Two-Step versus Simultaneous Estimation of Survey-Non-Sampling Error and True Value Components of Small Area Sample Estimators

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1 Two-Sep versus Sulaneous Esaon of Survey-Non-Saplng Error and True Value Coponens of Sall rea Saple Esaors a PVB Sway, TS Zeran b c and JS Meha a,b Bureau of Labor Sascs, Roo 4985, Massachuses venue, NE, Washngon, DC 0, E-al: paravasu_s@blsgov c and Zeran_aara@blsgov Deparen of Maheacs, Teple Unversy, Phladelpha, P 9 E-al: eha007@cosne bsrac The precson of a sall-doan saple esaor n be proved by esang sulaneously s rue value and saplng and non-saplng error coponens In prncple, hs sulaneous esaon s superor o any wo-sep esaon of he rue value and saplng error coponens, gnorng he non-saplng error coponen In hs paper, a e seres odel for sae eployen or uneployen s used o deonsrae he laons of a wo-sep ehod cross-seconal odel for sae eployen or uneployen s used o explan he advanages of sulaneously esang a rue value and he sus of saplng and non-saplng errors n wo or ore saple esaors of he rue value Key Words Indrec odel-dependen esae; doan ndrec; e ndrec; doan and e ndrec; coeffcen drver Inroducon ccurae esaes of eployen and uneployen a varous levels of geographc deal are needed o forulae good regonal polces, such as he deernaon of elgbly for and/or he alloon of Federal resources, wh a good undersandng of lol econoc condons Such esaes nno be produced fro survey daa colleced n each sall area beuse saple szes whn hose areas are ofen eher zero or oo sall o provde relable esaes If an area-specfc saple s avalable bu s no large enough o yeld drec esaes of adequae accuracy, hen her accuracy n be proved, usng ndrec odel-dependen esaors ha borrow srengh fro auxlary daa colleced n hs and oher sall areas and/or a ore han one e perod On a defnon suggesed by a passage n Rao (003, p ) ndrec esaors gh be classfed as doan ndrec, e ndrec, or doan and e ndrec dependng on wheher hey borrow srengh cross-seconally, over e, or boh In hs paper, we develop doan ndrec esaors ha faclae sulaneous esaon of he rue value and saplng and non-saplng error coponens of saple esaors In prncple, sulaneous esaon s superor o wo-sep esaon of hese coponens To llusrae he proposed echnques, we consder he proble of provng he accuracy of curren populaon survey () esaes of eployen and uneployen for 5 sall areas conssng of he 50 Uned Saes plus he Dsrc of Coluba sae esaes are wdely used and provng her accuracy s valuable The reander of hs paper s dvded no sx secons Secon descrbes he e seres odels used o esae eployen or uneployen for he saes, ajor eropolan areas, and correspondng balance of saes n he US Secon 3 shows he laons of a wo-sep ehod used o esae hese e seres odels The consequences of ncorrecly neglecng non-saplng errors are gven n Secon 4 Secon 5 uses cross-seconal esaes based on wo or ore saple esaors of sae eployen (or uneployen) o esae sulaneously he rue values of eployen (or uneployen) and he sus of saplng and non-saplng errors conaned n hose esaors Seasonal adjusen of he rue values of sae eployen (or uneployen) and esaon of he auocovarances of saplng errors are dscussed n Secon 6 Secon 7 provdes an exaple Secon 8 concludes Te Seres Models for he True Value Coponen of a Saple Esaor Saplng Model cknowledgen Coens by Chrsopher Bollnger are hghly apprecaed Dsclaer ny opnons expressed n hs paper are hose of he auhors and do no consue polcy of he Bureau of Labor Sascs

2 ˆ Le Y denoe he curren populaon survey () copose esaor of he rue populaon value, denoed by Y, of eher uneployen or eployen Le and j ndex saes, ajor eropolan areas, or balance of saes and le ndex onhs The decoposon of Yˆ no s unobserved coponens s ˆ Y = Y + e + u, =,, τ, =,, () where and represen saplng (or survey) and non-saplng errors (such as non-response or easureen errors), e u respecvely value, denoed by y, of Y obaned fro a specfc saple s an esae of Y ˆ The annual averages of he coponens of a ype of non-response raes for and 003 naonal esaes are gven n US Deparen of Labor (006, p 6-4) These averages show ha we nno assue ha u = 0 The ehods n Cochran (977, Chaper 3) gh have been already red o reduce he agnude of u Under he curren 000 desgn, he nuber of assgned households n onhly saples for dfferen saes ranges fro 700 o 5,344 These saple szes are no large enough o yeld Yˆ of adequae precson I s necessary o use odel-based approaches o prove he effcency of Yˆ The desgn-based approach o nference reas he Y as fxed quanes Unlke hs approach, he odel-based approach o saplng nference reas he Y as a rando saple fro a superpopulaon and assgns o he a probably dsrbuon pled by a odel for Y 3 Snce he rue odel for Y s unknown, we n use one of s approxaons n place of Y used n () bu hen we should be prepared o refully nvesgae wheher such a use proves he precson of Yˆ 4 Le E p and V p denoe he expecaon and varance operaors wh respec o he probably dsrbuon nduced by he 000 saplng desgn condonal on Y, respecvely Le E and V denoe he expecaon and varance operaors wh respec o a probably dsrbuon assgned o he Y by an assued odel for, respecvely (see Lle, 004, p 547) Y Survey Error (SurE) Model Suppose ha u = 0 for all and Then a odel for e pled by he desgn s σ e, e = e () σ e, where for, j =,, and =,, τ, E ( p e Y ) = 0, [( Covp e, e j ) Y, Yj] = 0 f j, Vp( e Y) = σ e =, + D σ wh = beween-prary Saplng Un (PSU) varance coponen of he oal desgn σ be,, e, e, e σ be,, varance of arsng fro selecng a subse of all non-self-represenng (NSR) PSUs n sae, D σ = whn- PSU varance coponen of he oal desgn varance of e arsng fro saplng of housng uns whn seleced PSUs of sae, D = he rao of he whn-psu varance coponen of he oal desgn varance of, assung he e, saple desgn, o he oal desgn varance, σ = ( N / ) ( ( / ) ) e, n Y Y N, of an unbased esaor of Y, assung sple rando saplng (desgn effec), N = he cvlan non-nsuonalzed populaon 6 years of age and older for he sae, ( N / n ) = he sae saplng nerval (see US Deparen of Labor, 006, p 3-6) and follows a xed nd order e e e, e, Here eployen s defned on a place-of-resdence bass, snce he s a household survey We srcly adhere o Cochran s (977, p 6) dsncon beween precson and accuracy 3 Rao (003, p 77) lls hs odel a lnkng odel 4 The rue odel for s an unknown funcon, Y f ( x,, x, x,, x ) Y K K +, L, =, of all of he deernans of wh he correc bu unknown funconal for and whou any oed deernans or seasured varables ny of s esable approxaons s sspecfed f has an ncorrec funconal for and suffers fro oed-varable and easureenerror bases (see Freedan and Navd, 986) Y

