φ Φ Ψ = 0 0 Σ Σ 4k 4k The estmtes of ( ) Σ re the covrnces of the mens of the ttrbutes for ech group n the Φ = cov β s the pproprte estmtor of the prmeter covrnce mtrx whch my nee to be correcte to ccount for heteroskestcty, commonly encountere problem n the estmton of wge equtons, or my be the prouct of mxmum lkelhoo estmton n the cse tht the ernngs t re not prove n contnuous recors. In orer to estmte the vrnce of the mesure of scrmnton we use the elt metho whch results n: (35) vr " ( D ) ( D θ) ( D θ) = Ψ θ θ Consequently ths estmte requres the efnton of the grent of D wth respect to the prmeters n θ. For the prevously efne set of scrmnton mesures efne n Secton 1 of ths pper, s etermne by the weghtng mtrx Ω (s summrze n Tble 1), we fn the followng estmte of the vrnce: 12

14 (36) " ( ) ( vr D = β ) ( β Σ β β ) ( + β ) ( β Σ β β ) ( X ) X ( X X) + Ω Φ Ω ( ( ) I ) ( I ) ( ) + X Ω X Φ X Ω X In the cse of the extreme vlues of D tht we hve erve n Secton 2 we o not efne unque vlue for the weghtng mtrx Ω. Thus β s not lner functon of the prmeter estmtes for ech cse ( β n β ) consequently we nee to erve fferent expresson for the pproxmte vrnce bse on the equton (25) gven s: (37) D = β + c γρ X Σ β + c γρ X " 1 vr( ) 1 H H c X c X β +γρh Σ β + +γρh X ½ ¼ ½ ¼ X X X γ ρ H β Ω γ ρ H β X ½ ¼ ½ ¼ X +γ ρ β Ω X X +γ H ρ H β gn where γ 1 = 1 n γ 2 = 1. In ton, we cn efne the pproxmte covrnce of both of the extreme vlue prmeters ( β n β ), s efne n equton (22) s: 1 2 (38) " cov( β ) =ρ QH Σ +Σ H Q {[ I+ ] [ ] [ ] G I G I G [ I G] } + ¼ γ Φ + γ + γ Φ γ where Q= I X X H 1, = ¼ H XH X, n 1 G =π H β X H 2 1, ρ ( β β)( ) 1 ½ ( 1 ) ( ) π = β H β XH X. ½ 13

15 5. Bootstrppng stnr errors n confence ntervls for D An lterntve to constructng the Wl tests usng the pproxmte vrnces efne n (37) n (38) s to employ Efron s (1982) bootstrp to construct lterntve stnr error estmtes n confence ntervls tht re not bse on ny prtculr strbuton. The bootstrp hs been pple n the computton of scrmnton mesures most notbly by Slber n Weber (1999) where they compre the vlues for the scrmnton mesures efne n Tble 1 for the fferences between Esterners n Westerners n the Isrel lbor mrket. The bootstrp nvolves the recomputton of multple vlues of the coeffcents of nterest D β by rwng wth replcement from the t use. Snce Efron s orgnl ( n ) contrbuton number of enhncements hve been propose to the bootstrp methoology. In fference to Slber n Weber who employ the nve percentle pproch on the mesure of scrmnton, we follow Horowtz s (2001) vce to bse the bootstrp only on pvot sttstc. We use contonl bootstrp for the regresson coeffcents s propose n Freemn n Peters (1984) n whch the moel s ssume but the regresson errors re smple wth replcement. The confence ntervls re constructe usng bootstrp-t technque s escrbe n Efron n Tbshrn (1993) whch s equvlent to usng the symptotc t-sttstc s our pvot. The smplng wth replcement s conucte usng secon-orer blnce resmple metho propose by Dvson, Hnkley n Schechtmn (1986). Ths mens tht the verge chrcterstcs of ech group ( X n X ) re both resmple usng the sme smple s the resuls use to recompute the prmeter estmtes ( β n β ). In ton, these smples re rwn n such wy to nsure tht the frequency of choosng ech observton s equl. 14

