__ a __ BE JOURNALOF Ekonometrics REIVESLE lanruoJ fo scirtemonocE 46 )4991( 381 602- seyaB ecnerefni ni noisserger sledom htiw AMRA ,p( srorre q) ahtrahddiS ,“-*bihC drawdE bgrebneerG devieceR( lirpA ;3991 lanif noisrev deviecer tsuguA )3991 tcartsbA eW poleved lacitcarp dna tcaxe sdohtem fo gnizylana ,p(AMRA 4) noisserger rorre sledom ni a naiseyaB krowemarf yb gnisu eht sbbiG gnilpmas dna sgnitsaH-siloporteM ,smhtirogla dna ew evorp taht eht lenrek fo eht desoporp vokraM niahc relpmas segrevnoc ot eht eurt .ytisned ehT serudecorp nac eb deilppa ot erup AMRA emit seires sledom dna ot enimreted serutaef fo eht doohilekil noitcnuf yb gnisoohc etairporppa esuffid .sroirp Our stluser era lanoitidnocnu no eht laitini .snoitavresbo eW osla wohs woh eht mhtirogla nac eb rehtruf deifilpmis rof eht tnatropmi laiceps sesac fo yranoitats )p(RA dna elbitrevni )q(AM .sledom evisruceR snoitamrofsnart depoleved ni siht repap ot ezilanogaid eht ecnairavoc xirtam fo eht srorre dluohs evorp lufesu ni .noitamits etsitneuqerf selpmaxE htiw detalumis dna actual cimonoce data era .detneserp ~JPK :slruo~~ sbbiG ;gnilpmas sgnitsaH-siloporteM ;mhtirogla Data ;noitatnemgua emiT ;seires AMRA ;sessecorp vokraM ;niahc naiseyaB scitsitats LEJ :noitacifiss.al.c lC 1; C15; C22 1. Introduction noissergeR sledom htiw detalerroc errors evah neeb the sucof of elbaredisnoc noitnetta ni scirtemonoce dna .scitsitats hguohtlA koobtxet snoitatneserp gnidnopserroC* .rohtua eW evah detifeneb morf eht stnemmoc fo mailliW ,lleB nafetS ,kinttiM mlehliW ,dniefeueN na etaicossa dn a,rotide owt suomynona .seerefer sihT si a noisiver fo seyaB‘ ecnerefnI aiv sbbiG gnilpmaS ni noissergeR sledoM htiw )p(RA dna )q(AM ,’srorrE lirpA ,2 .2991 00.70$/49/6704.4030 0( 4991 reiveslE ecneicS .A.S llA sthgir devreser SSDI 385103967044030 8 yllausu tcirtser noitnetta ot evissergerotua )RA( dna gnivom egareva )AM( ,sledom the rettal netfo of the tsrif order, the dexim evissergerotua dna gnivom egareva )AMRA( ledom si ylraelc the tsom gnitseretni case. ,yletanutrofnU the lanoitidnocnu doohilekil noitcnuf for the lareneg yranoitats dna elbitrevni ,p(AMRA 4) error ledom si etiuq detacilpmoc dna nac tneserp suoires -atupmoc lanoit .smelborp ,eroferehT etipsed the approaches ot mumixam doohilekil noitamitse depoleved ni dlobweN (1974), nagaP dna sllohciN (1976) xoB dna snikneJ (1976), yelsnA (1979), dna rendraG et .la (1979) software packages are dezinagro dnuora the dohtem of raenilnon tsael serauqs or sti ,tnelaviuqe the lanoitidnoc mumixam doohilekil ,yevraH( 1981). rehtonA enil of yriuqni has neeb detcerid at elbisaef dezilareneg tsael serauqs ,srotamitse tsom ylbaton ni Otto et .la (1987) dna htiarblaG dna hslaW-edniZ (1992). tnesbA ot a egral tnetxe morf siht erutaretil si the naiseyaB sisylana of noisserger sledom htiw ,p(AMRA 4) errors. hguohtlA a naiseyaB evitcepsrep for emit seires has neeb ylevitca ,deusrup a lluf tnemtaert for hcus sledom si ton .elbaliava hcuM of the ylrae work si detartnecnoc no evissergerotua sledom (see ,renlleZ 1971) elihw