3 auoregressve and 7 h order ovng average, or RM(, 7), odel wh V ( e ) = σ, as shown by Tller (99, 005) Ths RM(, 7) odel s approxaed by a 5 h order auoregresson or R(5) odel 3 Lnkng Model Durbn and Koopan (00) work wh Harvey s (989) srucural e seres odels Followng he, s assued ha he fne populaon s generaed accordng o he superpopulaon odel = + + (3) Y T Y, S Y, I Y, T S, I Y, where Y s he rend-cycle, he seasonal, and he rregular coponen, whch are reaed as he unobserved, Y coponens of Y 5 ll hese coponens are assued o be ndependen of each oher for all and Ths assupon ay no be rue f he decoposon of Y n equaon (3) s no unque To accoun for hs non-unqueness, Havenner and Sway (98) assue ha,, and are correlaed wh each oher for all and I s assued ha T Y, T Y, S Y, I Y, follows a rando walk odel wh one perod lagged value of a rando walk drf Durbn and Koopan (00, p 39) ll hs he lol lnear rend odel The seasonal coponen,, s expressed n a rgonoerc for o ake follow Durbn and Koopan s (00, (36), p 40) quas-rando walk odel The rregular coponen, 6, s assued o be serally ndependen and norally dsrbued wh ean zero and consan varance I Y, I s known ha eployen has a srong endency o ove cycllly, downward n general busness slowdowns and upward n expansons These cycles are asyerc beuse eployen decreases a a faser rae han ncreases The behavor of uneployen over e s he oppose of eployen s behavor and s lled asyerc couner-cycll behavor Mongoery, Zarnowz, Tsay and Tao s (998, p 487) resuls ply ha he e seres odels assued above for he coponens of Y n (3) ay be sspecfed beuse hey ay no exhb he asyerc cycles of eployen and uneployen More specflly, her resuls are: () frs order auoregressve odel n frs dfferences, denoed by RIM(,, 0), ha fed he US quarerly uneployen rae seres for que well was no able o accuraely represen he asyerc cycles of uneployen durng hs perod; () n RIM (,, 0)(4, 0, 4) odel wh a ulplve seasonal facor, denoed by RIM(4, 0, 4), under-predced he US uneployen rae durng he rapd 7 ncrease of 98 and exhbed foress ha flucuaed a grea deal ore durng sable perods of uneployen 4 Covarae Model Consder an exenson of odel (3) o allow one or ore of he unobserved coponens of Y o be relaed o correspondng coponens n anoher seres lled a covarae The uneployen nsurance clas (UI Clas) fro he Federal-Sae Uneployen Insurance Syse are used as a covarae seres f Y represens uneployen, and nonagrculural payroll eployen esaes fro he Curren Eployen Sascs (CES) survey are used as a covarae seres f Y represens eployen e, S Y, 3 Two-Sep Mehod of Esang he Paraeers of Survey Error and Lnkng Models We ll he odel conssng of equaon () whou u, equaon (), and equaon (3) wh covaraes he cobned odel For each, he unknown paraeers of odel () are: he desgn varances, σ, =,, τ e,, and 7 paraeers of R(5) odel For each, he unknown paraeers of odel (3) are he error varances of he odels for,, and I Y,, whch are four n nuber (see Tller, 005, p 7) When he y for =,, T Y, S Y, τ, are augened wh he avalable 5 Tller (005) also allows for eporary or peranen shfs n he level of he seres, { Y }, =, + 6 Soees s assued ha I Y, follows a lower-order auoregressve process (see Tller, 99, p 5) 7 I n be shown ha Y conans a rando walk coponen f conans a saonary coponen afer beng frs dfferenced d es I s precsely hs condon ha s volaed when he nonlnear odels ha accuraely represen he asyerc cycles of eployen and uneployen are consdered

4 observaons on he covaraes, he nuber of he unknown paraeers of (3) wh covaraes ncreases o 0 for each (see Tller, 005, p 0) ll hese paraeers of he cobned odel are no denfable on he bass of he avalable esaes, y, =,, τ, =,,, and observaons on he covaraes alone Consequenly, for each, a wo-sep procedure s used: Frs sep For each, he avalable esaes ( y, =,, of () ndependenly of (3) wh covaraes; Second sep Gven he esaes ( y wh covaraes are esaed, holdng he paraeers of () and R(5) fxed a her esaed values τ ) are used o esae he paraeers, =,, τ ), he paraeers of (3) 3 Survey Error Varance Esaes Le and be fxed so ha he coponens of n (3) are fxed Le be absen Then he desgn varance,, n () Y n be esaed usng he generalzed varance funcon (GVF) (see Woler (985, p 03), Len (99, 994), and US Deparen of Labor (006, p 3-6)), ˆ Vp ( Y Y ) = ( b / N ) Y + by (4) where b = k ( N / n ) D, k = / e, ( p ), p = σ / σ, and D, N,,, and be,, e, e, n σ be,, ˆ σ ( = V ( ) are as defned below () e, p Y Y ) u σ e, ehod of esang he b s suarzed n he followng seps: ll survey sascs are dvded no several groups wh odel (4) fed ndependenly n each group If b dffers aong he sascs n a group, hen hey are no conssenly esable beuse he nuber of unknown paraeers ncreases wh he sze of he group (see Lehann and Casella, 998, p 48) To avod hs dffcul ncdenal paraeer proble, re s aken o group ogeher all survey sascs ha follow a coon odel such as (4) wh fxed and Ths ay nvolve groupng ogeher he sascs wh slar desgn effecs; he sae characerscs for seleced deographc or geographc subgroups Exaples of such groupngs are gven n Woler (985, p 09) Le g ndex he sascs n a group fored n Sep The desgn varances, ˆ Vp( Y ( g) Y( g) ), are copued for several sascs of hs group usng ˆ ˆ ˆ ˆ ( b / N )( Y ( g)) + by ( g), where bˆ ˆ ˆ = k ( N / n ) D, kˆ ˆ, e, = /( p ) pˆ ˆ ˆ = σ / σ, ˆ σ = +, s obaned by ulplyng he 000 decennal census be,, e, e, ˆ σ ˆ be,, D ˆ σ ˆ σ e, e, be,, esae of beween-psu varance by he square of he rao of he annual average of ˆ Y ( g) for he curren year o he 000 census esae of Y ( g), s copued fro he naonal whn-psu desgn effecs by adjusng for D ˆ e, dfferences n a ceran nonnervew rae, see he eorandu (007) fro Khandaker Mansur, and σ = σ e, e, when Y ( g) = ˆ Y ( g) Le Vˆ ( ˆ p Y ( g) Y( g) ) denoe he esaor of ˆ Vp( Y ( g) Y( g) ) 3 Fng odel (4) o he daa ( ˆ Y ( g), ˆ ˆ Vp( Y ( g) Y( g) )) fro Sep gves he esae of b for he group The odel fng echnque s an erave weghed leas squares procedure, where he wegh s he nverse of he square of he predced value of ˆ ˆ ˆ Vp( Y ( g) Y( g))/( Y ( g)) (see Woler, 985, p 07, (54)) Le b denoe an erave weghed leas squares esaor of b 4 n esae of he desgn varance of a survey sasc, say ˆ Y ( g), for whch he successve dfference replon ehod of Fay and Tran (995) s no appled s now obaned by evaluang odel (4) a he pon ( ˆ Y ( g) ; b ) I s lled a GVF esae The esaor, b, obaned n Sep 3 ay no have any desrable sasl properes beuse of he effecs of he errors, ˆ Y ( g ) - Y ( g), and n be very precse f all sascs whn a group do no behave accordng o odel (4) These probles ay ge resolved f odel (4) s replaced by he less resrcve odel, ˆ ˆ ˆ ˆ Vp ( Y ( g) Y ( g)) = bg { Y ( g) (/ N )( Y ( g)) }, g =,,, G (5) wh b ˆ ˆ g = δ0kg ( N / n ) D e, g + ξ0g (6) ˆ

