AN ABSTRACT OF THE THESIS OF

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2 AN ABSTRAT OF TH THSIS OF Dl K. Yg for th dgr of Dotor of Phlooph Sttt prtd o Dmr 9. Ttl: Propt Sor Adjutmt Ug ovrt Orvtol Stud Atrt pprovd: Vrg M. Lr Al I. Gtlm I th th dvlop thortl frmork for th dtfto of tuto hr th qul frqu F or qul vr V ulfto m produ lor d/or vr of th tmtor. W odut multo tud to m th F d V pproh udr dffrt tp of modl mpfto. W ppl to ghtg hm our multo: qul ght W d vr vr IV ght. Our multo rult dt tht udr th qudrt trm mpfto th F-IV tmtor provd th lot d root m qur rror omprd to th ordr lt qur tmtor d othr propt or tmtor. Our thorm dvlopmt dmotrt tht f hghr vrto our th lrgr for th ul trtmt fft tmt th th F-IV tmtor h mllr ovrll th th F-W tmtor. W ho tht th F-IV tmtor l h mllr vr th th F-W tmtor. W lo propo ovl mthod of ulfto tht fou o rtg homogou propt or ul to produ tmtor th rdud d om rumt. W fl our rrh otrut to th fld of propt or djutmt provdg thorm to ompr th ovrll d vr t dffrt propt or tmtor.

3 oprght Dl K. Yg Dmr 9 All Rght Rrvd

4 Propt Sor Adjutmt Ug ovrt Orvtol Stud Dl K. Yg A THSIS umttd to Orgo Stt Uvrt prtl fulfllmt of th rqurmt for th dgr of Dotor of Phlooph Prtd Dmr 9 ommmt Ju

5 Dotor of Phlooph th of Dl K. Yg prtd o Dmr 9. APRROVD: o-mjor Profor rprtg Sttt o-mjor Profor rprtg Sttt hr of th Dprtmt of Sttt D of th Grdut Shool I udrtd tht m th ll om prt of th prmt ollto of Orgo Stt Uvrt lrr. M gtur lo uthor rl of m th to rdr upo rqut. Dl K. Yg Author

6 AKNOWLDGMNTS I ould lk to pr m dpt pprto d grttud to m dvor Dr. Vrg M. Lr d Dr. Al I. Gtlm for thr pt prt upport d gud durg th our of th drtto rrh. I pll thk Dr. Lr for hr trl mtorhp d dv throughout m dotorl progrm d for th opportut to ork o rl urv d mplg prolm th Surv Rrh tr. Wthout hr urrvd hlp d ourgmt th drtto ould ot hv fhd. I ot thk hr ough for gvg m th opportut to ttd d ft from th Jot Stttl Mtg Vouvr B. I lo rl thk Dr. Gtlm for hr vlul uggto ott dv d hlp durg th prprto of th drtto. I m dtd to oth of m dvr for thr mtorhp trordr hlpful d vllt durg m rrh d prprto of th drtto. I ould lo lk to hrtl pr m thk to Dr. Dvd S. Brk ho tught mot of th Sttt PhD l for h upport urrvd dv d hlp o m rrh d drtto prprto d for rvg o m ommtt. I offr pl thk to Dr. Dl W. Shfr for tkg th tm to rd th drtto d for rvg o m ommtt. It m plur to thk Dr. Kthr Gutr for rvg th drtto d for rvg m grdut oul rprttv. I ould lk to thk th tr fult tff d tudt of th Sttt Dprtmt t Orgo Stt Uvrt for thr otruto to m duto d rrh. I holhrtdl thk m fml for thr lov fth d upport pll m mothr. I m grtful for th rf h h md durg m tm grdut tudt. Hr uquvol ourgmt d dl upport h dpl to th uful omplto of m dotorl tud.

7 TABL OF ONTNTS Pg hptr. INTRODUTION.... Norpo d t mhm.... Th rpo propt or djutmt Th propt or mthod Summr of th otruto d orgto of th th...9 hptr. LITRATUR RVIW.... Th gl ovrt qutl ulfto djutmt.... Th propt or qul frqu ulfto djutmt Applto of propt or djutmt Propt or mthg pproh qul frqu ulfto d qul ght pproh Propt or qul frqu ulfto djutmt for urv orpo Propt or djutmt udr modl mpfto; qul vr ulfto otruto of th th qul vr ulfto mthod udr modl mpfto Thortl vtgto o propt or ulfto djutmt tmtor N pproh to propt or ulfto Orgto of th th...33

8 TABL OF ONTNTS otud Pg hptr 3. SIMULATION STUDIS OF PROPNSITY SORS MTHOD Smultg dt Sro volvg to dpdt ovrt Sro volvg gl ovrt d t qurd trm tmto of rgro d propt or modl Propt or ulfto Trtmt fft tmt OLS tmtor d propt or tmtor Murg th prform of trtmt fft tmt Smulto rult orrtl pfd modl volvg ovrt Mpfd modl th ovrt omo ludg Modl th qudrt trm mpfto omttg Summr of multo...5 hptr 4. THORTIAL INVSTIGATION OF PROPNSITY SOR STIMATORS pro of B V d Lmm udr lr rgro modl Thorm omprg dffrt ghtg hm Dord d oord omprg t to propt or tmtor omprg vr t IV d W tmtor omprg vr t F-IV d V-IV tmtor...68

9 TABL OF ONTNTS otud Pg hptr 5. PROPNSITY SORS BALANING SUBLASSIFIATION PSB ulfto mthod A multo tud udr orrtl pfd rgro d propt or modl Smultg dt volvg to dpdt ovrt tmto of rgro d propt or modl Murg th prform of trtmt fft tmtor A multo tud th ml th lot ul Smultg dt d tmtg th trtmt fft Smulto rult Summr of PSB multo tud hptr 6. ONLUSIONS Duo Futur ork...9 BIBLIOGRAPHY...9

10 LIST OF FIGURS Fgur Pg Fgur. qul frqu ulfto o th propt or... Fgur. Propt or dtruto of th trtmt pod to th trtmt or rk ftor ujt d otrol upod ujt...3

11 LIST OF TABLS Tl Pg Tl 3. Fttd modl d trtmt fft tmtor...37 Tl 3. ompro t multg d fttg th outom d propt or modl...4 Tl PRMB for th OLS d thr propt or tmtor ug orrtl pfd propt or modl d orrtl pfd rgro modl for ovrt...46 Tl PRMB for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...48 Tl PRMB for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...5 Tl 5. Modl pfto fttg th outom or th propt or modl...77 Tl 5. Th of th m vr d RMS for th trtmt fft tmtor ug orrtl pfd propt or modl d orrtl pfd rgro modl for ovrt...79 Tl 5. 3 Th of th m vr d RMS for th trtmt fft tmtor udr th odto tht proporto of otrol ujt hv propt or lor th th mmum propt or mog trtd ujt....83

12 LIST OF APPNDIS Pg APPNDIX... A. Glor... A3. Arom Notto d multo produr dgrm... A3.3. mpl of V ol produ l th fv ul...9 A3.5. orrtl pfd modl volvg ovrt... A3.5. Mpfd modl th ovrt omo ludg...6 A3.5.3 Modl th qudrt trm mpfto omttg... A3.5.4 Ug tru propt or...6 A To dpdt ovrt...6 A A gl ovrt d t qudrt trm...34 A4. Notto thor dvlopmt dgrm d Lmm...4 A4.. Prprg Lmm to dvlop B d V udr lr rgro modl...4 A4.. ptto of th th ul trtmt fft tmtor...44 A4..3 Vr of th th ul trtmt fft tmtor...46 A4..4 ovr of to th ul trtmt fft tmtor...5 A4.4 Thor dvlopmt to to multpl ovrt...5 A4.4. ptto of th th ul trtmt fft tmtor...5 A4.4. Vr of th th ul trtmt fft tmtor...54 A4.4.3 ovr of to th ul trtmt fft tmtor...55 A4.4.4 Thor dvlopmt to udr th m ulfto...56