the retal work no dexim AMRA sledom was derrups yb the approach of nahanoM (1983) hcihw si tsom lufesu for redro-wol processes. gnilemeorB dna yawraahS (1984) egralne the scope of naiseyaB emit seires sisylana yb gninoitidnoc no laitini seulav of elpmas-erp errors dna other -ilpmis snoitacif that ecalper the nwonknu errors gniraeppa ni the doohilekil noitcnuf htiw setamitse deniatbo yb raenilnon tsael .serauqs nI tnecer sraey ynam of the deviecrep seitluciffid of gnitnemelpmi the naiseyaB mgidarap evah ylevitceffe deraeppasid hguorht the ecnegreme of vokraM niahc etnoM olraC (MCMC) noitalumis sdohtem hcus as the sbbiG relpmas (see rennaT dna ,gnoW ;7891 dnafleG dna ,htimS 1990) dna sgnitsaHssiloporteM (MH) smhtirogla (see siloporteM et ,.la ;3591 ,sgnitsaH ;0791 dna ,yenreiT 1993). esehT sdohtem are lufrewop sloot for gnitalumis elbatcartni tnioj snoitubirtsid that yler no the ecnegrevnoc of a ylbatius -curtsnoc ted vokraM niahc ot the tnioj noitubirtsid of .tseretni ehT tuptuo of the -alumis noit si a elpmas of draws that nac eb desu for suoirav ,sesoprup for elpmaxe ot etupmoc roiretsop stnemom dna .selitnauq ehT eulav of these sdohtem for gnizilanoitarepo naiseyaB ecnerefni for emit seires ,noisserger yllaicepse htiw evissergerotua processes denoitidnoc no laitini ,snoitavresbo was dezingocer ylrae yb bihC )3991( hcolluCcM dna yasT (1993) dna treblA dna bihC (1993). nI siht paper we eunitnoc siht enil of attack tub sucof no a erom lareneg ssalc of ,sledom ,yleman noisserger ,sledom perhaps htiw deggal tnedneped ,selbairav whose errors wollof a yranoitats dna elbitrevni ,p(AMRA q) process of yna deificeps order. ,eromrehtruF ruo stluser are lanoitidnocnu no the laitini .snoitavresbo oT tup ruo work ni ,evitcepsrep llacer that the tseuq ni sbbiG gnilpmas si ot express the tnioj roiretsop ytisned of the sretemarap ni a mrof that sdnel flesti ot ,noitalumis yllausu revo a kcolb of sretemarap at a ,emit denoitidnoc no the gniniamer .skcolb gniveihcA siht ni the tnerruc txetnoc setatissecen the esu of lareves detaler seigetarts dna the tnempoleved of lareves wen .stluser ,tsriF we ecudortni a set of lanoitidda sretemarap otni the ,noitalumis na elpmaxe of data .noitatnemgua esehT selbairav are ton the p + 4 elpmas-erp errors that are desu ot enifed the lanoitidnoc ,doohilekil tub rather m = ,p(xam q + 1) snoitcnuf of these errors deniatbo morf the state space noitatneserper of the .ledom ,dnoceS we show that two snoitamrofsnart of the data nac eb yletarapes desu ot ezilanogaid the ecnairavoc xirtam of the error. morF the demrofsnart -avresbo snoit we niatbo the lluf lanoitidnoc snoitubirtsid of the noisserger ,sretemarap the evissergerotua ,stneiciffeoc dna the error .ecnairav ,drihT we enibmoc vokraM niahc ,seigetarts as has neeb enod ni roirp work yb relliiM (1993), tub htiw a tnereffid ssalc of gnitareneg-etadidnac .seitisned ,htruoF we niatbo the lluf lanoitidnoc noitubirtsid of the demrofsnart elpmas-erp errors yb iramlaK .gnihtooms ,htfiF we ezilaiceps the sisylana for )p(RA dna )q(AM