5 ˆ e, g where D and ˆg are he esaors of D and kg gven n Sep, and ξ 0g s a rando varable wh ean zero k e, g and consan varance Chang, Sway, Hallahan and Tavlas (000) ehod n be used o esae odel (5) under assupon (6) Esaor (5) s precse, snce ˆ Y ( g ) s an precse esaor of Y ( g) 3 Survey Error uocorrelaon Esaes In Zeran and Robson (995), he varances and auocovarances of he e n () are esaed fro he separae panel (roaon group) esaes of Y panel s defned as he se of saplng uns jonng and leavng he saple a he sae e Each panel s a represenave saple of he populaon The saple consss of 8 such panels n every onh To esae odel varance and auocovarances of he copose esaor fro he esaes of odel varances and auocovarances of he panel esaors, Zeran and Robson (995) and Zeran (007) consder he followng odel: Yˆ = μ + θ + β +e (7) j j j ˆ j Yˆ j where Y = esaor of Y fro panel j, ej = saplng error of e j μ = overall ean, θ j = onh-n-saple effec, β = e effec, and Le denoe e n () for panel j Model (7) s sspecfed f he e seres odels for he coponens of Y n (3) are correcly specfed If hs s so, odel (7) ncorrecly esaes ej as ˆ Y ˆ ˆ ˆ j μ θj β nsead of as ˆ ˆ ˆ ˆ Yj Ty The ncorrecly esaed nno gve accurae esaes of he paraeers of odel () j, j Sy j, j Iy j, j e j oher han even f odel (5) gves accurae esaes of Thus, he Frs sep--of esang he survey error σ e, σ e, odel paraeers fro desgn based nforaon ndependenly of he e seres odels for he coponens of Y n (3)-- n lead o very naccurae esaes of he survey error odel paraeers If he condon, holdng he paraeers of he survey error odel fxed a her esaed values, eans he condon, he paraeers of he survey error odel are se equal o he esaes obaned n he Frs sep, hen he naccurae esaes obaned n he Frs sep n lead o naccurae esaes of he paraeers of odel (3) wh covaraes n he Second sep There wll be no conssency beween he esaes obaned n he Frs sep and hose obaned n he Second sep Furherore, he rando walk odels assued for soe of he coponens of Y lead o a predcor of Y ha s uncondonally nadssble relave o quadrac loss funcons beuse he predcor does no possess fne uncondonal ean They do no provde he predcors of Y wh good condonal and uncondonal properes Brown (990, p 49) shows he porance of workng wh such predcors 33 Sae-Space For of he Cobned Model Le { x } be a covarae seres Le y x = ( y, x ) Le he cobned odel be wren n sae-space for, as n Durbn and Koopan (00, p 38) Hold he paraeers of he survey error odel fxed a he esaes obaned n he Frs sep For lculang n he Second sep loglkelhood funcon and he axzaon of wh respec o he paraeers of odel (3) wh covaraes, he jon densy of he saple observaons, y x, y x,, yx τ, pled by he cobned odel s wren as τ py (,, y ) = py ( y,, y ) (8) x xτ x x x = where s assued ha py ( x y x0) = py ( x ) wh fne ean (see Durbn and Koopan, 00, p 30) The ruh of hs assupon s usually unknown For hs reason, s convenen o assue ha Y X s dsrbued wh densy py (,, ) x yx yx for all ; 8 unforunaely, hs assupon conradcs he assupon ha py ( x y x0) = py ( x ) wh fne ean Ths s beuse he forer assupon says ha under he rando walk odels assued for soe of he coponens of Y, Y does no possess fne uncondonal ean for all and he laer assupon says ha Y possesses X X 8 Here we dsngush a rando varable fro s value by a lde For exaple, y x s he value aken by he rando varable Y = ( Yˆ, X ) X

6 fne uncondonal ean for = Even f s assued ha Y X s dsrbued wh densy py (,, ) x yx yx for all >, he assupon ha py ( x y x0) = py ( x ) wh fne ean nno be rue for all hose daa ses for whch he value = occurs a dfferen pons on he e axs One ay use l (> 0) non-saonary eleens n he sae vecor whch deernes he nuber of observaons requred o for prors of hese eleens Whou hs or any oher nalzaon, s no possble o apply he Kalan fler o saespace for of he cobned odel n ()-(3) (see Durbn and Koopan (00, pp 7-30, 99-04) and Maddala and K (998, pp )) The nal value y x0 s usually unknown and any assupon abou ay be quesoned Incorrec assupons abou he x0 e profles y n lead o he esaes of Y, =,, 4 Consequences of Incorrecly Neglecng Non-Saplng Errors τ, wh ncorrec nal values and hence wh ncorrec In he prevous secon, we have consdered equaon () wh suppressed even hough s no equal o zero wh probably ddng o hs equaon a lnkng odel of he general lnear xed (GLM) odel s ype for, whch ples ha Y possesses fne second oen for all and, he bes lnear unbased predcor (BLUP) of Y n be found 9 Ths BLUP n be expressed as a weghed average of Y and he regresson-synhec esaor of Y pled by he lnkng odel Under ceran regulary condons gven n Rao (003, p 7), he BLUP of Y concdes wh ha s no affeced by he sspecfons n he lnkng odel as he desgn varance of Y goes o 0 Tha s, when s desgn-conssen, he BLUP of Y n also be desgn-conssen Ths resul s he bass of Lle s (004, p 55) saeen ha one way of lng he effecs of odel sspecfon s o resrc aenon o odels ha yeld desgn-conssen esaors (see also Rao, 003, p 48) Ths observaon s of no use o us when Yˆ s subjec o non-saplng errors, n Yˆ whch se s no desgn conssen (see Lle, 004, p 549) u ˆ ˆ u Y Yˆ Yˆ In Rao s (003) ernology, he BLUP of Y becoes an eprl BLUP (EBLUP) when he varance paraeers nvolved n he BLUP are replaced by her respecve saple esaors second-order approxaon o he ean square error (MSE) of EBLUP nvolves hree ers, as Rao s (003, p 04, (63)) very elegan dervaon shows 0 Ths dervaon furher shows ha an unbased esaor of he MSE of EBLUP o a desred order of approxaon ay nvolve four ers (see Rao, 003, p 05, (637)) The value of he su of hese four ers obaned fro a specfc saple n exceed he value of he desgn varance of Yˆ, showng ha he EBLUP of Y n be less effcen han Yˆ whch s self precse beuse of he sallness of he saple on whch s based The MSE of he BLUP of Y conans wo ers and only one of hese ers s saller han he desgn varance of Yˆ, as shown by Rao (003, p 7, (77)) These resuls rase he queson: n wha sense does he lnkng odel for Y borrow srengh fro s explanaory varables n akng an esae of? Under ceran regulary condons, wo of he hree ers of a second-order approxaon o he MSE of an EBLUP Y of Y go o zero and he reanng er reans below he desgn varance of as he nuber of observaons on he explanaory varables of he lnkng odel goes o nfny (see Rao, 003, p 7) Thus, n he l he MSE of an EBLUP of Y n nvolve only one er ha s saller han he desgn varance of Yˆ ny lnkng odel for Y n borrow srengh fro s explanaory varables n he l as he nuber of observaons on s explanaory varables goes o nfny f Yˆ s desgn-conssen and soe regulary condons are sasfed Ths resul holds even when he lnkng odel for Yˆ 9 Sway, Zeran, Meha and Robson (005) exend Rao s (003, pp 96-98) dervaon of he BLUP fro a GLM odel for a saple esaor o ake accoun of oed-varable, easureen-error and ncorrec funconal-for (or sply specfon) bases n he GLM odel 0 Ths dervaon has been exended o ake accoun of specfon bases n lnkng odels of he GLM ype n Sway, Yagh, Meha and Chang (006) The gan n effcency assocaed wh he use of Rao s (003, pp 6-7) BLUP ay ge reduced f he lnkng odel ha provdes he BLUP suffers fro specfon bases (see Sway e al, 005)