16 In the cse of the mesures of scrmnton D we use the t-rto of the estmte to the estmte stnr error s efne n (36) n (37) to form the pproprte pvot sttstc. A sttstc efne s t-sttstc s compute for ech bootstrp smulton whch s efne s: t = D D vr( D ) (39) ( ) " b b b where the D b enotes the estmte scrmnton mesure for bootstrp smulton (b) n D s the pont estmte bse on the t. These sttstcs re then rescle to generte bootstrp-t vlue of the scrmnton mesure esgnte s ( b ) (40) D# = t vr( " D ) + D 6. A Smple Exmple b D # b whch s efne s: The fferences n verge wges for men n women n the US hs been well ocumente. A number of ppers hve shown how ths fferentl hs chnge over tme n the US nctng tht the fferentl hs been ecresng over tme (see Polchek n Robust 2001). The exmple we use here computes the vrous mesures of scrmnton s we hve efne n the context of mles s the vntge group n women s the svntge group. We use smll rnom subset of the 1985 Current Populton Survey (245 women n 289 men) from Bernt(1991) ( CPS85 from the t for chpter 5). Two regressons re estmte by gener, wth the log of ncome s the epenent vrble n the yers of eucton n potentl experence (s pproxmte by the number of yers snce left school) s the nepenent vrbles. The men n stnr evton of the t re lste n Tble 2. The regresson prmeter estmtes re lste n Tble 3. From these regressons we fn tht men re compenste t lmost ouble the rte for ther potentl experence thn women (.0163 versus.0089) lthough eucton seems to be better ccounte for n women. In Tble 4 we lst the vrous mesures of scrmnton (n terms of the log of the ncome). The fferences of the mens of the log of wges whch nclues both the 15

17 enowment fferences n the fference ttrbutble to scrmnton s foun to be From the rest of the rows n Tble 4 we fn tht ll of the pont estmtes of the mesures of scrmnton re lrger thn ths vlue whch woul ncte tht the enowment hs negtve effect on the wge fference. Ths tble nclues the pont estmte n the 3 column n the pproxmte stnr error n column 4. In ton, we hve nclue the bootstrppe vlues of the men, stnr error, n the 95% confence bouns. Note tht for the trtonl mesures of scrmnton the D to D n mesures the pont estmte n the men of the bootstrp estmtes re very close nctng lttle bs. Also the symptotc stnr error estmtes re lmost exctly equl to the bootstrp vlues. In the bootstrps performe here we use 10,000 replctons once we etermne tht more replctons not effect the results obtne to ny sgnfcnt egree. Tble 5 lsts the extreme bouns for the prmeter estmtes ( ) β long wth the symptotc stnr error estmtes. We see tht the non-scrmntory wge prmeters tht mxmze the scrmnton re those tht result n prmeters for potentl experence tht re smll n for whch we coul not reject the hypothess tht they re equl to zero. An for the mnmum set of non-scrmntory prmeters re those tht hve the gretest prmeter for the nfluence of potentl experence n for eucton s well. In the lst two rows of Tble 4 we lst the scrmnton mesures bse on the bouns of the nonscrmntory wge prmeters ( β ). Note tht D < [ D D ] < D, the upper n lower 1 2 boun estmtes ct s the lmts on the estmtes of the ll the lterntve scrmnton mesures. In ths exmple, the extreme mesures the symptotc n bootstrp vlues ffer more thn for the other mesures. The verge of the bootstrppe vlues nctes tht the pont estmte of postvely bse n D 1 (bse on the mnmum for the scrmnton mesure) my be D 2 (bse on the mxmum for the scrmnton mesure) my be negtvely bse, though n nether cse s the estmte bs more thn 5%. From the 16