sledom dna show that hcum of the sisylana nac eb .deifilpmis ,yllaniF we yllamrof evorp the ecnegrevnoc of the CMCM mhtirogla ot the derised tnioj roiretsop noitubirtsid of the .sretemarap nI the proof, we hsilbatse a tluser of tnednepedni tseretni that states that the set of sretemarap that dael ot ytiranoitats dna ytilibitrevni si .detcennoc-cra nI tnerrucnoc dna tnednepedni work ttoirraM et .la (1992) poleved a tnereffid approach ot the noitamitse of AMRA sledom that si desab no gnilpmas snoitcnuf of the laitrap .snoitalerrocotua A eutriv of rieht approach si that eno-rof-eno draws of each laitrap noitalerrocotua nac eb deniatbo tub at the cost of a erom xelpmoc .mhtirogla roF the tsom part, that paper sesucof no the tnatropmi data citylana seussi detaler ot ,gnitsacerof gnissim ,seulav dna ledom .ycauqeda nI ,tsartnoc we ylticilpxe wolla for a noisserger ,erutcurts evired exact smrof for etelpmoc lanoitidnoc ,snoitubirtsid tcudnoc the gnilpmas htiw skcolb of sretemarap ot evorpmi the ecnegrevnoc of the vokraM ,niahc dna yfirev lamrof ecnegrevnoc snoitidnoc for ruo proposed .mhtirogla ehT nalp of siht paper si as .swollof noitceS 2 stneserp the ledom dna the roirp .snoitubirtsid noitceS 3 sniatnoc the snoitamrofsnart denoitnem ,evoba the lluf lanoitidnoc ,snoitubirtsid the sliated of the CMCM ,mhtirogla dna a meroeht that the mhtirogla .segrevnoc noitceS 4 takes pu the )p(RA dna )q(AM laiceps cases. lareveS laciremun selpmaxe desab no detalumis dna lautca data are detneserp ni noitceS 5, elihw noitceS 6 sniatnoc gnidulcnoc .skramer ehT xidneppA sniatnoc a proof of noitisoporP 2. 2. Model dna roirp snoitpmussa redisnoC the gniwollof naissuaG ledom ni hcihw the noitavresbo at emit ,t ,,y si detareneg yb Yt = B;x + ,tE f = ,n,...,l (1) 186 S. Chih. .E Greenberg/Journal fo Econometrics 46 (1994) 602-381 where X, si a k x 1 rotcev of ,setairavoc p si the k x 1 rotcev of noisserger ,sretemarap dna E, si a modnar error. esoppuS that E, swollof na ,p(AMRA 4) process ,-,E&=& 1~,U1u+,U+,~,E,~+...+ ,,_,u,u+...+ (2) hcihw si expressed ni smret of a laimonylop ni the tfihskcab operator L as 4(L)&, = ,,u)L& (3) where 4, # ,0 ,0 # 0, ,U 2“ N(0, a’), a2 > 0, JV setoned the lamron ,noitubirtsid 4(L) = 1 - 4, L -... - qbpLp, dna U(L) = 1 + U1 L + ... + H,Lq. ,yltnelaviuqE the ledom ni (1) dna (2) nac eb expressed ni state space mrof (see ,yevraH 198 1) as :swollof 1’1 = B;X + ,IC’Z (4) x, = ,_,EG +fir,, (5) where z = (1, 0, . . . :‘)O, zn x 1. x, = ,,,lA( :‘),tmz m x 1, m = ,p(xam q + ,)l 14 1 42 ! G= 43 f IIf-1 m x m, . . . . . . , . . . . . . . mbq f 0 “. 