Chapter 49 ARCH MODELS” TIM VELSRELLOB Northwestern University and N.B.E.R. TREBOR .F ELGNE University of California, San Diego and N.B.E.R. LEINAD .B NOSLEN University of Chicago and N.B.E.R. Contents tcartsbA 2961 1. noitcudortnI 2961 .1.1 snoitinifeD 1692 .2.1 laciripmE seitiraluger fo tessa snruter 3692 .3.1 etairavinU cirtemarap sledom 7692 .4.1 HCRA ni naem sledom 2192 .5.1 cirtemarapnoN dna cirtemarapimes sdohtem 2192 2. ecnerefnI serudecorp 2974 .1.2 gnitseT rof HCRA 4192 .2.2 mumixaM doohilekil sdohtem 1792 .3.2 mumixam-isauQ doohilekil sdohtem 3892 .4.2 noitacificepS skcehc 4892 ehT“ srohtua dluow ekil ot knaht nebroT .G ,nesrednA kcirtaP ,yelsgnilliB mailliW .A ,kcorB cirE ,slesyhG sraL .P ,nesnaH werdnA ,yevraH ekalB ,noraBeL dna oehT ro fnamjiN lufpleh .stnemmoc laicnaniF troppus morf eht lanoitaN ecneicS noitadnuoF rednu stnarg 7082209-SES ,)velsrelloB( -SES 6502219 ,)elgnE( dna 1310119-SES dna 3860139-SES ,)nosleN( dna morf eht retneC rof hcraeseR ni secir PytiruceS ,)nosleN( si yllufetarg .degdelwonkca seiriuqnI gnidrager eht atad rof eht kcots tekram laciripme noitacilppa dluohs eb desserdda ot rosseforP .G mailliW ,trewhcS etaudarG loohcS fo ,tnemeganaM ytisrevinU fo ,retsehco R,retsehcoR NY ,72641 .ASU ehT MTSSUAG edoc desu ni eht kcots tekram laciripme elpmaxe si elbaliava morf ytisrevinU-retnI muitrosnoC rof lacitiloP dna laicoS hcraeseR ,)RSPCI( .O.P xoB ,8421 nnA ,robrA MI ,60184 ,ASU enohpelet .0105-367)313( redrO ssalC“ ”5 rednu siht s’elcitra .eman Handbook ofEconometrics, Volume IV, Edited by R.F. Engle and D.L. McFadden 0 1994 Elseuier Science B.V. All rights reserved T. Bollersku et al. 2989 3. yranoitatS dna cidogre seitreporp 9892 .1.3 tcirtS ytiranoitats 0992 .2.3 ecnetsisreP 4. suounitnoC emit sdohtem 2992 .1.4 HCRA sledom sa snoitamixorppa ot snoisuffid 4992 .2.4 snoisuffiD sa snoitamixorppa ot HCRA sledom 6992 ,3.4 HCRA sledom sa sretlif dna sretsacerof 1992 2999 .5 noitagerggA dna gnitsacerof .1.5 laropmeT noitagergga 9992 .2.5 tsaceroF rorre snoitubirtsid 1003 6. etairavitluM snoitacificeps 3002 .1.6 rotceV HCRA dna lanogaid HCRA 3003 .2.6 rotcaF HCRA 5003 .3.6 tnatsnoC lanoitidnoc snoitalerroc 7003 .4.6 etairaviB HCRAGE 8003 .5.6 ytiranoitatS dna ecnetsisrep-oc 9003 7. ledoM noitceles 3010 8. evitanretlA serusaem for ytilitalov 3012 9. laciripmE selpmaxe 3014 .1.9 .S.U kramhcstueD/ralloD egnahcxe setar 4103 .2.9 .S.U kcots secirp 7103 10. noisulcnoC 3030 secnerefeR 3031 .hC 49: ARCH Models 2961 Abstract sihT chapter setaulave the tsom tnatropmi laciteroeht stnempoleved ni HCRA epyt gniledom of gniyrav-emit lanoitidnoc .secnairav ehT egarevoc edulcni the -acificeps noit of etairavinu cirtemarap HCRA ,sledom lareneg ecnerefni ,serudecorp -idnoc snoit for ytiranoitats dna ,yticidogre suounitnoc emit ,sdohtem noitagergga dna gnitsacerof of HCRA ,sledom etairavitlum lanoitidnoc ecnairavoc ,snoitalumrof dna the esu of ledom