13 LIST OF APPNDIX FIGURS Fgur Pg Fgur A3. Dgrm udr ro volvg to dpdt ovrt...5 Fgur A3. Dgrm of V ulfto udr Sro volvg...6 Fgur A3. 3 Adjutg oudr th ordr of t 5 th d d 4 th ul...7 Fgur A3. 4 V ulfto produr do ot produ rult for fv ul...9 Fgur A4. Dgrm of thor dvlopmt...4 Fgur A5. PSB Sulfto produr...6 Fgur A5. PSB Sulfto produr rtrtg th of th lot ul...6 Fgur A5. 3 Smulto tud dgrm for to dpdt ovrt...63

14 LIST OF APPNDIX TABLS Tl Pg Tl A3. Arom... Tl A3. Notto...3 Tl A3. 3 Prtg rltv mur formul of δˆ...8 Tl A3. 4 Vlu of prmtr...8 Tl A3.5.. PRBM for th OLS d thr propt or tmtor ug orrtl pfd propt or modl d orrtl pfd rgro modl... Tl A3.5.. PRSD for th OLS d thr propt or tmtor ug orrtl pfd propt or modl d orrtl pfd rgro modl... Tl A PRRMS prtg rltv RMS for th OLS d thr propt or tmtor ug orrtl pfd propt or modl d orrtl pfd rgro modl...4 Tl A3.5.. PRBM for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...6 Tl A3.5.. PRSD for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...8 Tl A PRRMS for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...

15 LIST OF APPNDIX TABLS otud Tl Pg Tl A PRBM for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl... Tl A PRSD for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...4 Tl A PRRMS for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl...5 Tl A PRBM for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl for...7 Tl A PRMB for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl...9 Tl A PRSD for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl...3 Tl A PRRMS for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl...3 Tl A PRBM for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl for...34 Tl A PRMB for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl...35 Tl A PRSD for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl...36 Tl A PRRMS for th OLS d thr propt or tmtor ug tru propt or d orrtl pfd rgro modl...38 Tl A4. Notto...4

16 Propt Sor Adjutmt for ovrt Orvtol Stud hptr. INTRODUTION Th hptr provd grl troduto of th th. I to. dr mg dt d th orpo prolm. I to. trodu th rpo propt or djutmt mthod to djut for orpo. I to.3 provd rf rv of propt or mthod d to.4 ummr th otruto d orgto of th th.. Norpo d t mhm Mg dt ommo prolm th dt ollto pro h th dgtd formto of trt ot olltd for porto of th mplg ut Lttl d Ru. I urv mplg th prolm lld orpo. For mpl ml urv vld mlg ddr or mpld ujt hoog ot to r th qutor ll oth ld to orpo. I oom ttu urv ujt m tv d rlutt to r quto rltg to om lvl hh ld to orpo. I mdl thrp tud ptt m drop out durg th tud prod du to dvr d fft or lk of trtmt ft ldg to mg dt. Igorg th orvto ld to d rult. Th th vtgt to djut for orpo.

17 I urv mplg thr r to tp of orpo: ut orpo d tm orpo. Ut orpo mpl tht th tr ut or ujt mg. I tm orpo th ujt h ot rd ll th quto kd. Ol porto of ompltd quto r otd for tht pf ujt. Norpo or mg dt lfd to thr mhm. Th r MAR mg omplt t rdom MAR mg t rdom or gorl d ogorl formtv mg Lohr 999 Lttl d Ru. I ordr to dr th mhm lt Y th omplt dt mtr for th outom vrl of th mpl d t prttod Y Y o Y m hr Y o rprt th orvd outom d Y m rprt th mg outom; lt X th dt mtr of ovrt; lt R vtor of rpo/orpo dtor; lt θ d φ to t of dtguh prmtr orrpodg to Y o R rptvl. Th thr mhm drd follo. Udr MAR th orpo r dpdt of th outom vrl d th ovrt. Th mpl tht rpodt r rprttv of th ltd mpl. Th odtol prolt dt futo pdf. of R mplfd f φ R Y X f φ R hh mpl th jot pdf. f θ φ Y o R f φ Rf θ Y o. For mpl f vr thr odto mk t mpol for urv trvr to mt th mpld ujt or ltro glth u om rord to lot mdl tud th kd of mg m hv o rlto to th outom or ovrt. Thrfor th prolt of rpo dpdt of th orvd outom d th rrhr

18 3 l th ompltd dt th o djutmt dd. Th mpl rult r rprttv of th populto. Udr MAR th orpo dpd o th ovrt ut ot th mg outom vrl Lohr 999. Th odtol pdf. of R th f φ R Y X f φ R X. For mpl th ougr mplg ujt td to rfu prtpto mor oft th oldr ujt. I om ujt th hghr lvl of duto r mor llg to prtpt th ujt th lor lvl of duto. Th tp of mg mhm pld th orrpodg ovrt.g. g duto. MAR lo rfrrd to gorl. If th mg h o rlto to th mg outom th th ovrt ud to mk orpo djutmt. O th djutmt mplmtd th mg mhm rml MAR. Th ogorl mg mhm th mot dffult tuto u th orpo mhm dpd o th mg outom d ot full pld th ovrt. Th odtol pdf. of R f φ R Y o Y m do ot hv mpl pro th mhm of orpo mor ompl. A mpl of th llutrtd tt t urv. Uppr om lvl houhold m l lkl to rpod to th urv d rvl thr tul om th mddl or lor om houhold v though th hv mlr dmogrph g tgor r d gdr. For ogorl mg trog modlg umpto r r to out for orpo. Th l d ol o th ompltd dt or d o th

19 4 orvd outom d ovrt ll grt th tmt Lttl d Ru.. Th rpo propt or djutmt Rtl om holr uggt tht MAR th mhm of orpo oft foud mot tuto h mg dt our Bjørtd 7 d hmr 7. W ll m mthod to djut for th ut orpo udr th MAR gorl mhm. Lohr 999 dr thr trdtol mthod to dl th MAR urv mplg: ghtg l djutmt WA pot-trtfto d 3 mputto. For th ghtg l djutmt dmogrph formto vll o th tr mpl. Wghtg l r rtd oft ug th dmogrph formto. Th ghtg l djutmt um tht th ghtg l th rpo otd from th rpodg ujt d orpodg ujt r mlr. I ddto thr dffrtl rpo rt ro dmogrph group. Th d of th mthod to dvlop djutmt ght for th rpodt ordr to rg th l of th dmogrph of th mpl mor l to th populto dmogrph. S th rpo from th orpodg dvdul r ot vll th WA mthod u dmogrph formto otd for th orgl mpl rpodt d orpodt to form o-ovrlppg ut or ghtg l. For mpl g ud to rt o-ovrlppg ghtg l. Rpo rt oft dffr ro g group th oldr dvdul g mor lkl to rpod th ougr dvdul. Lt π th lto prolt of ujt th

20 5 orgl mpl th th mplg ght of ujt / π. Lt ghtg l Ζ +.. potv tgr; lt um r dot th ummto of mplg ght for rpodt ghtg l d lt um dot th ummto of ght for th orgl mpl ghtg l. Th rpo prolt for h ghtg l tmtd pˆ um r / um r. H th WA mthod djut th ght of rpodt ghtg l multplg / pˆ Lohr 999. Th pot-trtfto djutmt PSA ud to rg th mpl mor l th th populto ht do th th WA. Hovr th o dmogrph formto vll o th orpodt. Populto formto otd from trl our ud to djut th ght. For mpl u dt oft ud tol populto urv to lrt th mpl rult to populto umr. Pot-trt r rtd d o dmogrph formto olltd o th qutor from th rpodt log th formto vll o th m dmogrph vrl from th u. For mpl g group ud to rt th pot-trt. Th umr of dvdul th mpl m out of l th proporto of th g group th populto. Th pot-trt r rtd to mk ghtg djutmt to rg th mpl mor l th th populto proporto. Dtl o th djutmt r provdd Lohr 999. Th mputto thqu ud to g or mput vlu to th vlu or tm tht r mg. For mpl f th om vrl h om mg vlu th