0 dna f = (1, 8, , . , 0,)‘. nI gnitirw G dna f we yolpme the snoitnevnoc that 4s = 0 for s > p, 8, = 0 for r > q, dna ,,I( = 1. eW ekam the gniwollof :snoitpmussa noitpmussA M :)ledoM( ehT data J’ = ,,y( . . . ,yJ are detareneg yb (1) dna (2), htiw p dna q .nwonk noitpmussA S :)ytiranoitatS( llA roots of 4(L) eil edistuo the tinu .elcric noitpmussA I :)ytilibitrevnI( llA roots of U(L) eil edistuo the tinu .elcric .S ,bihC .E lanruoJ/grubnerrG ’fo scirtemonocE 46 )4991( --381 602 781 where r$ = ,r$~( . . . ,4J, 0 = (or, . . . , ,‘)qJif $A. .) si the etairav-s lamron -ubirtsid ,noit YY( .) si the detrevni ammag ,noitubirtsid AI si the rotacidni noitcnuf of the set ,A ,S si the set of $C that seifsitas noitpmussA ,S dna BS si the set of 3l that seifsitas noitpmussA .I ehT sretemaraprepyh ,Oif BO, ,& ,,,@ ,OI( O,,, ,Ov dna & are .nwonk A few stnemmoc tuoba these snoitpmussa are ni order. gnimussA ytiranoitats does ton timil ni yna tnatropmi yaw the ytiliba ot ledom yranoitatsnon data, ecnis x, yam niatnoc deggal seulav of ,y whose stneiciffeoc are .detcirtsernu nI the ecnesba of deggal ,Cy the selbairav ni x, yam eb )toor-tinu( yranoitatsnon ni hcihw case noitpmussA S stnuoma ot na noitressa that y dna x are .detargetnioc ehT nur-gnol noitaler neewteb a yranoitatsnon y dna sti setairavoc dluow esiwrehto kaerb ;nwod y ni effect dluow eb a erup emit seires process, dna there dluow eb on tseretni ni gnitamitse 8. noitpmussA I si decudortni for -acifitnedi noit .sesoprup htiW respect ot noitpmussA ,P ti lliw eb deton that the lausu detrevni-lamron ammag noitubirtsid has neeb demussa for II/ dna a2, elihw those for $C dna 0 are etairavitlum lamron detacnurt ot rieht yranoitats dna elbitrevni ,snoiger .ylevitcepser eugaV roirp noitamrofni nac eb deniatretne yb gniretnec these snoitubirtsid at zero dna gnittes each roirp noisicerp xirtam lauqe ot :I semit na ytitnedi ,xirtam where E si a llams .rebmun A ylhgih evitamrofni roirp egral( )noisicerp no a retemarap nac take the ecalp of a ,tniartsnoc ybereht gnittimrep the sisylana of lanosaes AMRA processes. roF the laitini state ,rotcev the ytiranoitats noitpmussa seilpmi that zO, denoitidnoc no 13, 4, ,I( dna ,’T( swollof a lamron noitubirtsid htiw sretemarap E(Q) = 0 dna )bxor(E = Q, where )2S(cev = l(20 - G @ .)’ff(cev‘))G (7) ,yllaniF the lamron dna detacnurt lamron sroirp that we emussa are elbisnefed no lareves ,sdnuorg yliramirp lacitylana ytilibatcart dna .ytilibixelf ,sselehtenoN fi derised a tnereffid ssalc of sroirp nac eb deyolpme esuaceb ti si elbissop ot elpmas morf dradnatsnon snoitubirtsid (as we do woleb for 0) nihtiw a vokraM niahc mhtirogla yb gniyolpme the sgnitsaHssiloporteM .mhtirogla yllauqE ,tnatropmi tuptuo gnidnopserroc ot tnereffid roirp ,snoitubirtsid for elpmaxe those ton ni the evoba ,ssalc nac eb deniatbo yb a dethgiew partstoob deilppa ot the delpmas draws. eW kramer rehtruf no these stniop .woleb 3. niaM stluser ehT laog of the paper si ot enimreted stnemom dna other serutaef of the roiretsop noitubirtsid of $ = ,I/( ,4 ,0 a’) rednu snoitpmussA M, ,S ,I dna .P yB seyaB ,meroeht the roiretsop ytisned si nevig yb )vl$(’f, XT ,)$Iy(’J)$(cr where rc($) si the roirp ytisned dna )$ly(’f, si the doohilekil .noitcnuf ehT tcerid noitaluclac of the exact doohilekil noitcnuf si .elbatcartni tI si llew ,nwonk ,revewoh that nevig the elpmas-erp errors .i = (c,, . . . .c +,, ,, ,,,u , +qmu ,), the ytisned of y nevig ($, L) nac eb expressed as ,$lV_(’l. ).j = if (27ra2)) ‘.2 