noitceles airetirc ni na HCRA .txetnoc ,yllanoitiddA the chapter sniatnoc a noissucsid of the laciripme seitiraluger gniniatrep ot the laropmet noitairav ni laicnanif tekram .ytilitalov detavitoM ni part yb tnecer stluser no lamitpo ,gniretlif a wen lanoitidnoc ecnairav ledom for retteb gniziretcarahc stock nruter ytilitalov si osla .detneserp 1. Introduction litnU a decade ago the sucof of tsom cirtemonoceorcam dna laicnanif emit seires gniledom deretnec no the lanoitidnoc tsrif ,stnemom htiw yna laropmet -neped seicned ni the rehgih order stnemom treated as a .ecnasiun ehT desaercni ecnatropmi deyalp yb ksir dna ytniatrecnu snoitaredisnoc ni nredom cimonoce ,yroeht ,revewoh has detatissecen the tnempoleved of wen cirtemonoce emit seires seuqinhcet that wolla for the gniledom of emit gniyrav secnairav dna .secnairavoc neviG the tnerappa kcal of yna larutcurts cimanyd cimonoce yroeht gninialpxe the noitairav ni rehgih order ,stnemom ylralucitrap latnemurtsni ni siht tnempoleved has neeb the evissergerotua lanoitidnoc citsadeksoreteh )HCRA( ssalc of sledom decudortni yb elgnE (1982). lellaraP ot the sseccus of dradnats raenil emit seires ,sledom gnisira morf the esu of the lanoitidnoc susrev the lanoitidnocnu ,naem the yek thgisni offered yb the HCRA ledom seil ni the noitcnitsid neewteb the conditional dna the unconditional dnoces order .stnemom elihW the lanoitidnocnu ecnairavoc xirtam for the selbairav of tseretni yam eb emit ,tnairavni the lanoitidnoc secnairav dna secnairavoc netfo dneped yllaivirt-non no the past states of the .dlrow gnidnatsrednU the exact erutan of siht laropmet ecnedneped si yllaicurc tnatropmi for ynam seussi ni scimonoceorcam dna ,ecnanif hcus as elbisreverri ,stnemtsevni noitpo ,gnicirp the mret erutcurts of tseretni rates, dna lareneg cimanyd asset gnicirp .spihsnoitaler ,oslA morf the evitcepsrep of cirtemonoce ,ecnerefni the ssol ni citotpmysa ycneiciffe morf detcelgen yticitsadeksoreteh yam eb ylirartibra egral ,dna nehw gnitaulave cimonoce forecasts, a hcum erom etarucca etamitse of the forecast error ytniatrecnu si yllareneg elbaliava yb gninoitidnoc no the tnerruc noitamrofni set. 1 .I. snoitinijeD Let })O(,E{ etoned a etercsid emit citsahcots process htiw lanoitidnoc naem dna ecnairav snoitcnuf dezirtemarap yb the etinif lanoisnemid rotcev OE 0 s ,”R where 2962 T. Bolfersleu et al. 8, setoned eht eurt .eulav roF lanoitaton yticilpmis ew shall yllaitini assume that )O(,s is a ,ralacs htiw eht suoivbo snoisnetxe ot a etairavitlum krowemarf detaert ni noitceS 6. ,oslA tel _,E ).(r etoned eht lacitamehtam ,noitatcepxe lanoitidnoc no eht past, fo eht ,ssecorp gnola htiw yna rehto noitamrofni elbaliava at emit t - 1. ehT }),@(,E{ ssecorp is neht denifed ot wollof an HCRA ledom fi eht lanoitidnoc naem slauqe ,orez ~1-1MRJ)) = 0 t= 1,2,..., )1.1( tub eht lanoitidnoc .ecnairav 44J = -,raV lWo)) = L ))&(:~(l t= 1,2,..., )2.1( sdneped yllaivirt-non no eht dleif-amgis detareneg yb eht past ;snoitavresbo .e.i hW.