21 6 rrhr rpl th mg vlu uttutg th m of th orvd om m mputto. Aothr pproh uttut th mg vlu th prdtd om vlu from rgro modl of th orvd om v. orvd ovrt. For trtd rdr Lohr 999 d Hrg t l. provd mor dtld drpto out th mputto thqu. Dvd t l. 983 provd th fudmtl umpto for rpo propt hh tht th rpo or orpo mhm gorl. Lttl 986 dvlop th rpo propt or mthod to djut th tmtd m. H df th rpo propt or th odtol prolt of ujt rpodg to th urv gv th orvd ovrt. Th rpo propt or mthod vd ltrtv to th WA; t h modl-d lmt th rpo propt or r grll tmtd logt modl o ll orvd ovrt. Th th ghtg l.k. ul r formd o th tmtd rpo propt or. Th mjort of rrhr urrtl hoo to form qull-dvdd ghtg l or ul o th tmtd rpo propt or d to ppl qul ght W to djut th orpo h tmtg th outom..3 Th propt or mthod To udrtd propt or mthod ll provd grl rv. ohr 968 uggt tht for orvtol tud th ol o ovrt d r trtmt dtor.g. trtmt v. otrol fv-%-qutl qul

22 7 frqu F ulfto mthod fft produr to lmt 9% of th of th tmtd trtmt fft dud th ovrt. Hovr th produr m ot fl f thr r multpl ovrt olltd tud. Lt th totl umr of ovrt p hr p Ζ +. If thr dhotomou ulfto F th to ul for h ovrt ug ohr 968 pproh p ul r produd. A p r th dt m ot l to upport th totl umr of p ul. For multpl ovrt o lo odr ug rgro djutmt to tmt th trtmt fft ut Ru 979 quto tht th rgro tmt m ot pproprt f th lr modl orrt. Roum d Ru 983 dt tht th ptto of th qudrt trm of th odtol ptd qurd for th rgro tmt of th trtmt fft r f th ovrt vr or ovr mtr of th trtmt d otrol group r dffrt. Furthr Roum d Ru 983 df ujt propt or th odtol prolt for h ujt to rv th trtmt gmt gv th orvd ovrt. Th provd th thor of th propt or mthod to djut th of th tmtd trtmt fft du to th orvd multpl ovrt. I Roum d Ru 984 th provd othr thorm orrpodg to ohr 968 rult. Th thorm mpl tht ug fv-%-qutl ulfto group th qul ght F-W o th tmtd propt or rdu 9% of th of th tmtd trtmt fft ud ll orvd ovrt orvtol tud th r trtmt.g. trtmt v. otrol prtpt v.

23 8 o-prtpt. Roum 987 trodu propt or mthod for pottrtfto djutmt. If tud h ovrhlmgl lrgr umr of otrol ut th trtmt ut rrhr m t to lt porto of th otrol ut to ompr th th trtmt ut to tmt th trtmt fft. Roum d Ru 985 dvlop th propt or mthg pproh to lt otrol ut hr thr tmtd propt or r mlr to trtmt ut d o rt rtr. Ru d Thom 99 dr th propt or mthg pproh ug drmt l d logt rgro to tmt th propt or. Drk 993 trodu multo pproh to vlut th propt or tmtor d th ordr lt qur OLS tmtor omttg dpdt ovrt d mpfg qudrt trm. Dhj d Wh 999 propo to furthr plt F ul utl th dffr of th m of th propt or t th dhotomou trtmt group plt ul homogou or ld Im 4 from to-mpl t-tt. Hullk d Lou propo qul vr V ulfto o th propt or hh volv vr vr IV ghtg of th trtmt fft tmtor. ldo d Kopg 8 provd rt rv for th propt or mthodolog ludg: mthg rt ghor lpr d rdu ulfto.k.. trvl mthg lokg or trtfto d vr propt or ghtg.

24 9.4 Summr of th otruto d orgto of th th Th ojtv of th th ork to dvlop thortl frmork to dtf tuto hh F or V pproh mght ttr trm of rdug th d/or vr of th tmtor d to propo ovl pproh of formg th ul th homogou propt or. Th frt otruto of th th to vlut th qul vr ulfto mthod udr modl mpfto. Th od otruto thortl vtgto o propt or ulfto djutmt tmtor to dtf t ht odto th F-IV tmtor h mllr th th F-W tmtor d t ht odto th F-IV tmtor h o lrgr vr th th V- IV tmtor. Th thrd otruto th ovl propt or lg ulfto pproh. W propo pproh of ulfto th homogou or dtl propt or t trtmt ujt d otrol ujt t lt for mot of th ul. Th pproh td to produ lrg umr of ul hr th dt ould llo ordr to rdu mor hh lo rflt th uggto ohr 968 Im 4 d Mr d Lou 7. Th orgto of th th th follog: I hptr To ll provd ltrtur rv of th kgroud of propt or mthodolog rpo propt or djutmt o urv orpo dtl of th otruto d orgto of th th. I hptr Thr ll provd multo tud to

25 vlut th propt or djutmt tmtor th rpt to F d V ulfto pproh; ll u dffrt ghtg mthod uh W or IV ght of th trtmt fft. I hptr Four ll provd th thortl ork of th vr d root m qur rror RMS of th propt or ulfto djutmt tmtor th dffrt ghtg mthod. I hptr Fv propo our ovl propt or lg PSB ulfto pproh. W th mult tuto th proporto of otrol ujt tht hv lo tmtd propt or. W lo provd vluto for th PSB pproh d othr propt or tmtor trm of vrg vr d orrpodg RMS. I hptr S ll ummr th mjor fdg from our multo rult thor ork d our l propod PSB ulfto pproh d du futur rrh trt.

26 hptr. LITRATUR RVIW Thr r oft rpodt d orpodt urv. If rpodt provd dffrt rpo tht ptd from orpodt th lg ol th ompltd dt ll produ d rult. If th orpo mhm pld th orvd ovrt th l hould tk to out th orvd ovrt to djut for orpo. Th vd log to orvtol tud tht lud trtmt group d otrol group. Th ojtv of th th to k djutmt mthod ug ovrt formto vll orvtol tud or urv tht outr orpo. To du th djutmt for urv orpo ll frt m djutmt mthod th ott of orvtol tud th trtmt group d otrol group. W ll th td th mthod to urv orpo djutmt. Th hptr rv th ltrtur o gl ovrt ulfto mthod ll propt or ulfto mthod d thr pplto to urv orpo. I to. dr th qul frqu ulfto djutmt o gl ovrt. I to. du th djutmt ug propt or ulfto. Follog th dvlopmt to.3 rv pplto of th propt or mthod ludg urv orpo djutmt. Sto.4 rv th vluto of th propt or qul frqu ulfto udr modl mpfto d th qul vr ulfto pproh. Sto.5 outl th otruto of th th d to.6 dr th orgto of th th.

27 . Th gl ovrt qutl ulfto djutmt I orvtol tud th trtmt group d otrol group rrhr r oft trtd omprg th m of th outom t th trtmt group d th otrol group to tmt th trtmt fft. Thr m ml tho ovrt ro th trtmt group d th otrol group. A drt ompro of th m of th outom t th trtmt group d th otrol group m d tmtg th trtmt fft du to th ml. Th fft of th ovrt ot prtd from th fft of th trtmt o th outom. I urv th dtruto of om dmogrph.g. g or duto lvl m vr dffrt t th rpodt d orpodt group. If th ml ot tk to out th tmt for th rpodt r d th tmt lo rflt dmogrph dffr. ohr 968 tll md th prolm orvtol tud volvg trtmt group d otrol group d o ovrt rltd to th outom. H trtd dvlopg djutmt to out for th ml du to th ovrt. H oluto drd th gl ovrt qul frqu F ulfto djutmt. Th mthod qull dvd th ovrt log th prtl of t mprl dtruto d o prdtrmd umr of ul. Th ul r o-ovrlppg. Wth h ul th dtruto of th ovrt rltvl mlr t th trtmt group d th otrol group. Th dffr th m