exp[ ]:u$- I=, = if )’an2( pxe‘“ - & ,v_( - _tIP_ ]2i, (8) =, I l where ,_,,,j = j/ix + (4(L) - ,y()l - )j/:x + (O(L) - ,u)l si the daeha-pets-eno noitciderp of JJ, nevig noitamrofni pu ot emit t - ,.I eW therefore poleved na approach that seiler no (8) tub sselehtreven sedivorp the roiretsop ytisned for the exact .doohilekil ,tsriF we show that the lanoitidnoc doohilekil nac eb expressed ni smret of ylno IM elpmas-erp ,selbairav ton lla the p + q stnemele ni _i sihT gnisirprus tluser si yllautca a ecneuqesnoc of the state space mrof of the AMRA ledom dna appears ot evah neeb dekoolrevo ni the .erutaretil ruO noitartsnomed snigeb yb gniredisnoc the doirep t = 1. nehT morf (8) o,,?_ = +B;x C/),X0 +...+ ,+pL& + LlrJ + oll,U +...+ .,+,_u,u yB gnivlos the state space mrof for z, morf the mottob ,pu we dnif that #,c, + +‘“ &L,, + , + (I, L4” + ..’ + o,u_,+, = c/),x,, + rA20. ,eroferehT f, 0I = j/ix + c$, ,,,CC + ;~~1~ ,.e.i the stnemele of i retne ylno hguorht rows of Q,. sihT si eurt for lla seulav oft as the gniwollof tnemugra .sevorp roF =)L(,denifed,p<t<l 1 nehT.‘~’L,~I~-...-L,b(- ,yU(r4 - )B;x = 4,(‘%‘% = &)L(&’Z = z’(z, - ,_,?,hg -.“- f$_,‘X,) (9) = a,, - 4,X,.,P, -...- .*,xl& yB yldetaeper gnisu the noisrucer ,,CC = ,_~,,(c& + +,x ,-1,, + Or-, ,,u (9) nac eb nettirwer as ’ erehT IS on deen ot ecudortni eht elpmas-erp srorre fi eht ledom seod ton niatnoc a gnivom egareva .tnenopmoc nI taht esac a tcerid hcaorppa nac eb desab no eht noitpmussa taht eht tsrif p -avresbo snoit emoc morf eht yranoitats .noitubirtsid sihT tnatropmi laaeps ,esac dna ruo tnemtaert fo ,ti si debircsed ni noitceS .1.4 morf hcihw ti swollof that -,lt!i 1 = B;+ + -,Y(I$ 1 - )B,-;x +...+ 1‘!(,& - )p;x + ,-,u,H +...+ ,ur_,B + 0lX& + .o.r+,(c )OT( suhT tlj (1 < t < p) sdneped no i ylno hguorht x0. nopU gniylpitlum yb 4(L), the emas epyt of tnemugra shows that ,-,,rj = if;X + i i-ry(i$ - )l/j-iX ,=i + 1-,UlO +...+ e,-,u, + ‘&+I.(), t = p + I,...,& (11) where we evah desu the snoitnevnoc that jU = 0 j( > )y dna ~.,(c = 0 (r > m). ecniS elpmas-erp errors retne _,,,;? 1 ylno hguorht rows of x0, we evah -batse dehsil the tluser that ,$Iy(f. 1”) = ,$ly(f x0). eroferehT we edulcni ,j[ 4,H, c2, dna a0 as stnemele ni ruo CMCM ,mhtirogla gnitalumis these sretemarap morf the gniwollof lanoitidnoc :seitisned ,yIP(n $_,]. x0), ,yl4(cr ,+_!,I ,)~LC ,),(~,,~,t,yIU(rr .~~_$,yl~a(~r x0), dna ,yl~~(cr $), where, e.g., s-+r setoned lla sretemarap ni I!,I other naht .I/ oT evired these ,seitisned we proceed yb gniton that each si lanoitroporp ot the tnioj roiretsop ytisned for the detnemgua retemarap rotcev ,/II( ao) nevig yb ,it(n )YIO~ x ~(~)~MW’(Yl~~ >I“% (12) where ,$l~~(f, )or si the lanoitidnoc ytisned of y ees( )woleb dna the other seitisned are nekat morf noitpmussA .P gniyfilpmiS (12) si the txen order of .ssenisub 3.2. Full conditional distributiorls owT stluser are lartnec ot the sisylana that won .swollof eW show that the ytisned Iy(‘f, $, x0) nac eb dezilanogaid yb evisrucer snoitamrofsnart of the data ot ecudorp a noisserger pihsnoitaler for 3/ dna 4. esehT elpmis snoisrucer for the lareneg AMRA melborp evah ton appeared erehwesle dna yam eb lufesu for tsitneuqerf .noitamitse Drfinition 1. Let the sccllars ~1~= y ,F = 0 und the vectors x, = x d = 0, s < 0, and let a,.0 = 0, r > m. For t = 1, . . . , n, dt$ine lj _* J’t -2’t- 3-rYS4 - i OiY,*_i - X,+1,0, i=l :x = ,x - i s.