18 bootstrppe confence ntervls we fn tht the 2.5% lower boun for the mnmum vlue of the scrmnton mesure s.1545 n the 97.5% upper boun for the mxmum of the scrmnton mesure s Thus we cn boun the estmte of the scrmnton mesure lthough these probblty sttements gnore the probblty of choce between the two extremes n ny vrton tht my be ue to lterntve moel specfctons. An equvlent metho for emonstrtng the probblty bouns for the scrmnton mesure s by exmnng the ensty of the two extreme mesures. Fgure 1 splys two kernel ensty estmtes s etermne by the 10,000 stuentze bootstrp vlues for ech mesure. Note tht the ensty estmte for the lower boun ppers to be estmte wth greter precson thn the upper boun s ws the cse for the bootstrppe vrnce estmte s borne out by the bootstrp estmte of the stnr evton for stnr evton estmte for D 1 s oppose to the D 2. However t s pprent from ths fgure tht the exmnton of the mnmum scrmnton mesure results n n unmbguous concluson tht scrmnton s non-zero n ths cse. In other wors we coul reject the hypothess tht scrmnton ws zero wth very low probblty of mkng n error. Thus by usng the mnmum mesure of scrmnton n the lowest boun we stll fn tht scrmnton s postve. A cvet for ths pplcton s n orer. The moel specfcton my crete lrger egree of mesure scrmnton ue to the lck of more etl s to eucton type, occupton, chrcterstcs of the employer, fmly crcumstnces, n the proxy for experence. In prtculr, the use of potentl experence lone for both men n women s probbly responsble for ncresng the mesure scrmnton ue to the nequcy of ths vrble to ccount for the fferentl n ccumulte humn cptl tht hs been shown to expln such lrge proporton of the gener wge gp (see Polchek 1995). Fler (1993) emonstrtes emprclly tht ths s n npproprte proxy for comprble experence 17

19 mesure for both men n women by emonstrtng how other proxes chnge the gener fferentls n coeffcents. Specfclly potentl experence oes not ccount for potentl gps n experence whch re more prevlent for mrre women n women wth chlren thn for men. By mesurng less ctul experence for women thn for men t s expecte tht the prmeter n wge equton woul be less s well. 7. Conclusons It s well known tht the vrous wge fferentl ecompostons trtonlly one n nlyzng scrmnton rely hevly on the ssumpton regrng the non scrmnton wge structure β (see equton (7)). Severl uthors hve ttempte to motvte the specfcton of ths "no scrmnton" wge structure bse on the objectve functon of the employer n prctcng scrmntory behvour. The purpose of ths pper hs been to show tht the wge structure tht woul prevl n the bsence of scrmnton cn n fct be boune when we ssume tht the nformton to estblsh ths wge structure s weghte verge of the wge structure for the vntge n the svntge groups. Bse on theorem from Chmberln n Lemer (1976) we showe n ths pper tht the nonscrmnton wge prmeters (β ) must le wthn n ellpso efne by the t n the regresson results for ech group. By usng ths metho we re ble to select the β whch wll mxmze (mnmze) the level of the scrmnton n the lbor mrket. In ton to ervng the formuls for the estmte prmeters for the nonscrmnton wge structure tht mnmzes the level of scrmnton we lso specfy the pproxmte stnr errors. The pont estmte n the pproxmte stnr errors cn be use to efne pvot sttstc whch cn be use to bootstrp the scrmnton mesures. Thus t s possble to construct n estmte of the ensty of the scrmnton mesures whch cn then be use to mke probblty sttements concernng the presence of scrmnton. In the exmple use here we foun tht the mesure of scrmnton tht 18

20 ws constructe ws unmbguously postve s efne by the strbuton of both the mnmum scrmnton mesure. 19

21 REFERENCES Blner, A. S., "Wge Dscrmnton: Reuce Form n Structurl Estmtes," Journl of Humn Resources 8, (Fll 1973), Chmberln, G. n E. Lemer, "Mtrx Weghte Averges n Posteror Bouns," Journl Royl Sttstcl Socety, Seres B, 38, (1976), Cotton, J., "On the Decomposton of Wge Dfferentls," The Revew of Economcs n Sttstcs 70, (My 1988), Efron, B., The Jckknfe, the Bootstrp n Other Resmplng Plns, Socety for Inustrl n Mthemtcs, (1982). Fler, Rnll K., The Usefulness of Precte Vlues for Pror Work Experence n Anlyzng Lbor Mrket Outcomes for Women, The Journl of Humn Resources, 28 (3), (1993), Freemn, D.A. n S.C. Peters, "Bootstrppng Regresson Equton: Some Emprcl Results", Journl of the Amercn Sttstcl Assocton, 79, (1984), Gwne, K., Are U.S. Nontrff Brrers Retltory? An Applcton of Extreme Bouns Anlyss n the Tobt Moel, The Revew of Economcs n Sttstcs, 77, (1995), Lemer, E. E., Let s Tke the Con Out of Econometrcs, The Amercn Economc Revew, 73, (1983), Levne, R. n D. Renelt, A Senstvty Anlyss of Cross-Country Growth Regressons, The Amercn Economc Revew, 82, (1992), McAleer, M. A. R. Pgn n P. A. Volker, Wht Wll Tke the Con Out of Econometrcs?, The Amercn Economc Revew, 75, (1985), Men, Dv, Towrs broer explnton of mle-femle wge fferences, Apple Economcs Letters, 7, (2000), Men, J. F., "Dscrmnton A Mnfestton of Mle Mrket Power?" n Cynth B. Lloy (e.), Sex, Dscrmnton, n the Dvson of Lbor (New York: Columb Unversty Press, 1975). Neumrk, D., "Employers' Dscrmntory Behvor n the Estmton of Wge Dscrmnton", The Journl of Humn Resources. 23, (1988), Oxc, R., "Mle-Femle Wge Dfferentls n Urbn Lbor Mrkets," Interntonl Economc Revew, 9, (Oct. 1973),