28 3 of th outom t th trtmt group d th otrol group otd for h ul. Th ul-pf m r omd th ght to omput th ovrll tmt of th trtmt fft. A ommo ghtg hm ud th pplto th qul ght W pproh hh th ght th rprol of th umr of ul. Blo rf drpto of th gl ovrt qul frqu ulfto djutmt dvlopd ohr 968 prtd. Notoll lt j dot th outom for ujt j th otrol group hr j ; th umr of ujt from th otrol group >. Lt dot th outom for ujt th trtmt group hr ; th umr of ujt from th trtmt group >. Lt... d... dot th orrpodg ovrt th otrol group d th trtmt group rptvl. For o um tht oth th outom d ovrt r otuou. Suppo th ovrt rlt to th outom uh tht α + u + j α + u j + j hr u rgro futo.g. u β u j β j d β rgro offt. Hr α d α r th tru m of th trtmt group d th otrol group rptvl d d j r th orrpodg ro m dpdt rdom rror trm of th trtmt group d th otrol group. Th gol to tmt th trtmt fft α - α. Th udjutd m of th outom of th trtmt group d th otrol group otd

29 4 u u + + α α hr / / u u u u j j. Thu drt ompro of th m ll ld to th ud th ovrt u u. For th k of mplt lt th umr of ul fv d lt d th ul. Wth ul th m of th outom of th trtmt group d th otrol group otd u u + + α α hr / / j j u u u u ; d r orrpodg ul-pf umr of ujt from th trtmt group d th otrol group rptvl; d >. Th ul-pf ud th ovrt th - u u. Suppo th ght of ul dotg th gl ovrt qul frqu ulfto djutmt. Th th ovrll ghtd ud th ovrt ] - u u. Thrfor th proporto of rmovd th gl ovrt ulfto djutmt mthod from th ovrt ] ]/ - u u u u.. If ug qul ght th /5.

30 5 To llutrt th flu of th orvd ovrt lt u look t mpl ohr 968. Thr tud r odutd th U.K. d d th U.S. to vtgt th mpt of mokg o ptt ftlt rt. h tud volv thr group: o-mokr grtt mokr d gr/pp mokr. Th udjutd tmt from th thr tud ho tht th ftlt rt of o-mokr d grtt mokr r mlr hl gr/pp mokr hv hghr ftlt rt. Aro ll thr tud th vrg g of th gr/pp mokr hghr th th vrg g of th grtt mokr d o-mokr. Ovoul ptt g pll for th ldrl ptt lo rltd to hghr ftlt rt. A djutmt to out for th ovrt.. g ould dvtgou th mpl. Ug qul ght th djutd tmt from ll thr tud dt tht th ftlt rt of grtt mokr r ottl hghr th o-mokr. For gr/pp mokr ftlt rt ho o r h omprd to o-mokr. It pol tht u of hghr g of gr/pp mokr othr mdl ftor.g. ptt lvtd holtrol umr hro omplto t. m lo otrut to thr ftlt. Thu ddtol ovrt from th tud m dd to furthr djut th ftlt rt of gr/pp mokr. If ug qul ght quto. ov vd futo of th ovrt. Tht th prtg of rduto ug th ovrt qul frqu ulfto djutmt ll dpd o th ovrt ot o th outom. Thrfor quto. qutf ho fftv th gl ovrt qul frqu ulfto djutmt rmovg th trodud th ovrt. Th

31 6 vluto of th mthod hvd ohr multo hr rt mprl dtruto r umd for th ovrt: Norml t h-qur d Bt. Th rult provdd ohr 968 dt tht for orvtol tud th trtmt group otrol group d gl ovrt rltd to th outom vrl ug fv ul th qul ght th gl ovrt qul frqu ulfto djutmt rmov 9% of th trodud th ovrt. For jut o ovrt th qul frqu ulfto djutmt ork ll to lmt th dud th ovrt. Hovr h odrl umr of ovrt r orvd orvtol tud th mthod mprtl to mplmt. I om thr drtll rd umr of ul. For mpl Roum d Ru 983 du oror rtr p urgr tud of ptt ludg 74 ovrt. If fv F ul for h ovrt r ltd th totl umr of 5 74 ul r dd. If jut to F ul for h ovrt r ltd th totl umr of ul ll 74. Udr thr ro th mthod of ulfto djutmt o h ovrt ould vr dffult to mplmt du to th lmtd umr of ujt h ul.. vr mll mpl. Thrfor for orvtol tud th lrg umr of ovrt mthod orportg th orvd ovrt to uvrt mur drl. Th propt or mthod dvlopd to dl th th tuto.

32 7. Th propt or qul frqu ulfto djutmt I orvtol tud hr m ovrt r olltd Roum d Ru 983 propo th propt or mthod to tmt th trtmt fft. Th propt or dfd th odtol prolt of ujt g gd to th trtmt group gv th orvd ovrt. I rdomd prmt th propt or of h ujt ko to th rrhr t th dg tg. For mpl rdomd prmt volvg qul umr of dvdul oth th trtmt group d th otrol group th propt or / for h ujt. I orvtol tud hovr trtmt ot rdoml gd to ujt. Suqutl thr m ml th ovrt ro th trtmt group d th otrol group. omprg th outom m t th trtmt group d th otrol group m ofoudd th ovrt. Th propt or ummr th formto from ll orvd ovrt to uvrt murmt. B odtog o th propt or th ujt pld to ul th mlr ovrt formto ro th trtmt group d th otrol group. Thu th trtmt fft tmtd th ul thr rdug th flu of th orvd ovrt. For orvtol tud othr mthod to out for th ovrt h tmtg th trtmt fft multpl lr rgro orportg ovrt. Ru 979 dt hovr tht th tmt ug multpl rgro m pproprt f th lr modl ot orrt. Roum d Ru 983 dr

33 8 ovrt ml. For mpl f modl for th m outom ot lr futo of th ovrt d th ovrt vr ovr mtr t th trtmt group d th otrol group r dffrt th thr ovrt ml. Udr th odto th dt tht th ovrt ml r th ptto of th qudrt trm of th odtol ptd qurd of th rgro tmt of th trtmt fft. Wh om of th ovrt r mld th propt or mthod opto to odr rmovg th ofoudg fft from th orvd ovrt to dqutl tmt th trtmt fft. To furthr pl th propt or mthod provd vrl dfto. A mpl lld ld f th ujt of th trtmt group d th otrol group ulfd uh tht th ofoudg fft of ovrt o th trtmt fft r rmovd. For h ujt th trtmt gmt dtor dfd th follog: lt dot tht th ujt th trtmt group d dot tht th ujt th otrol group. Lt th vtor of orvd ovrt for o ujt. Th propt or of ujt dfd Pr{ }. A Roum d Ru 983 tt lg or futo of th ovrt uh tht th odtol dtruto of gv th m for trtmt d otrol ujt. Th flu of th ovrt o th trtmt fft h rmovd ulfg th ujt d o th lg or. Th

34 9 provd th plt to ot tmt of th trtmt fft thout th flu of th ovrt. Aothr dfto ud h dug th propt or th trog gorl trtmt gmt umpto. Wh th outom of trt odtoll dpdt of th trtmt gmt dtor gv th orvd ovrt th umpto mt. Th umpto th log to th mg t rdom MAR umpto hptr O th tuto h du urv d orpo. Roum d Ru 983 provd four mjor thorm to gv th thortl foudto for propt or djutmt to rdu du to orvd ovrt orvtol tud. Th frt thorm tt tht gv th propt or th orvd ovrt d th trtmt gmt r odtoll dpdt. Th ml orvd ovrt orportd to th propt or gl murmt. Th thorm lo mpl tht f ul of ujt or mthd trtmt-otrol pr homogou or dtl th propt or th th ujt of th trtmt group d th otrol group th ul or mthd pr ll hv th m dtruto of ovrt. Thrfor th ml dtrutol dffr of th ovrt t th trtmt group d th otrol group r lmtd th th ul or mthd pr. Th od thorm mpl tht th propt or lg or. I th thrd thorm f th trtmt gmt

35 trogl gorl gv th orvd ovrt th th trtmt gmt trogl gorl gv th propt or. Th fourth thorm tt tht f um trogl gorl trtmt gmt th th dffr t th odtol ptto of th outom gv th propt or for th trtmt group d th otrol group qul th ptto of th vrg trtmt fft gv th propt or. Tht udr th umpto of th trog gorl trtmt gmt odtog o th propt or dqut to provd rltvl ud.. ohr 9% rduto tmt of th trtmt fft pr-mthg or ulfg th propt or. Sulfto mplmtd formg ul th homogou propt or. Wth h ul th trtmt fft tmtd omprg th outom t th trtmt group d th otrol group. A grph dpl of qul frqu ulfto o th propt or prtd Fgur..