7 Y s sspecfed, provded Yˆ does no conan non-saplng errors I s of led use for us beuse large saple heores of esaon are rrelevan o sall area esaon and he sources of error oher han rando saplng varaon gven n Cochran (977, p 359) are presen n he and CES survey There s usually no jusfon o gnore he possble sspecfons n he e seres odels for he coponens of n equaon (3) wh covaraes 5 Sulaneous Esaon of Sus of Survey and Non-Saplng Errors and he Correspondng True Values ˆ Le us now reconsder Y Suppose ha an addonal esaor, denoed by Y, of Y s avalable Ths addonal esaor wll be gven by UI Clas daa f Y represens uneployen and by daa fro he CES survey or by exrapolaed daa fro he Quarerly Census of Eployen and Wages (QCEW) progra f ˆ Yˆ ˆ Y represens eployen We assue ha each of Y and Y s subjec o boh saplng and non-saplng errors and wre Y = Y + ε = + u and = Y + e ε wh ˆ ˆ Y ˆ ε = +, where he vecors, (, e ) and (, e u e u u ε wh ), denoe he saplng and non-saplng errors of ( Y, ˆ Y ), respecvely Le y denoe a value of Y obaned fro a specfc saple We now fx and le vary o focus on cross-seconal varaons n ε and ε These ypes of varaons n e and e are dfferen fro her rando saplng varaons ha are presen beuse only pars of he populaon have been easured usng soe saplng desgns 3 Modelng assupons are needed o analyze he non-saplng errors, u and u, and cross-seconal varaons n he saplng errors, e and e The proble of choosng beween Yˆ and Yˆ n only be solved f we n derve fro he a sngle esae ha s closer o Y han eher esae We show n hs secon ha such a sngle esae n be found even when y - Y > y - Y 5 Rando Coeffcen Regresson Model Wrng ˆ α = ( ε / Y ), a funcon of ˆ Y and ε, we ransfor he saplng odel, ˆ Y = Y + ε, no he lnkng odel, Y = ˆ α Y Replacng Y n he saplng odel n () by ˆ α Y gves he odel: ˆ Y = α0 + α Ŷ, =,, (9) where α0 = ε, α0 = π00 + π0bp + π0hp + 0 (0) BP HP α = π + π + π + () ( TP ) ( TP ) 0 where BP = he black populaon 6 years of age and older for he area, HP = he Hspanc populaon 6 years of age and older for he area, TP = he cvlan non-nsuonalzed populaon 6 years of age and older for he area, all he π s are fxed, he superscrp,, of he π s and s s shorhand for regresson of Y ˆ on ˆ BP, and he varables, BP, HP, ˆ Y ( TP ), ˆ Noe ha f Y s an esaor of eployen, hen s correc decoposon s Y = Y + ( Y - Y ) + +, where s a place-of-work eployen ha s dfferen fro he place-of-resdence eployen, Y, he rue-value coponen of To adop he place-of-resdence concep, he dfference, Y - Y, s added o u so ha Y = Y + ε Mahowez and Dunn (988) adop he place-of-work concep o sudy response errors n rerospecve repors of uneployen 3 Here we gnore he auocorrelaons of e pled by he saplng desgn beuse wh fxed and varyng, e exhbs only cross-seconal varaon Our purpose n hs secon s o odel hs varaon n he presence of non-saplng errors n advanage of havng cross-seconal daa alone on wo or ore esaors of s ha hey n provde nforaon abou he su of he saplng and ˆ Y ˆ non-saplng errors of Y, whereas e-seres daa alone on Y uddle he wo errors, wh no prospec of esang even her su The forer resul follows fro he analyss of hs secon and he laer resul follows fro he analyss of Secon 3 ˆ ˆ e u Yˆ Y

8 HP and ( ) TP, are lled he coeffcen drvers 4 ddonal coeffcen drvers ay be ncluded n (0) and () f hey are hough o be approprae We esae equaons (9)-() under he followng assupons: For =,, and fxed, () he (, ) are condonally and ndependenly dsrbued wh ean vecor 0 and consan covarance arx 0 σ Δ gven he coeffcen drvers () ( ˆ P α 0, α Y, BP, HP, TP ) = where P( α0, α BP, HP, TP ) P() s a jon probably dsrbuon funcon of he varables n (9)-() The coss of assupons () and () are consderably less han hose of assupons underlyng he cobned odel n Secon 3, as he followng dscusson shows: () Desgn-based nference s srcly napplble o suaons where u 0 (see Lle, 004, p 540) Hence odel () ha s needed for desgn-based nference s no eployed, snce such suaons are consdered n hs secon () Wha we have done n (9) s ha we reaned he saplng odel () and avoded all he sspecfons n, and he denfon probles wh, odels () and (3) by replacng he lnkng odel (3) by s alernave, Y = ˆ α Y No uch s known abou he rue odel for Y n foonoe 4 Therefore, he correcness of any specfed odel for Y ay be quesoned However, he sspecfons n he e seres odels for he coponens of Y n (3) seeed o us o cry ou for alernave lnkng odels reful coparson of he resuls produced by odel (3) and s alernave, ˆ Y = α Y, gh expose he weaknesses of boh odels () The condonal ndependence assupon () s weaker han he uncondonal ndependence assupon, ˆ P( α0, α Y ) = P( α0, α ), whch s false beuse of he nonzero correlaons beween ( ˆ Y, e, u ) and α We nclude he coeffcen drvers n (0) and () o avod hs false assupon (v) Snce (9) s a relaon beween wo esaors of eployen or uneployen, he range of s nercep s uch wder han ha of s slope To accoun for hs dfference n ranges, s nercep s ade o depend on BP and HP, whch BP ake values over wder nervals han and HP, on whch s slope s ade o depend TP TP (v) Sulaneous esaon of he nercep and slope of (9) pers sulaneous esaon of all he coponens of Yˆ n (), snce α0 = ε and Y = ˆ α Y In prncple, hs sulaneous esaon s superor o he wo-sep esaon of Y and e, gnorng u, as n Secon 3 The sulaneous esaon based on (9)-() elnaes he nconssences nheren n he wo-sep esaon of Secon 3 The Chebychev nequaly shows ha f he varance of 0 s sall, hen he dsrbuon of α0 s ghly concenraed abou he rgh-hand sde of (0) wh he error suppressed (see Lehann, 999, p 5) In hs se, he rgh-hand sde of (0) wh he error suppressed gves a good deernaon of he su of saplng and non-saplng errors n ˆ Y Subracng a good esaor of e + u fro Yˆ gves a good esaor of Y 5 Thus, under a resrcon on he varance of 0, equaons (9)-() per accurae sulaneous esaon of all he coponens of Yˆ These resuls reveal he advanages of usng equaons (9)-() nsead of he cobned odel n Secon 3 (v) Sway, Meha and Chang (006) show ha superor esaes of Y are unlkely o be obaned when he dependen and explanaory varables of (9) are nerchanged Therefore, we do no ake such an nerchange n hs paper For =,, and fxed, he Y are assued o be a rando saple fro he superpopulaon defned by Y = ˆ α Y and are assgned a odel condonal probably dsrbuon of ( ˆ - ) = ˆ α Y gven Yˆ = y and he coeffcen Y α0 4 Sway, Meha and Chang (006) exend odel (9) o nclude ore han wo esaors of and also o sub-sae areas for whch daa are eher oo sparse or unavalable 5 The porance of havng a good esae of eployen n be seen fro Rowhorn and Glyn ((006) and Magnac and Vsser (999) Y