-txs4 - i .i_*,x,H =s. 1 i=l 091 .S ,hih’C .E ~grehwrG lanruoJ ’fo ~.krtrnromcE 46 (1994J 1x3 602 sihT noitinifed seilpmi the gniwollof :ammel Lernrnu 1. Lety* he the n x 1 cector rhtfc :1~ and let X* be the n x k matrix htiw Xl *’ as sti htt .wor Then ,$l*Y.(‘f. )0x = pxe*~“~)2ac72( - $ (_,J* - x*b)‘(J’* - x*/q 1 1 $corP yfireV that 1~ T - x if’f = ~4~ dna proceed yb ,noitcudni gnikam esu of (10) dna (11). n morF the noitinifed of ::’J dna sti ecnaraeppa ni ,$I*s(’f, x0) we see how z0 sretne the lanoitidnoc .ytisned ,revoeroM the noisserger pihsnoitaler *y = X*p + ,u where u - .O(;N. ,),,l‘~ yletaidemmi sdleiy the lluf lanoitidnoc -irtsid noitub of p dna a2. eW eunitnoc yb gnicudortni a noitamrofsnart that swolla su ot enimreted the lluf lanoitidnoc noitubirtsid of 4. noitinifqD 2. roF s < 0, let the scalurs y., = J,, = :U,, = 0 and the vectors ~,x = 0, and let x rO = 0, r > m. roF t = 1, . , n, define htiW siht noitinifed we nac evorp the gniwollof :ammel Lemma 2. Let ij he the column rector ’fo eht Jt and let 2: n x p he gioen by n x 1 ’ X10 0 . . . . . 0 x , 0 0 XI0 2u- x , !I10 ... 0 =x .xp- 1 z-,/x. ... ... x10 ,_nf. 2mnu_ J_,X ... ,2 p Then . 1 f(jl$, )oz = pxe2‘”m)2az2( - $ j( - XC$)‘(P - X4) [ Proof yfireV that Ij - x ; j[ = ,lu where x ‘, si the tsrif row of x, dna proceed yb ,noitcudni gnikam esu of (10) dna (11). n A yralloroc of siht tluser si that y = $c?T + ,u where u - ,Y^,(O, 0’1,). tA siht ,tniop we ecudortni noitaton for noitisoporP 1, hcihw si detneserp .woleb eW tel ,B = 0B + ,*X‘*X‘-a ,,@ = Q0 + ,?T‘?J‘-o dna enifed the -cnuf ,0,4(pnoit a’) = ,I#c(Q~~‘“-)’Jc( ,~(Qbn)2a2/l(-[pxe‘“~l)U ,],,x‘-)N hcihw si the roirp ytisned ,j[l,(c(cr 4, ,0 .)’0 roF a nevig eulav of(0, 02), the rettal noitcnuf si detoned asp, (4), dna for a nevig eulav ,+(fo 02), ti si detoned as p,(B). oslA tel ,il = *‘).I/ - I/*X 2II dna d, = abQ(4, O)- ‘x0. ,yllaniF ,,,Olj dna njoR are the naem dna ecnairavoc of the lluf lanoitidnoc noitubirtsid of q,, hcihw are deniatbo morf the snoisrucer (see ,yevraH 1981) nQ = lQ + ,+,?c(,B ,,I - ,),ltljG ,,,R = R,lt + B,(R,+ ,in - R,, ,il)B:, t = n - 1, n - 2. . . . ,O, dna ,B = R+GR,+ ,I~, 0 < t < n - 1, where ‘&t,,s dna R,,, for s < t are the forward retlif setamitse dna R +, I I t si the esorneP-erooM 2.esrevni eW are won ni a noitisop ot tneserp the lluf lanoitidnoc snoitubirtsid that are desu ni the noitalumis for the noisserger ledom htiw ,p(AMRA q) errors. Proposition I. Under Assumptions M, S, I, und P, the jiill corlditional distrihu- tions,for fl, 4, 02, x0, und 0 me gicrn by 6) Plu, ks, 0a _ ,BqA ‘(&/lo + ,)*y‘*x’K ,B ‘), (4 4I.k ti-+ m.x P~(~)xJV~(@~‘(@~~~ + ,)Y’X2mo ,4s1)1,@ (iii) 021y, ,2,_/,1 cto _ J%((vo + n + ,2/)m (6, + d, + d2)/2), (iv) ~YIo~ /II - &AK- hnloR IV( ,Yl~fd ,0-$ )03 cT )0(2p x fi ]2)U(,u)202/1(-Cpxe I=1 x exp[ o($- - flo)‘OO(O - OO)IIS~. Proof (i) dna (iii) wollof morf noitpmussA P dna ammeL ;1 (ii) swollof morf noitpmussA P dna ammeL ;2 (iv) swollof morf noitpmussA P dna the noitinifed of the namlaK gnihtooms ;snoisrucer dna )u( swollof morf the noitinifed of the lluf lanoitidnoc .noitubirtsid n ehT‘ roirp noitubirtsid fo Oxc sretne siht noisserpxe hguorht Role = cov(r,), erehw eht ecnairavoc si taht fo eht roirp .noitubirtsid ehT esorneP-erooM esrevni si deriuqer esuaceb ,,R Il, semoceb ralugnis rof egral .t ,revoeroM ecnis ,lx = ,Y - ,lf;x R,I, si syawla .ralugnis tuB fo erom ecnatropmi si taht ,i,R + 0 as t --* w sihT seilpmi taht ton lla fo eht n snoitavresbo niatnoc noitamrofni tuoba ,a os taht eht retlif nac eb detanimret rof egral hguone .t nI gnissap we noitnem that the smret ,)+(rp pz(0), dna m dna d2 ni( the detrevni ammag )noitubirtsid esira morf the roirp no ~(c hguorht sti ecnedneped no the sretemarap (4,0, 0’). 3.3. /nydementcrtion tiott~s eW evah won nwohs ni noitisoporP 1 that the lluf lanoitidnoc snoitubirtsid of >p c2, a0 are drawrofthgiarts ot ,etupmoc gnoleb ot dradnats seilimaf of -ubirtsid ,snoit dna are ylidaer .detalumis ,yltnedivE the noitautis htiw Q, dna 0 si erom etacirtni ,dna therefore, a short noissergid si ni order. Mftvopolis-Hustings (MH) Algorithm: esoppuS p(z) si a ytisned noitcnuf of a lanoisnemid-itlum 2 that si ot eb .detalumis nA CMCM mhtirogla that secudorp a elpmas of draws morf p( .) proceeds as .swollof esoppuS that ”’Z si the tnerruc draw ni the .niahc oT niatbo the txen draw ,’r+‘(Z tsrif draw a etadidnac ’Z morf a elbatius ytisned ’”Z(q , ,)z hcihw si dellac the -etadidnac gnitareneg .ytisned ehT etadidnac draw si won detcejbus ot a rehtruf -modnar noitazi dna si accepted htiw ytilibaborp ,‘”Z(q/)’Z(p Z’) ,)‘(Z(x 2’) = nim 1 i Z(ql)‘”Z(p I ? ’))i(Z I fI ’Z si ,detcejer iicZ r’ SI set lauqe ot .“(Z sihT process si .detareti tI dluohs eb deton that siht erudecorp does ton eriuqer the gnizilamron tnatsnoc of p(z). roF erom sliated see yenreiT (1993). ,ylraelC lufsseccus noitatnemelpmi of the MH ,mhtirogla htiw a hgih -tpecca ecna rate of etadidnac draws, seriuqer a elbatius gnitareneg-etadidnac .ytisned ,yletanutroF hcus seitisned are elbaliava for htob d, dna 0. esoppuS we tel q(c,b(“, 4) eb the ytisned of J$(@; 1(@oq50+ CTe 2x’y), @, 1)ls4, dna tel 4’ eb a draw morf siht 3.noitubirtsid nehT the sgnitsaH-siloporteM step stnuoma ot na noitcejer--ecnatpecca of 4’ htiw ytilibaborp roF 0, a elbatius q( -, -) ytisned si the detacnurt lamron noitamixorppa ot ,~_lO(rr ,Bk Q) nevig yb q(Qs, p, 4, f12, )+I( = q(0) x exp [ - i lC[ - ’])+O(m -)+O(V ’ O[ - m(O+)]] ,,,sI (13) A‘ tneinevnoc ygetarts si ot elpmas eht detacnurtnu lamron dna niater eht gniward fi ti seil ni .,S sihT ygetarts yam eb tneicilfeni fi eht ssam fo eht roiretsop si ton detartnecnoc revo eht yranoitats ,noiger hcihw yam osla etacidni taht eht ledom si .deificepssim