36 otrol F trtmt prtl: Sul : Fgur. qul frqu ulfto o th propt or Rll tht ohr 968 foud tht th gl ovrt ulfto djutmt ug fv ul d qul ght rdu 9% of th otrutd th ovrt. Roum d Ru 984 provd ddtol thorm to td ohr rult to th orvtol tud th multpl ovrt. Th ov thorm provd th foudto of th propt or djutmt mthod ug prmthg or ulfto. Th pplto of th propt or mthod ll trodud th t to.

37 .3 Applto of propt or djutmt.3. Propt or mthg pproh I orvtol tud um mll umr of ujt th trtmt group d lrg umr of ujt th otrol group. I th otg th outom of trt for ll otrol ujt m too pv.g. lotg lrg umr of otrol ujt m dffult d murg thr outom m lo vr otl. Roum d Ru 985 dr th mthg mthod for ltg ut of otrol ujt tht r mlr to th ujt th trtmt group th rpt to orvd ovrt. Th propt or mthg mthod mth ujt th trtmt group th ujt th otrol group ho h mlr propt or. Propt or mthg mthod hv dl ppld ll d oomtr tud. D'Agoto Jr. 998 m th ff of rmovg ovrt ml propt or mthg mthod for ll tud. I tud ug mthg Ru d Thom trodu to of propt or mthg for ll tud; th mthg mthod u oth th tmtd propt or d ollto of orvd ovrt tht r lol rltd progot to th outom of trt. Smlrl Dhj d Wh llutrt propt or mthg mthod o lor progrm dt to tmt th mpt of th trg.

38 3.3. qul frqu ulfto d qul ght pproh Th propt or mthg pproh lt ut of th ujt th otrol group to ompr th th ujt th trtmt group. Thu mthg u rdud umr of mpl hh ot l th murg ll otrol ujt. Hovr om tud rrhr m trtd ug th full mpl. Udr th rumt th propt or ulfto mthod opto to odr. Th propt or ulfto lo ko trvl mthg lokg or trtfto ldo d Kopg 8. Th propt or ulfto mthod oprmtr produr hh do ot dpd o pfd rgro futo rltg th outom to ovrt. Ru 997 tt tht th ftur of th propt or ulfto dvtg tmtg th trtmt fft omprd to multpl rgro u th rgro tmt m ot rll f t modl pfto orrt. O tp of th propt or ulfto Roum d Ru propt or qul frqu qutl ulfto mthod. Th drd th follog D'Agoto Jr. 998: tmt th propt or logt rgro modl of th trtmt gmt dtor o th orvd ovrt; Sort th tmtd propt or dg ordr;

39 4 3 U qutl of th tmtd propt or oudr to form fv ul. If qul ght r ud th th ovrll trtmt fft tmt qul to th vrg of th m dffr of th outom t th trtmt d th otrol group mog th fv ul. Lt d th ul d um th ulpf m of th outom r for th trtmt group d for th otrol group th th tmtd trtmt fft / 5. 5 Th frt thorm of th propt or mthod Roum d Ru 983 tt tht gv th propt or th orvd ovrt d th trtmt gmt r odtoll dpdt. Th mpl tht f ul of ujt homogou th propt or th th ujt of th trtmt group d th otrol group th ul ll hv th m dtruto of ovrt. Thrfor th dgr of ml of th ovrt t th trtmt group d th otrol group vlutd ttg th m dffr of th propt or t th trtmt group d th otrol group th h ul. Dhj d Wh 999 ppl to-mpl t-tt to hthr th m tmtd propt or th h ul r dtl or ld Im 4 ordr to rt th mml t of dtl ul. Th ld to furthr plttg th propt or ul. Th plttg produr drd four tp: Form fv qul frqu ul o th tmtd propt or;

40 5 Appl to-mpl t-tt to hk hthr th m of th tmtd propt or t th trtmt group d th otrol group r dtl th h ul; 3 If th t-tt tttll gft th dt th tmtd propt or tht ul r ot ld th th F ul ll furthr qull plt; 4 Rpt tp d 3 utl th t-tt r ot tttll gft for ll furthr plt F ul ddtol dtl of th produr r provdd Dhj d Wh 999. Propt or qul frqu ulfto qul ght F-W mthod hv rodl mplmtd vrou tud. Lttl d Ru du th pplto of th propt or F-W pproh ll trl d pdmolog. I tud pplg qul frqu ulfto Akvk u F ul to vlut Norg lor trg progrm..3.3 Propt or qul frqu ulfto djutmt for urv orpo W ppl th propt or mthodolog to urv tuto. Mot urv ht orpo ot vr ltd ujt rpod to omplt qutor rqut. Itd of trtmt group d otrol group tht dud rlr for orvtol tud hv rpodt group d orpodt group.

41 6 Dvd t l. 983 provd th fudmtl umpto for rpo propt hh tht th rpo or orpo mhm gorl. Th umpto logou to th trogl gorl trtmt gmt umpto of th propt or mthod orvtol tud th trtmt group d otrol group. Igorl m tht th rpo mhm rltd to th outom of trt pld th orvd ovrt. For mpl pul opo urv th rpo mhm h othg to do th th opo olltd lthough t otd th ko ovrt uh g. Th rpo propt or dfd th odtol prolt of ujt rpodg to or prtptg th urv gv th orvd ovrt. Th ulfto ppld o th tmtd rpo propt or orvtol tud. Wh om dmogrph.g. g om r mld t th rpodt group d th orpodt group th propt or ulfto mthod h th dvtg of rdug th orpo ut ot rqurg ulfto o ll ovrt dud Sto.. It hd uggtd tht mg t rdom MAR th mhm of orpo oft foud tuto h mg dt our Bjørtd 7 d hmr 7. Th trog gorl trtmt gmt umpto of th propt or mthod lo log to th MAR umpto. Th duo provd om kgroud of th rpo propt or ulfto djutmt tmtor.

42 7 Th rpo propt or mthod h dvlopd d utld vrl rrhr. Lttl 986 trodu rpo propt or mthod to djut th tmtd m. Buldg o Lttl rrh th rpo propt h ppld to vrt of tud th urv orpo. I th propt or r ud to djut th m for orpo. ltg d Yh 997 u propt or qul frqu ul to form orpo djutmt l ll. Aothr tud ug th rpo propt or Smth t l. tmt th vto rt for th U.S. rlo d Wllm ppl th rpo propt or mthod to houhold urv hl D-T t l. u th mthod o ph urv dt. Vrtvr d Lttl 3 trodu jot lfto d o th rpo propt or d th prdtd m from rgrg th rpodt outom o th ovrt to mprov ff d to rdu th orpo. Furthrmor Hrrod d Lr 7 propo rpo propt or modl to dl th th tuto h umpl of urv orpodt olltd..4 Propt or djutmt udr modl mpfto; qul vr ulfto Drk 993 provd multo tud to vlut th mpt of ovrt omo d qudrt trm mpfto o th propt or qul frqu ulfto djutmt udr lr rgro modl. Th multo rult dt tht for ovrt omo th propt or modl th tmtd propt or qul frqu ulfto du mlr th ordr

43 8 lt qur OLS tmtor. For th qudrt trm mpfto th propt or modl th tmtd propt or qul frqu ulfto tmtor h l th th OLS tmtor. Oft udr propt or qul frqu ulfto th lor ul td to ot fr umr of trtmt ujt d th uppr ul td to ot fr umr of otrol ujt. A oqu th th ul trtmt fft tmt m hv hgh vrto. v though om ul m hv mor vrto qul ght r grll gd to ul ovrll trtmt fft tmto. Hullk d Lou trodu th propt or qul vr V ulfto mthod to qul th th ul vr. Thr pproh ppl trto produr to ulf th propt or d o th vr of th th ul trtmt fft tmtor hh r ppromtl quvlt qul vr. Th vr of th th ul trtmt fft r tmtd rgro modl of th outom o th ovrt th h ul. Hovr ftr th qul vr ul hv formd th th ul trtmt fft tmtd rgro modl of th outom o th trtmt gmt dtor ol. S th vr of th quld vr IV of th th ul trtmt fft r lo ud th ght for th ovrll trtmt fft tmt th ght r thortll qul mog ul. Th uggt tht ug th mmum umr of propt or ul hr dt ould llo ould l th propt or ulfto mthod to rmov