9 drvers ssupons (0) and () ply ha ε (= α ) and ε = Y - = ( - α ) follow a bvarae condonal Yˆ 0 ˆ Y frequency dsrbuons gven = y and he coeffcen drvers, as vares for fxed Le he odel eans of hs dsrbuon be denoed by ˆ E( ε Y = y, BP, HP) = μ and ( ˆ BP HP E ε Y = y,, ) = μ Then μ and μ ˆ ˆ Yˆ TP TP represen he bases n Y and Y, respecvely The connecon beween ( μ, μ ) and he ers n equaons (0) and () s as follows: π = Consan bas n ha affecs all or he consan er of he bas, μ, n ; π 00 BP Yˆ + π HP = Varable coponen of he bas, μ, n ; = Flucuang coponen of he error, ε ; BP HP 0 y TP TP Yˆ [( π ) π ( ) π ( )] = Varable coponen of he bas, μ, n ; y = Flucuang coponen of he error, ε Thus, α presens no dffcul probles even hough s a non-lnear funcon of and above decoposon of he saplng and non-saplng errors of Yˆ s relaed o Cochran s (977, p 378) decoposon of an error of easureen on a un based on hs odel of easureen error 6 I should be noed ha wh fxed, he e seres shocks of Y ˆ and ˆ Y are fxed and ge subsued no μ and μ, respecvely Yˆ Yˆ Yˆ ε f assupon () s rue The Inserng (0) and () no odel (9) gves Yˆ = ( π 00 + π 0 BP + π0 HP + 0 ) BP + { ˆ 0 ( HP π + π ) + π ( ) + }, =,, () Y TP TP Noe ha n hs odel, he neracons beween soe of he coeffcen drvers and he explanaory varable of odel (9) appear as addonal explanaory varables and he dsurbances, ˆ 0 + Y, are heeroscedasc Ths eans ha he forulaon n () s uch rcher han odel (9) wh fxed nercep and fxed slope whch s sspecfed 7 In (), crossseconal varaon n he par, ( Y ˆ, ˆ Y ), s odeled by geng he naonal nercep, π 00, and slope, π 0,--whch are coon o all areas--odfed by ndvdual area coponens In oher words, he pars, ( Y, ), are odeled sulaneously so ha odel () for each area consss of he coon naonal nercep, π 00, and slope, π 0, and her devaons fro he naonal level Model () recognzes ajor aspecs of he saplng desgns yeldng Yˆ and Yˆ beuse he coeffcen drvers are seleced o explan hgh proporons of spaal and eporal varaons n ˆ ˆ ˆ Yˆ and he sus of saplng and non-saplng errors of Y and Y On a defnon suggesed by a senence n Hwang and Depser (999, p 98) aepng o recognze ajor aspecs of saple desgn, eporal varaon, and spaal varaon aouns o geng he scence rgh Model () ges he scence rgh n hs sense The sspecfons n () wh hese feaures n be less serous han hose n (3) Under assupons (0) and (), odel () ples ha for =,, and fxed, gven = and he coeffcen Yˆ drvers, he are ndependenly and condonally dsrbued wh odel ean equal o μ -μ + y and odel varance equal o ˆ ˆ V{( 0 + Y ) Y = y, BP, HP, TP} The basc fng algorh for () s an Ieravely Re-Sled Generalzed Leas Squares (IRSGLS) ehod of Chang, e al (000) where n every eraon, he weghng of he sus of squares and cross producs of observaons on he dependen and explanaory varables n () by he eleens of he nverse Yˆ y Y 6 Esaon of hs odel requres repled daa whch we do no have 7 The reaen of he coeffcens of odel (9) as consans s a sspecfon beuse gnores () varaons n he saplng and nonsaplng errors of Y ˆ, () varaons n he saplng and non-saplng errors of ˆ Y, and () he correlaons beween α and he eleens of ( ˆ, e, u ) Y

10 of he covarance arx of he heeroscedasc dsurbances, ˆ, wll be perfored These odel-based weghs as well as he desgn-based weghs ebedded n Y and affec he esaes of he coeffcens and he error ers of () Ths s how he saplng echanss generang Y and Y are odeled n () odel-based approach ha gnores he saplng echans s no vald unless he saplng dsrbuon does no depend on he survey oucoes (see Lle, 004, p 548) + y 0 Yˆ ˆ ˆ 5 Esaon of Model () The nuber of he unknown paraeers of odel () s 0: sx π s, hree dsnc eleens of Δ, and one σ Gven he cross-seconal daa, y, y, BP, HP, TP, for =,, and fxed, an IRSGLS ehod n be used o oban good approxaons o he nu varance lnear unbased esaors (MVLUEs) of he coeffcens and he BLUPs of he errors of odel () These approxaons are denoed by ( ˆ π kh, k = 0,, h = 0,,, ˆ k, k = 0, ) For k = 0,, ˆ k s an EBLUP of k Usng hese approxaons n place of her populaon counerpars used n (0) and () gves he esaors of and α, denoed by ˆ and ˆ α, respecvely The correspondng esaor of σ Δ s denoed by ˆ ˆ α0 α0 σ Δ pproprae forulas for copung he sandard errors of hese esaes have been worked ou n he ppendx Karya and Kuraa (004, pp 4 and 73) prove ha he IRSGLS esaors of he coeffcens and he EBLUPs of he errors of odel () possess fne second-order oens under very general condons To hese condons he condon ha 6 6 should be added Wh hs addonal condon, wo or ore degrees of freedo rean unulzed afer he esaon of he coeffcens of (), σ, and he eleens of Δ Suffcen condons for he conssency and asypoc noraly of he IRSGLS esaors of he coeffcens of odel () are gven n Cavanagh and Rohenberg (995) We need assupons () and () beuse a necessary condon for he conssency of he IRSGLS esaors of he coeffcens of () s ha he odel condonal expecaons of 0 and, gven ˆ Y = y and he coeffcen drvers are zero 53 Bas- and Error-Correced Verson of he Esaor Snce α0 = ε represens he su of saplng and non-saplng errors n ˆ Y = Y + ε, sasfyng equaon (0), follows ha he forula ˆBEC ˆ ˆ ˆ ˆ ˆ ˆ ˆ Y = Y α0 = Y π00 π0bp π0hp 0 (3) BEC gves bas- and error-correced (BEC) esaor of Y value of hs esaeor s denoed by y The sandard error (SE) of Y ˆ BEC s he square roo of an approxaely unbased esaor of he odel MSE, ˆ BEC E ( Y Y ) Is dervaon s gven n he ppendx Cochran s (977, p 4) copuaons show ha he effec of he ˆ ˆ bas n Y due o u on he probably of he error, Y - Y, of ore han (or less han) 96 (or -96) es he sandard devaon of Yˆ s apprecable f he absolue value of he bas s greaer han one enh of he sandard devaon of Yˆ In hese ses, he bas correcon n (3) s desrable f he absolue value of he bas reanng n Y ˆ BEC afer he correcon s less han one enh of he sandard devaon of Y ˆ BEC Such desrable bas correcons ay frequenly occur wh Y ˆ BEC pplons Use he successve dfference replon ehod descrbed n US Deparen of Labor (006, Chaper 4) o oban a new esae of he whn-psu varance conrbuon o he oal desgn varance, ( ˆ Vp Y Y ), of Yˆ n () BEC pply an IRSGLS ehod o equaons (5) and (6) afer nserng hs new esae and y every place an esae of he whn-psu varance and a esae of Y are used n hese equaons, respecvely Ths applon n gve proved esaes of desgn effec and used n equaon (4) 54 Iproved ddonal Esaor b