22 Oxc, R. n M. Rnsom, "Serchng for the Effect of Unonsm on the Wges of Unon n Nonunon Workers," Journl of Lbor Reserch, 9, (Sprng 1988), , "On Dscrmnton n the Decomposton of Wge Dfferentls," Journl of Econometrcs, 61, (Mrch 1994), 5-21., "Clculton of Approxmte Vrnces for Wge Decomposton Dfferentls, Journl of Economc n Socl Mesurement, 24, (1998), Polchek, Solomon W., Humn Cptl n the Gener Ernngs Gp: A Response to Femnst Crtques, n Out of the Mrgn: Femnst Perspectves on Economcs, ete by Eth Kuper n Jolne Sp, Routlege, 1995, Polchek, Solomon W. n John Robust, Trens n the Mle-Femle Wge Gp: The 1980s Compre wth the 1970s, Southern Economc Journl, 67(4), (2001), Remers, C., "Lbor Mrket Dscrmnton Agnst Hspnc n Blck Men," The Revew of Economcs n Sttstcs, 65, (Nov. 1983), Slber, Jcques n Mchl Weber, Lbour mrket scrmnton: re there sgnfcnt fferences between the vrous ecompostons?, Apple Economcs, 31, (1999), Temple, J., Growth Regressons n Wht the Textbooks Don t Tell You, Bulletn of Economc Reserch, 52, (2000),

23 Tble 1 The propose vlues of the weghtng mtrx Ω. Weghtng Mtrx Author Ω O = I, or 0 Oxc (1973) Ω R = ½I Remers (1983) Ω C = (N /N) I Cotton (1988) Ω N = (X X + X X ) -1 (X X ) Neumrk (1988) Tble 2 The chrcterstcs of the smple exmple. Gener Vrble Men SD Men (289 obs) nturl logrthm of verge hourly ernngs potentl yers of experence (AGE-ED-6) yers of eucton Women (245 obs) nturl logrthm of verge hourly ernngs potentl yers of experence (AGE-ED-6) yers of eucton Tble 3 Result of smple moel regresson Gener Vrble β SE t-sttstc Men (R 2 =.232, σ=.469) Women (R 2 =.262, σ=.423) (Constnt) potentl yers of experence (AGE-ED-6) yers of eucton (Constnt) potentl yers of experence (AGE-ED-6) yers of eucton

24 Tble 4. Mesures of scrmnton wth bootstrppe sttstcs bse on smple moel. Vrble Reference Est Asymptotc Bootstrppe vlues Prmeters St Dev, Men St Dev 2.5% 97.5% ln( Y ) D β β +β D r ½ ( ) D β nβ+ nβ n+ n D c ( ) ( ) D n β D 1 β D β Tble 5 Extreme Bouns comprson prmeter estmtes ( β ) Boun Vrble β SE (sy) t-sttstc Mn of D ( β 1 ) Mx of D ( β 2 ) (Constnt) potentl yers of experence (AGE-ED-6) yers of eucton (Constnt) potentl yers of experence (AGE-ED-6) yers of eucton

25 Fgure 1. A comprson of the estmte enstes of the t-bootstrppe vlues of D 1 n D 2 24