44 9 mor. Th trd-off ould hgh vrto of th th ul trtmt fft tmt. Mr d Lou 7 dt tht th propt or qul vr ul oudr ould rltvl fr from th qul frqu ul oudr. I ordr to qul th vr mog ul th lor d ul m d to dr. Although dr ul m hv mllr vr t m produ lrgr. Udr qul frqu ulfto d udr qul vr ulfto Mr d Lou 7 dtrm th umr of ul tht produ th mllt m qur rror MS of th trtmt fft tmtor. Th olud tht udr mpl lr modl th propt or qul frqu ulfto h dvtg ovr th qul vr pproh. Th lo uggt rg th umr of ul to rmov mor utl th ovrll trtmt fft tmtor om mlr or utl th ovrll vr of th tmtor uttll r..5 otruto of th th.5. qul vr ulfto mthod udr modl mpfto Hullk d Lou dd ot vlut ho th qul vr ulfto pproh ould prform udr propt or modl

45 3 mpfto mplmtd Drk 993. No follo-up pulto o th top hv t foud th ltrtur. Ug multo ll th qul vr ulfto mthod udr modl mpfto. Addtoll Mr d Lou tt tht th ulfto tmtor th vr vr ghtg grll udrtmt th vr of th ovrll tmtd trtmt fft. W ll vtgt th vr d th of th trtmt fft tmt ug vr vr ght for th qul frqu ulfto d th qul vr ulfto udr oth orrtl pfd modl d mpfd modl..5. Thortl vtgto o propt or ulfto djutmt tmtor W hv trt vlutg th u of propt or mthodolog o vr d. Th propt or qul frqu ulfto pproh m produ trtmt fft tmt th hgh vrto om ul. Th mthod u qul ght th ovrll trtmt fft tmto. Prtl udr qul frqu F ulfto to ghtg hm qul ght d vr vr ght ppld; though o vluto h provdd o hh ghtg hm ould mor fft to rmov th of th ovrt. For t r th lor d d th uppr d qutl ul of th propt or th mpl of th trtmt group or th otrol group td to mll. Thu th vr of th r-d ul m hghr th tho of othr ul. W ll ho tht udr qul frqu ulfto th vr

46 3 vr F-IV ghtg tmtor h mllr th th qul ght F-W tmtor udr rt odto. W ll lo ho tht udr qul frqu F ulfto th F-IV tmtor l h vr o lrgr th th F-W tmtor. W ll vtgt th qul vr V ulfto pproh th vr vr ghtg hm..5.3 N pproh to propt or ulfto rtg propt or for orvtol tud th trtmt d otrol ujt h hllgg rt odto. Rrhr lud th otrol ujt ho tmtd propt or r lor th th mmum of th tmtd propt or from th trtmt group d th trtmt ujt ho tmtd propt or r hghr th th mmum of th tmtd propt or from th otrol group Dhj d Wh 999. For mpl Stürmr t l. 6 provd hr th lot d hght propt or r dltd h ol o of th to group.. thr trtmt or otrol prt. Aftr th luo ol ut of ujt r ud for l Fgur. Stürmr t l. 6. Th luo lo ko propt or trmmg Stürmr t l. 7.

47 3 Fgur. Propt or dtruto of th trtmt pod to th trtmt or rk ftor ujt d otrol upod ujt I urv dopt propt or to djut for orpo. Th tmtd propt or m our t th lor d of th dtruto; ougr ujt rfu to prtpt mor oft th oldr ujt. Th m rult om ougr ujt g trmmd from th l. Hovr om rrhr m trtd lg th tr mpl. Thrfor ltrtv propt or ulfto mthod drd to dl th th tuto.

48 33 W propo pproh of ulfto ttmptg to l th propt or for mot ul. Our mthod ll form ul tht h homogou tmtd propt or th trtmt group d th otrol group. Thu th dtruto of th ovrt th h ul dtl. Th pproh td to rt lrg umr of ul to rdu mor uggtd ohr 968 Im 4 d Mr d Lou 7..6 Orgto of th th I hptr Thr ll provd multo tud to vlut th propt or djutmt tmtor of th trtmt fft th rpt to qul frqu d qul vr ulfto mthod ug to ghtg hm: qul ght or vr vr ght. W ll lo propt or djutmt tmtor udr modl mpfto th multo. I hptr Four ll drv vr d th root m qur rror RMS of th propt or ulfto djutmt tmtor umg qul ght d vr vr ght. Our thortl dvlopmt ll um th outom grtd lr rgro modl. W th dvlop lmm thorm d orrpodg orollr. W ll td th thortl rult to th multpl ovrt tuto.

49 34 I hptr Fv propo ovl propt or lg ulfto th PSB mthod. W mplmt multo to ompr th rult of th PSB tmtor th othr tmtor ludg OLS F-W F-IV d V-IV. W lo mult propt or trmmg tuto. W ll um thr o trtmt ujt dt r th lor d of th tmtd propt or. W ll m our PSB mthod d th F d V pproh udr th tuto. I hptr S ummr th mjor fdg from our multo rult thor dvlopmt d our propod PSB ulfto pproh. W lo gv propol for futur rrh ork.

50 35 hptr 3. SIMULATION STUDIS OF PROPNSITY SORS MTHOD I th hptr m th prform of vrl trtmt fft tmtor trm of d root m qur rror RMS udr dffrt tp of modl mpfto. Spfll m th rgro or ordr lt qur OLS tmtor d thr propt or tmtor F-W F-IV d V-IV. Th dffr mog th propt or tmtor ot of to lmt. O lmt th formg of th propt or ul thr formg qul frqu F ul or qul vr V ul. Th othr lmt th ghtg hm ppld to om th th ul tmtor to omput th ovrll tmt for th trtmt fft thr pplg qul ght W or vr vr IV ght. W u th ttg of orvtol tud th trtmt group d otrol group for th multo. Both th multo modl d th tp of th modl mpfto r doptd from Drk 993. For oth th outom vrl d th trtmt dtor vrl th dpdt ovrt r grtd prdtor for to ro. I th frt ro th trtmt dtor multd dpdt Broull rdom vrl ug logt modl volvg to dpdt ovrt. Th outom multd ug rgro modl hvg to dpdt ovrt ddto to th trtmt dtor. I th od ro th trtmt dtor multd dpdt Broull rdom vrl ug logt modl volvg gl

51 36 ovrt d t qurd trm. Th outom multd ug rgro modl tht lud th trtmt dtor gl ovrt d t qurd trm. For th frt ro ll ot th OLS tmtor d propt or tmtor udr orrtl pfd modl d mpfd modl. Udr orrtl pfd modl th OLS tmtor otd fttg th rgro modl th to dpdt ovrt. For th propt or tmtor th orrt pfto om from lo fttg th modl of th propt or th to dpdt ovrt. Udr mpfd modl th OLS tmtor modl mpfto volv fttg th rgro modl ludg o of th to dpdt ovrt. Lk for th propt or tmtor th mpfto om from fttg th modl of th propt or thout o of th to dpdt ovrt. Rll tht th outom grtd from rgro modl ludg qudrt trm. I th od ro udr th mpfd modl th OLS tmtor d propt or tmtor r otd from fttg th modl through th luo of th qurd trm. Tl 3. provd ummr of multo modl d trtmt fft tmtor. I Hullk d Lou o mt provdd rgrdg th prform of th V ulfto pproh udr th tp of modl mpfto uggtd Drk 993. W hv foud o follo-up pulto o th prform top th rt ltrtur for V-IV tmtor udr modl mpfto.