11 Usng ˆ α and y n place of α and ˆ Y used n Y = ˆ α Y, respecvely, gves ˆ I BP ˆ ( ˆ ˆ ˆ ˆ HP Y = α y = π + π + π + ) y (4) 0 TP TP where Y ˆ I denoes an proved addonal (I) esaor of Y The SE of Y ˆ I s he square roo of an approxaely unbased esaor of he odel MSE, ˆ I E( Y Y) Is dervaon s gven n he ppendx 55 Coparson of BEC and I Esaors proof of he resul, Y ˆ BEC = Y ˆ I wh probably (wp), s gven n he ppendx The followng pror nforaon helps us assess he relave accuraces of Y ˆ, Yˆ, Y ˆ BEC, and Y ˆ I Pror nforaon () The overall saple sze s suffcen o produce naonal-level onhly eployen or uneployen esaors ha sasfy prespecfed precson requreens () However, relavely sall sae saple szes do no per he producon of relable onhly eployen and uneployen esaes for he saes () The and CES survey beng he household and esablshen surveys provde nforaon abou place-of-resdence and placeof-work eployen, respecvely (see US Deparen of Labor, 997, p 45) Consequenly, Yˆ ay conan a larger agnude of non-saplng error han Yˆ when s vewed as an esaor of place-of-resdence eployen, parcularly for eropolan areas lke Washngon, DC where a good any suburbanes work (see foonoe ) (v) Exrapolaed QCEW daa generaed fro a e seres odel ay conan large errors beuse he bes nonlnear odels ha accuraely represen asyerc cycles of sae eployen are unknown (v) Boh a he naonal and sae levels UI Clas are very naccurae esaes of uneployen Ths pror nforaon ples ha a he naonal level, esaes are ore accurae han he correspondng addonal esaes; ha ay or ay no be he se for sae level daa For hs reason, Yˆ beng a sae level esaor ay no be ore precse han Y or Y ˆ I and Y ˆ ay no be ore precse han ˆ Le y be an esae of Y based on eher ˆ BEC ˆ BEC Y or Y ˆ I Then Resrcon I y shall le whn an esaed confdence nerval of y Le be he nuber of saes or areas whch geographlly exhaus he enre US Then Resrcon II he su y shall equal he esae of he naonal eployen or uneployen, Y = = y The dea of Resrcon I s o l he devaon of fro Ths lng reduces he loss n effcency due o he sspecfons n odel () for soe saes whou losng he benefs of correcly specfed odel () for oher saes (see Jang and Lahr, 006, p 36) Pror nforaon () saed above jusfes Resrcon II The y s do no sasfy Resrcons I and II f he coeffcen drvers n (0) and () are napproprae Sway, Meha and Chang (006) found ha when Yˆ based on he ern Couny Survey was used n (9), he coeffcen drvers n (0) and () were adequae n he sense ha hey produced he values of Y ˆ BEC ha sasfed Resrcon I for all =,, I s a good pracce o experen wh all possble cobnaons of coeffcen drvers on whch daa are avalable fer everyhng ha n be done n hs way has been done, f soe of he resulng y s do no sasfy Resrcon I, hen hs resrcon ay be posed exernally Tha s, use y f y les whn an esaed confdence nerval of y and use an esaed lower or upper confdence l of whchever s closer o oherwse (see Rao, 003, p 8) Hopefully, ajor changes n hese resrced y s are no needed o sasfy Resrcon II 6 Te Seres of Cross-Seconal Esaes y y y Y

12 Esang odel () separaely for each onh denoed by =,, τ (= n( τ,, τ )) gves (,, =,, τ, where BEC BEC y, 0 a ), =, y and a, are he values of Y and α, respecvely, obaned fro cross-seconal 0 ˆ BEC ˆ 0 daa n () for each =,, τ To produce seasonally adjused esaes of he ˆ σ e, Y, he esaes, BEC y, =,, τ, wll be adjused exernally by X- RIM for each =,, Le be an esae of Such esaes are a ˆ 0 e, σ e, copued by he Bureau of Labor Sascs (BLS) n R (5) odel ay be fed o / σ, =,, τ, for each =,,, o esae he varances and auocovarances of e pled by he saplng desgn 7 Exaple ˆ ˆ CES In hs secon, Y denoes oal eployen, Y = Y, he CES survey esaor of Y, denoes January 006, ndexes he 50 Uned Saes and he Dsrc of Coluba, and = 5 Followng he IRSGLS ehod and usng he daa on he varables n () wh hese values of,, and, we oban he esaes of he coeffcens of () gven below: ( ˆ y = + BP HP + ) 0 (699) (04495) ( 087) BP HP ( ˆ CES ) y (5) TP TP (708) (05334) ( 0836) CES where y s he CES survey esae of Y, all he oher varables are as defned n (), and he fgures n parenheses below he coeffcen esaes are he -raos Equaon (5) shows ha he esae, , of he coon naonal nercep, π 00, s very close o zero and he 5 5 ) CES esae, 0403, of he coon naonal slope, π 0, dffers slghly fro ( y = 0664 Boh hese / y = = esaes are sgnfn The 5 esaes of he errors of (0) and () and he coeffcens of (9) le whn he ranges: ˆ 0 070, ˆ , ˆ α , and 03 ˆ α CES 38 The esaes of he varances of 0 and are: 985E+08 and , respecvely These esaes show ha he coeffcen drvers, and HP, ncluded n (0) explan only a very sall proporon of he varaon n α, =,, BP 5 We need o nclude addonal coeffcen drvers n (0) o reduce he varance of 0 o a sall nuber The pal equpen and he rao of pal o labor for each area n perod n serve as addonal coeffcen drvers Facors ha CES adjus he place-of-work nonfar eployen esaes, y, o a place-of-resdence bass, as n he, n also serve as addonal coeffcen drvers Unforunaely, daa on hese varables are no avalable The esae, y, for each of he 5 areas s gven n he colun, labeled, of Table The values of he BEC esaor n (3) for he 5 areas pled by he esaes n (5) are gven n he colun, labeled BEC, of Table They are he esaes of he rue value coponen, Y, of ˆ Y n () The esae of he error coponen, e + u, of Yˆ n () for each of he 5 areas s gven n he colun, labeled S&NSE, of Table Noe ha he esaes n he coluns, labeled BEC and S&NSE, are he resuls of sulaneously esang he coponens of Yˆ usng odel () The coeffcen drvers ncluded n equaons (0) and () are adequae for 36 areas n he sense ha he unresrced BEC esaes for hese areas sasfy Resrcon I We had o pose Resrcon I exernally on he BEC esaes for he reanng 5 areas The wo-sep esaes of he coponens, Y and e, of for each sae when u s gnored, are gven n he coluns, labeled, Sgnal and SurE, of Table, respecvely We have now hree dfferen esaes of Y They are denoed by, BEC, and Sgnal, respecvely The esaes are no sasfacory beuse her accuracy s nadequae, as we have already poned ou n Secon The Sgnal esaes are also no sasfacory beuse hey are pled by he ncorrec esaes of he paraeers of odels () and (3), as we have already shown n Secon 3 Suffcen condons for he MSE of he BEC esaor n (3) o be saller han he desgn varance (4) of he esaor, Yˆ 0

13 Y ˆ, n () are gven n he ppendx These condons n be sasfed n large saples even when an approxaely unbased esae of he MSE of (3) s larger han a saple esae of he desgn varance n (4), as we have explaned n Secon 4 If hese condons are no sasfed by odel () n sall saples, hen hey n be sasfed when we expand he se of coeffcen drvers n (0) The desgn varance of Yˆ gven n (4) nvolves unknown quanes and hence s unknown Is esae was obaned usng y and he BLS esaes of k and D n place of Y, k, and used n (4), respecvely I s gven for each of e, D e, he 5 areas n he colun, labeled varcps of Table n alernave esae of (4) was obaned usng he BEC esae, y, of Y and he IRSGLS esaes of δ 0 and ξ 0g n place of ˆ Y ( g), δ 0, and ξ 0g used n (5), respecvely I s gven for each of he 5 areas n he colun, labeled varbeccps of Table The varbeccps esaes n be ore accurae han he varcps esaes beuse he BEC esaor n (3) correcs for bas and error n ˆ Each value gven n he colun, labeled VarSgnal, of Table s an esae of he odel varance of he wo-sep esaor of Y for a sae I s an underesae of he rue odel varance beuse he paraeers of odel (3) are esaed, holdng he paraeers of odel () fxed a her ncorrecly esaed values 8 Conclusons Sulaneous esaes of he rue value and saplng and non-saplng error coponens of sall area saple esaors presened n hs paper are based on weaker assupons han her wo-sep esaes Msspecfons n lnkng odels n resul n sesaed coponens of saple esaors and desgn varances Specfon errors n he lnkng odels used n sulaneous esaon n be neglgble copared o hose n he lnkng odels used n wo-sep esaon ppendx Dervaon of he MSE of he BEC Esaor Le x be he -vecor, ˆ BP HP (, Y ), z be he 5-vecor, (, BP, HP,, ), Π be he ( 5) arx havng TP TP ( π00, π0, π 0,0,0) and ( π0,0,0, π, π ) as s frs and second rows, respecvely, and be he -vecor, ( 0, ), where he ranspose of a arx s denoed by a pre Usng hese defnons he equaons n () are wren ˆ Long Y = X z π + Dx () where Y ˆ s he -vecor, ( Yˆ,, Y ˆ ), X z s he ( 0) arx havng a Kronecker produc beween z and x, 8 Long denoed by ( z x ), as s h row, π s he 0-vecor gven by a colun sack of Π, D x s he ( ) arx, Long dag[ x,, x ], s he -vecor, (,, ), and here are zero resrcons on he eleens of π These Long resrcons n be saed as R π = 0, where R s he 4 0 arx of full row rank havng ones as s (, 4)-h, (, 6)-h, (3, 7)-h, and (4, 9)-h eleens and zeros elsewhere, and 0 s he 4-vecor of zeros Now a (6 0) arx C of full row rank n be found such ha R C = 0 Under assupons (0) and (), E( Dx Xz) = 0 and V ( Dx X z) = Dx( I σ Δ ) D x = Σ, where I denoes an deny arx and a subscrp s ncluded o nde s order, and s he 4-vecor havng σ and he dsnc eleens of Δ as s eleens Sway and Tnsley (980) explan how we go fro equaon () o equaon () Long Idenfon n he sense of Lehann and Casella (998, p 4) The coeffcen vecor, π, s denfable f X z has full colun rank The error vecor,, s undenfable beuse D x does no have full colun rank Ths resul ples ha s no conssenly esable (see Lehann and Casella, 998, p 57) The coeffcen drvers are used n (0) and () o Y 8 The defnon of a Kronecker produc we use s gven by Greene (003, p 84), aong ohers

14 reduce he undenfable porons of he coeffcens of (9) However, Dx s denfable and s predcor n be used o oban a conssen esaor of For known, applyng he dervaon n Greene (003, p 00) wh approprae odfons o odel () gves he Long Long MVLUE of π ha sasfes he resrcon R π = 0 Ths esaor s Long ˆ ˆ πr ( ) = C ( CΨ C ) CX z ΣY () where he subscrp of ˆ Long π R s shorhand for resrced and Ψ = ( X zσ X z ) I follows fro CR Rao (973, p77, Long Proble 33) ha C ( CΨ C ) C = Ψ - Ψ R ( RΨR ) RΨ Pos-ulplyng boh sdes of hs equaon by Ψ π Long Long Long gves C ( CΨ C ) CΨ π = π, snce Rπ = 0 Ths resul s needed o prove ha esaor () s unbased wh he odel covarance arx Long V ( ˆ π ( ) X ) = C ( CΨ C ) C (3) R z For known, he BLUP of s ˆ R ˆ ( ) = ( I ) D M Y (4) Δ Σ σ x where M = I - X zc ( CΨ C ) CX z Σ The arx, M, s depoen (hough no syerc) wh he propery ha M XzC = 0 I n be shown ha ˆ Long E ( ( ) X ) = 0, ˆ Cov ˆ [( πr ( ), R ( )) X z] = 0, and ˆ V( R ( ) Xz) = ( I ) Dx MDx( I ) R z σ Δ Σ σ Δ beuse M Σ M = M Σ Pre-ulplyng boh sdes of equaon (4) by D gves ˆ Long D ( ) = Y - X ˆ π ( ) whch proves ha Y ˆ BEC n x x (3) s equal o Y ˆ I n (4) wh probably for all =,, and fxed For known se, he BEC esaor of Y n (3) becoes ˆ BEC Y ( ) = ˆ Y ˆ α0 ( ) = ˆ Long ( ) ˆ ( ) ˆ Y z l π j D ( ), where s he -vecor havng as s h eleen and zeros elsewhere, l R ˆ R l R s he -vecor, (, 0), and D s he arx, ( I l ) l z R j The MSE of ˆ BEC Y ( ) s ˆ BEC E[{ Y ( ) Y} Xz] = ˆ [{ ˆ ˆ E Y α0 ( ) Y + α0 } Xz] = ( ) (5) where jd [( I σ Δ ) ( I σ Δ ) D Σ D ( I σ Δ )] D j + l x x l [ jd ( ) ( l I σδ D xσ Xz z l )] C ( CΨ C ) CX zσ Dx( I σδ ) Dl j (6) and g ( ) ( z l ) C ( CΨ C ) C( z l ) (7) Long s n Rao (003, p 99, (6)), he second er n (5) arses as a drec consequence of usng ˆ π ( ) nsead of Long π n (4) In he ses where he coeffcen drvers n (0) and () reduce he agnudes of he eleens of σ Δ o sall values, he frs er of g ( ) n be uch saller han Rao s (003, pp 99 and 7) g ( ) δ whch, n urn, s saller R han he desgn varance of when u s absen lso, g ( ) s saller han s frs er f s second er s negave e Now we n elaborae on our dscusson n Secon 4 Rao (003, p 7) proved ha he frs er n (5) s saller han he desgn varance of f () he non-saplng error,, s zero wh probably for all and, () he saplng errors, e u e, =,,, are ndependenly dsrbued wh known desgn varances, () for =,, and fxed, he Y follow a GLM odel of Rao s (003, p 6) ype wh no consrans on s coeffcens, (v) he errors of he GLM odel (or he s n Rao s noaon) are denlly and ndependenly dsrbued wh known odel varance, (v) he GLM odel error s v