52 37 Tl 3. Fttd modl d trtmt fft tmtor Modl Trtmt fft tmtor Rgro mthod Propt or mthod OLS ordr lt qur F-W F-IV V-IV orrt Spfto Rgro modl h to prdtor Propt or modl h to prdtor Mpfto Rgro modl h o prdtor Propt or modl h o prdtor I th frt to lo ll trodu ho dt r multd udr th to ro. I th od to ll dr th tmto of th rgro modl d th propt or modl. Th propt or ulfto mthod r drd th thrd to. I th fourth to trodu ho th OLS tmtor d th propt or tmtor r otd. Th ffth to prt th rult of th multo. I Appd Sto A3. provd th rom otto tl d flo hrt dgrm for h tp th multo produr. 3. Smultg dt I th to mplmt th multo grtg th trtmt dtor d outom vrl. Th dt r multd to ro. O ro volv to dpdt ovrt; th othr ro volv gl ovrt d t qurd trm.

53 Sro volvg to dpdt ovrt Df to dpdt ovrt d. N d dpdtl d. N. I th frt ro mult th trtmt dtor vlu dpdt Broull rdom vrl udr logt modl: Broullπ π Pr { + p-γ + γ + γ ]} - 3. W grt th outom vrl ug th modl: β + β + β + δ + 3. hr d. N th gv d dot dpd. For dtld formto Appd A3. Tl A Sro volvg gl ovrt d t qurd trm Dot gl ovrt d. N. W mult th trtmt dtor vlu dpdt Broull rdom vrl udr logt modl ug d t qur: Broullπ

54 39 π Pr { + p-γ + γ + γ ]} W u lr rgro modl th gl ovrt d t qudrt trm to grt th outom vrl: β + β + β + δ Th π dotd quto 3. or quto 3.3 th tru propt or for ujt. I orvtol tud π uko. Th tru propt or r ud to mult th trtmt dtor vrl udr oth ro. W lo um thr quto 3. or quto 3.4 to th tru outom modl. W u th prmtr provdd Drk 993 hr β β δ 3 β 3; γ γ.4 γ At h omto rdom mpl r multd. h mpl ot of rdoml grtd orvto. 3. tmto of rgro d propt or modl Udr h of th to ro drd Sto 3. ll ft th rgro modl to ot th OLS tmtor of th trtmt fft. W ll lo ft th logt rgro modl to ot th propt or tmtor of th trtmt fft. For oth th rgro modl d th propt or modl th orrt pfto of th modl mpl o omo for o of th to dpdt ovrt udr th

55 4 frt ro. Th mpfd modl lud ovrt omo udr th frt ro or qudrt trm omo udr th od ro. Tl 3. ompr th modl ud to mult d ft oth th outom modl d propt or modl urpt gord for mplt. Tl 3. ompro t multg d fttg th outom d propt or modl ovrt Tru propt or: OLS tmtor Outom modl: π { + p-γ + γ + γ ]} - β + β + β + δ + β + β + δ + Propt or tmtor F-W F-IV V-IV Propt or modl: { + p-γ + γ + γ ]} - Tru outom: β + β + β + δ + { + p-γ + γ ]} - OLS tmtor Tru propt or: π { + p-γ + γ + γ ]} - Outom modl: β + β + δ + Propt or tmtor F-W F-IV V-IV Propt or modl: { + p-γ + γ ]} - Tru outom: β + β + β + δ + Not: - orrtl pfd rgro or propt or modl - mpfd rgro or propt or modl.

56 4 3.3 Propt or ulfto Th propt or tmto pproh produ th tmtd propt or logt rgro thr th orrtl pfd propt or modl or th th mpfd propt or modl drd Tl 3.. Th tmtd propt or r th ortd dg ordr for pplg ulfto. Th umr of propt or ul t t fv. Thr r to mthod ud h formg propt or ul: qul frqu F d qul vr V. Th F d V ulfto produr r drd t. Th F ulfto u qutl of th tmtd propt or oudr to form fv djt ul dtl r hptr To. W mplmt trtv produr to ppl th V ulfto provdd Hullk d Lou. Th V ulfto trt th fv F ul o th tmtd propt or. Wth h F ul th produr ppl rgro modl of th outom vrl o th ovrt to tmt th vr of th th ul trtmt fft rgro tmtor. Th th ul oudr r djutd to ppromtl qul th tmtd vr of th th ul trtmt fft tmtor mog ul. For tr formto hptr To d Appd- Fgur A3..

57 4 I om th V ulfto produr do ot produ rult. Th m ttrutd to outlr rtg d ul rultg too f orvto rmg to form fv ul. Thu l th fv V ul r rtd th tuto. Fgur A3.4 th Appd llutrt tuto hr V ulfto ol produ thr ul. 3.4 Trtmt fft tmt Aftr formg th propt or ul th ul-pf trtmt fft tmtor d t tmtd vr r otd th h ul ordr to omput th ghtd ovrll propt or trtmt fft tmtor. For h of th to ro drd Sto 3. ll ot th OLS tmtor d propt or tmtor for modl udr orrt pfto or mpfto OLS tmtor d propt or tmtor I th of th OLS tmtor for th trtmt fft δ δˆ otd from th rgro modl drd Tl 3.. For th propt or tmtor frt ot th tmtd trtmt fft d t tmtd vr th h ul. Th ul-pf trtmt fft th ul tmtd ˆ δ 3.5

58 43 hr / /. For dtld formto Appd Tl A3.. Th tmtd trtmt fft th ul quto 3.5 lo otd fttg th rgro modl for ul dt ol: β + δ Hullk d Lou u th m rgro modl to ot δˆ d t tmtd vr Vˆ. Ug V ulfto th tmtd vr Vˆ hh r otd fttg quto 3.6 m ot rl qul mog ul. Th tmtd vr for δˆ ˆ ˆ V Vr ˆ δ Vr ˆ + Vr ˆ 3.7 tmt Th propt or tmtor of th ovrll trtmt fft ghtd 5 ˆ δ 3.8

59 44 umd th If qul ght r umd W /5. If vr vr ght r ˆ IV / V / Vˆ Addtol dtl r provdd th Appd tl A Murg th prform of trtmt fft tmt W grtd multo to th prform of th trtmt fft tmtor. W rtd multd mpl h th rdom orvto. Amog tho multd mpl otd δ m δ hr ˆ ˆ ˆ... ˆ δ δ δ δ. W omputd th prtg rltv of th m PRMB. W rptd th for rg of prmtr vlu th tl ho Sto 3.5. I ddto omputd th prtg rltv of th md RPBM of δˆ ho Drk 993 th prtg rltv tdrd dvto PRSD d prtg rltv RMS PRRMS. Th prtg rltv mur formul r ummrd th Appd Tl A3.3 d th tru vlu of th prmtr r Appd Tl A3.4. ˆ

60 Smulto rult Th prform of th trtmt fft tmtor r vlutd thr uto. I th frt uto du otg th OLS tmtor d propt or tmtor udr th orrtl pfd modl. I th od uto ot tmtor ug mpfd modl omttg dpdt ovrt for tmtg propt or d fttg th rgro modl. I th thrd uto u th qudrt trm mpfto for tmtg propt or d fttg th rgro modl. I h uto th PRMB r otd for th trtmt fft tmtor.

61 orrtl pfd modl volvg ovrt Tl PRMB for th OLS d thr propt or tmtor ug orrtl pfd propt or modl d orrtl pfd rgro modl for ovrt Prmtr PRMB of δ β γ OLS F-W F-IV V-IV ˆ δ Th rult Tl dt tht h ug orrtl pfd propt or th PRMB of ll propt or tmtor F-W F-IV d V-

62 47 IV r γ.. th flu of o th propt or r. Th pttr ott for h lvl of β.. th flu of o th outom. All propt or tmtor produ potv. Th PRMB of ll propt or tmtor t δ3 r mllr th th r t δ. Th u th PRMB omputd δ - δ/δ]% o t δ3 th domtor r ftor of thr omprd to h δ. Amog propt or tmtor th F-IV h lor PRMB th th othr to propt or tmtor hl V-IV h lor PRMB th F-W. Gv prdtrmd p-vlu lvl for h omto of δ β d γ vlu prd t-tt prformd to vlut hthr thr gft dffr of m t F-IV d V-IV. A mlr tt lo ppld to ompr m t V-IV d F-W. A ptd OLS h th lot PRMB roud ro mog ll tmtor th tmto modl orrtl pfd.