15 ndependen of e ˆ BEC for all and, and (v) he esaor, Y, s replaced by he BLUP of gven by he GLM odel Even when hese condons hold, he su of he wo ers n (5) ay no be saller han he desgn varance of e unless s suffcenly large and he regulary condons gven n Rao (003, p 7) are sasfed In any se, gnorng he non-saplng error, u, n resul n an nconssen and neffcen predcor of Y We now assue ha he error er of odel () s norally dsrbued ( ˆ E M Y ) = Yˆ ( I σ Δ ) D M ( M Σ M ) M (8) x where (MΣ M ) s a generalzed nverse of M Σ M and hs generalzed nverse s defned as n CR Rao (973, p 4) Sway and Meha (975, p 596) proved ha he rgh-hand sde of equaon (8) s equal o he BLUP n (4) Thus, when s noral, s BLUP s equal o s bes unbased predcor (BUP), snce ˆ E( M Y ) s he BUP of Le be a ( 6) arx of full colun rank such ha XzC = 0 Then ˆ E( Y ) = ( I σ Δ ) D x( Σ ) Yˆ (9) I follows fro CR Rao (973, p 77, Proble 33) ha ( Σ ) + Σ X zc ( CX z Σ X zc ) CX z Σ = Σ Inserng hs deny no (9) shows ha (9) s equal o he BLUP of Ths dervaon exends Jang s (997) proof o he ses where he coeffcens of odel () are subjec o lnear resrcons We now urn o he se where s unknown We use of σ Δ s a funcon of Yˆ o esae Yˆ Le ˆ be he 4-vecor havng Y σ Δ so ha our esaor, denoed by σ Δ, ˆ ˆ σ ˆ and he dsnc eleens of Δ ˆ as s eleens The BEC esaor, Y ˆ BEC, of Y n (3) n be wren as ˆ BEC Y ( ˆ ) = Y - ˆ Long α ˆ 0 ( ) = Y - ( z ˆ ˆ l ) π R ( ) - ˆ ( ˆ jd l ) R Suffcen condons for ˆ BEC E ˆ [ Y ( ) Y] = 0 are gven n Karya and Kuraa (004, pp 4 and 73) The MSE of ˆ BEC Y ( ˆ ) s ˆ BEC E ([ ( ˆ Y ) Y] Xz) = ([ ˆ BEC E Y ( ) Y] Xz) + ˆBEC ([ ( ˆ) ˆBEC E Y Y ( )] Xz) + ˆBEC ˆBEC ˆBEC E ˆ ([ Y ( ) Y][ Y ( ) Y ( )] Xz) (0) In (5), we have already evaluaed he frs er on he rgh-hand sde of hs equaon To show ha he hrd er on he rgh-hand sde of hs equaon vanshes, we frs noe ha ˆ BEC Y ( ˆ ) - ˆ BEC Y ( ) = - ˆ α ˆ 0 ( ) + ˆ α 0 ( ) = Long - ( z l ˆ ˆ ) π ( ) ˆ Long - jd ( ˆ ) + ( z l ˆ ) π ( ) ˆ + jd ( ) = ( z l ){ C ( CΨ C ) CX Σ - R l R ˆ R l R Long ˆ Ψ π R C ( CΨ ˆ C ) CX z Σ ˆ }[Y - X zc ( CΨ C ) C ( ) ˆ Long ] + jd {( ) [ ˆ l I ( )] σ Δ D xσ Y XzπR - ˆ ˆ Long ( I ˆ ) ˆ [ ˆ ( ˆ σ Δ D xσ Y XzπR )] } s a funcon of Yˆ beuse ˆ, ˆ Y - ( ) Long X ˆ zc CΨ C CΨ π R ( ), ˆ Long Y - X ˆ zπ R ( ), and ˆ Long Y - X ˆ ( ˆ zπ R ) are all funcons of Yˆ Furherore, he equaon, ˆ BEC Y ( ) Y = Long Long ( z l )[ π ( ) π ] [ ˆ jd ( ) ], s such ha he frs er on s rgh-hand sde s ndependen of ˆ R l R Yˆ beuse of he condon ha X z C = 0, and he las er on s rgh-hand sde n be shown o be equal o [ ( ˆ jd l E ) ] Y usng he resul n (9) Hence, he hrd er on he rgh-hand sde of equaon (0) s equal o ˆBEC ˆBEC Long Long ˆ E ˆ ˆ ([ Y ( ) Y ( )] E[ ( z l ){ πr ( ) π } Y, Xz]) + ˆBEC ˆBEC ˆ ˆ E ˆ ([ Y ( ) Y ( )][ j Dl ( E whch vanshes Ths proof exends he { E( Y ) } Y, Xz)]) proofs of Sway and Meha (969), Jang (00), and Rao (003, p 4) o he ses where he coeffcens of odel () are subjec o lnear resrcons ˆ ˆ z

16 Beuse of he second er on he rgh-hand sde of equaon (0), he MSE of ˆ BEC Y ( ˆ ) s always larger han ha of ˆ BEC Y ( ) n he noral se The ehod of approxang he MSE of Y ˆ BEC ( ˆ ) by he MSE of ˆ BEC Y ( ) could, herefore, lead o serous underesaon Unforunaely, he exac evaluaon of he second er on he rgh-hand sde of equaon (0) s generally no possble excep n soe specal ses, as Rao (003, p 03) has poned ou I s herefore necessary o fnd an approxaon o hs er We begn he dervaon of such an approxaon wh he assupons (see, eg, Lehann and Casella, 998, p 430) ha per an expanson of ˆ α ˆ 0 ( ) abou ˆ α0 ( ) wh bounded coeffcens Usng a Taylor approxaon, we oban ˆ α ˆ ˆ ˆ 0 ( ) α0 ( ) d( ) ( ) () where d( ) = ˆ α0 ( )/ and s assued ha he ers nvolvng hgher powers of ˆ are of lower order relave Long Long o d( ) ( ˆ ) Le b = jd l ( I ) σ Δ D xσ Then ˆ α0 ( ) = [( z ˆ l ) b X z ][ πr ( ) π ] + Long ( z l ) π + ˆ Long b ( Y X π ) Under noraly, z ˆ Long zπ d( ) ( b / )( Y X ) = d ( ), () Long Long snce he ers nvolvng he dervaves of ˆ π R ( ) - π wh respec o are of lower order Therefore, [ ( ) ( ˆ )] E d [ ( ) ( ˆ )] E d r[ E ( ( ) ( ) ) ( ˆ )] d d V = r[( b ˆ / ) Σ ( b / ) V( ) ] = g ( ) (3) 3 where V ( ˆ ) s he asypoc covarance arx of ˆ, and he negleced ers are of lower order I now follows fro ()-(3) ha ˆBEC ˆBEC E ([ ( ˆ) ( )] ) ([ ˆ ˆ ˆ Y Y Xz = E α0 ( ) α0 ( )] Xz ) g3( ) (4) Inserng (5) and (4) no (0) gves a second-order approxaon o he MSE of ˆ BEC Y ( ˆ ) as ˆ BEC E ˆ ([ Y ( ) Y] Xz) g( ) + g( ) + g3( ) (5) where he ers, g ( ) and g 3 ( ), arse as a drec consequence of usng he esaors raher han he rue values of Long π and, respecvely, n ˆ BEC Y ( ˆ ) and are of lower order han g ( ) The esaor ˆ BEC Y ( ˆ ) nno be rejeced Yˆ n favor of even when he desgn varance of s saller han he MSE n (5) The reason s ha hs desgn Yˆ varance undersaes he MSE of by gnorng u e B Esaon of he MSE of he BEC Esaor I follows fro Rao (003, p 04) ha E ˆ g ( ) g ( ) and E ˆ g3 ( ) g ( ) 3 o he desred order of approxaon, bu g ˆ ( ) s usually a based esaor of g ( ) wh a bas ha s generally of he sae order as g ( ) and g ( ) To 3 evaluae hs bas, we ake all he assupons ha per a Taylor expanson of g ( ˆ ) abou g ( ) wh bounded coeffcens (see Lehann and Casella (998, p 430)) Under hese assupons, g ˆ ( ) = g ( ) + ˆ ( ) g ( ) + ( ˆ ) g ( )( ˆ ) (6) where g ( ) s he vecor of frs-order dervaves of g ( ) wh respec o and g ( ) s he arx of second-order dervaves of g ( ) wh respec o The esaor ˆ s generally a based esaor of and hence he odel expecaon of he second er on he rgh-hand sde of equaon (6) s generally nonzero Consequenly, E ˆ ˆ g( ) g( ) + E( ) g( ) + r[ g( ) V( ˆ )] (7) If Σ has a lnear srucure, hen (7) reduces o E ˆ ˆ g( ) g( ) + E( ) g( ) g3( ) (8) Ths resul shows ha an esaor of he MSE of ˆ BEC Y ( ˆ ) o he desred order of approxaon s gven by