63 Mpfd modl th ovrt omo ludg Tl PRMB for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl Prmtr PRMB of δ β γ OLS F-W F-IV V-IV ˆ δ Th rult Tl dt tht for mpfd propt or th ovrt omo th PRMB of ll propt or tmtor r uttll γ r. Th pttr th m for ll propt or tmtor t h lvl

64 49 of β. Th mpfd OLS tmtor hh rgr ol o ovrt follo th m pttr. A γ d β r th PRMB of ll tmtor r uttll. Th PRMB of ll tmtor t δ3 r mllr th th r t δ. Th u th PRMB omputd δ - δ/δ]% o t δ3 th domtor r ftor of thr omprd to h δ. Ovrll o of th tmtor prform ll trm of PRMB h dpdt ovrt omttd from th propt or modl d from th rgro modl. Th mpl tht udr th odto rrhr hould ot u th tmtor th l. Th ot urprg u th omttd ovrt th o ud oth rgro modl to grt th outom d logt modl to grt th propt or. Th multo rult uggt tht rrhr hould tmt th propt or d o ll vll ovrt formto to r ur of th l. Alo th propt or modl pfto hould lud ll orvd ovrt.

65 Modl th qudrt trm mpfto omttg Tl PRMB for th OLS d thr propt or tmtor ug mpfd propt or modl d mpfd rgro modl Prmtr PRMB of δ β γ OLS F-W F-IV V-IV ˆ δ Th rult Tl dt tht udr qudrt trm mpfto th PRMB of ll propt or tmtor r γ r. Th pttr th m for ll propt or tmtor t h lvl of β. Th mpfd OLS

66 5 tmtor hh omt qudrt trm follo th m pttr. Hovr th F-IV h th lot PRMB of ll tmtor th multo hl th OLS tmtor h th hght PRMB. Th rult ho tht β r th PRMB of th OLS F-W d V-IV tmtor r. Hovr th rult otd from th F-IV tmtor do ot follo th m pttr. A Sto 3.5. gv prdtrmd p-vlu lvl for h omto of δ β d γ vlu prd t-tt ppld to ompr dffr m t V-IV d F-W d t V-IV d F-W. Th PRMB of ll tmtor t δ3 r mllr th th r t δ. Th ot urprg th PRMB omputd δ - δ/δ]% o t δ3 th domtor r ftor of thr omprd to h δ Summr of multo I th vtgto of dffrt trtmt fft tmtor foud tht udr orrtl pfd propt or modl th F-IV tmtor h th lot PRMB d PRRMS omprd to othr propt or tmtor ro th rg of prmtr vtgtd. W lo dovrd tht udr th qudrt trm mpfto th F-IV tmtor h th lot PRMB PRSD d PRRMS mog ll tmtor ludg th OLS tmtor Appd A3.5. A3.5. d A Th multo rult r ummrd lo. For propt or tmtor: th PRMB d th PRRMS grll r th flu of th od ovrt o th propt or γ r.

67 5 A th flu of th od ovrt o th outom vrl β r th PRMB d th PRRMS oft r. Udr orrtl pfd propt or modl d rgro modl ug to dpdt ovrt: Amog propt or tmtor F-IV h th lot PRMB PRSD d PRRMS. OLS h lor PRMB PRSD d PRRMS th th propt or tmtor. Th ptd th tmtg modl orrtl pfd. Udr th mpfd modl.g. omttg dpdt ovrt: th prform of th OLS vr mlr to th prform of propt or tmtor. All tmtor prform poorl trm of PRMB PRSD d PRRMS. Udr th mpfd modl.g. ludg qudrt trm: F-IV provd th lot PRMB PRSD d PRRMS. Th uggt tht t h th t prform mog ll tmtor th multo. V-IV h lor PRMB PRSD d PRRMS th F-W. OLS h th hght PRMB PRSD d PRRMS h omprd to th propt or tmtor. I th t hptr dvlop thortl drvto to pf udr ht odto lor ghtd ovrll or vr of th tmtor ould otd ug thr W or IV ght.

68 53 hptr 4. THORTIAL INVSTIGATION OF PROPNSITY SOR STIMATORS I th hptr dvlop thortl frmork to ompr dffrt propt or ulfto djutmt tmtor F-W F-IV d V-IV. I th frmork fd th follog: Udr th F ulfto djutmt f hghr vrto our th lrgr for th ul trtmt fft tmt th th ovrll of th IV ghtg tmtor mllr th tht of th W tmtor. Th F-IV tmtor l h o lrgr vr th th F-W tmtor. 3 If th vr of th trtmt fft tmtor th ul th V ulfto lrgr th th hrmo m of th vr of th F th ul trtmt fft tmtor th th F-IV tmtor h lor vr th th V-IV tmtor. Th ttg of our thortl frmork orvtol tud th trtmt dtor vrl d ovrt. W um th outom vrl grtd from lr rgro modl tht lud th trtmt dtor d th ovrt. Th frt to provd pro for B th du to th ovrt th ul; pro of th vr of th ul-pf trtmt fft tmt V ; d lmm gvg odto h B ogtv. Th od to dvlop thorm to ompr th d vr of th to tmtor udr F ulfto. Th thrd to dvlop thorm for omprg vr t th F-IV d V-IV tmtor. I th Appd Sto A4. provd

69 54 rom otto tl d flo hrt dgrm for h tp th thortl drvto. W td th thor ork to th tuto th multpl ovrt Appd A pro of B V d Lmm udr lr rgro modl W um th outom vrl grtd udr lr rgro modl tht lud trtmt dtor d gl ovrt: β + β + δ + 4. hr d. N σ th. For dtld formto Appd A4. Tl A4.. To tmt vr of th propt or tmtor um tht thr r t lt to orvto oth th trtmt group d th otrol group th h ul. B dfto Sto 3.4 th propt or tmtor gv ˆ δ ˆ δ 4. hr th dg th ul.

70 55 Hr lt dot th dtor of hthr ujt ul uh tht f d ol f othr hr r lor oud d uppr oud of ul rptvl; th δ β β δ β β. S mpl tht tht. Thrfor rdu to δ β β hr. Smlrl + + β β. Thu hv + + δ β 4.3 Tkg th ptto quto 4. gv ] + + δ β 4.4 Hr hould ot tht o of r dfd pt h from th trtmt group d th ul. Thrfor rt or mpl. Smlrl rt

71 56 rt d. B dfto futo of o tht mpl. Thrfor oth of d r ro. H hv β ] +δ 4.5 Th for ul th B β ] hr B + δ. H th ptto of th propt or tmtor om ˆ δ ] B +δ 4.6 d th ovrll of th propt or tmtor Bδ ˆ ] B 4.7 W dot th th ul vr of V Vr d th ovrll vr of th propt or tmtor Vrδ ˆ ] V 4.8

72 57 for dtl Appd A4.. A4..3 Nt gv Lmm tht provd odto udr hh B ogtv for ll ul. Lmm 4. Suppo P{ } o-drg futo of R. Th for for < R. Proof: Frt provd to dtt: f f P P f P f P d f f P P f P P ] f P ] f. P Lt f dot th odtol pdf of gv h rtrtd to th trvl. Th f. Smlrl lt f dot th pdf of h f f d rtrtd to th trvl.

73 58 Ug th rt th odtol ptto: ] ] d f d f d f d f d P f d P f d f d f d. ] ] ] ] ] ] ] d f d f d P f d P f d f d f

74 59 Nt o-drg futo of t rtl o-drg futo. B th ovr Iqult Thorm ll d Brgr hv ] ov. H. ] ] ] ] ] ]} ] ]{ ]} ]{ ] ] ] ] ] ] ] ] ] ] ] ov Thrfor. Th omplt th proof. 4. Thorm omprg dffrt ghtg hm W o ompr th of to propt or tmtor udr th m ulfto ut ug dffrt ght. W lo m th vr t th qul ght W tmtor d th vr vr IV ght tmtor. 4.. Dord d oord odr th vtor of ul from prtulr ulfto hm. To ghtg hm ll provd to dffrt vtor of ght. Ag th of to propt or tmtor ug th m ulfto ut dffrt ght