}...,)0e(2-,~r)O~(l-t~{ ne suoivbo morf eht ,txetnoc eht ticilpxe ecnedneped no eht ,sretemarap 8, lliw eb desserppus rof lanoitaton .ecneinevnoc ,oslA ni eht etairavitlum esac eht gnidnopserroc emit gniyrav lanoitidnoc ecnairavoc xirtam lliw eb detoned yb .,2f nI hcum fo eht tneuqesbus noissucsid ew shall sucof yltcerid no eht }ts{ ,ssecorp tub eht emas saedi dnetxe yltcerid ot eht noitautis ni hcihw }ts{ sdnopserroc ot eht snoitavonni morf emos erom etarobale cirtemonoce .ledom nI ,ralucitrap tel }),O(ly{ etoned eht citsahcots ssecorp fo tseretni htiw lanoitidhoc naem MwtP = 4 - JY(l t=l2 ) ....) )3.1( ,etoN yb eht gnimit noitnevnoc htob ~~(0,) and ),O(:a era elbarusaem htiw tcepser ot eht emit t - 1 noitamrofni .tes enifeD eht }),e(,s{ ssecorp yb au de,) = ,Y - t= 1,2,.... )4.1( ehT lanoitidnoc ecnairav rof }tc( neht slauqe eht lanoitidnoc ecnairav rof eht },y{ .ssecorp ecniS yrev wef cimonoce and laicnanif emit seires evah a tnatsnoc lanoitidnoc naem fo ,orez tsom fo eht laciripme snoitacilppa fo eht HCRA ygolodohtem yllautca llaf nihtiw this .krowemarf gninruteR ot eht snoitinifed ni snoitauqe )1.1( and ,)2.1( ti swollof that eht dezidradnats ,ssecorp 2,(e,) s E,(e,)a:(e,)- 2’1 t= 1,2,..., )5.1( lliw evah lanoitidnoc naem ,orez and a emit tnairavni lanoitidnoc ecnairav fo .ytinu sihT noitavresbo smrof eht basis rof tsom fo eht ecnerefni serudecorp that eilrednu eht snoitacilppa fo HCRA epyt .sledom fI eht lanoitidnoc noitubirtsid rof ,z is eromrehtruf assumed ot eb emit tnairavni Ch. 49: ARCH Models 2963 htiw a etinif htruof ,tnemom ti swollof yb s’nesneJ ytilauqeni that );&(E = E(zf’)E(a;) 2 2)fa(E);z(E = E(zp)E(q2, where the ytilauqe sdloh eurt for a tnatsnoc lanoitidnoc ecnairav .ylno neviG a lamron noitubirtsid for the dezidradnats snoitavonni ni noitauqe (1.5), the -idnocnu lanoit noitubirtsid for E, si therefore .citrukotpel ehT putes ni snoitauqe (1.1) hguorht (1.4) si ylemertxe lareneg dna does ton dnel flesti yltcerid ot laciripme noitatnemelpmi tuohtiw tsrif gnisopmi rehtruf snoitcirtser no the laropmet seicnedneped ni the lanoitidnoc naem dna ecnairav .snoitcnuf woleB we llahs ssucsid emos of the tsom lacitcarp dna ralupop hcus HCRA -alumrof snoit for the lanoitidnoc .ecnairav elihW the tsrif laciripme snoitacilppa of the HCRA ssalc of sledom were denrecnoc htiw gniledom yranoitalfni ,ytniatrecnu the ygolodohtem has yltneuqesbus dnuof yllaicepse ediw esu ni gnirutpac the laropmet seicnedneped ni asset .snruter roF a tnecer yevrus of siht evisnetxe laciripme erutaretil we refer ot velsrelloB et .la (1992). 1.2. Empirical regularities fo asset returns nevE ni the etairavinu case, the yarra of lanoitcnuf smrof dettimrep yb noitauqe (1.2) si ,tsav dna yletinifni regral naht nac eb detadommocca yb yna cirtemarap ylimaf of HCRA .sledom ,ylraelC ot evah yna hope of gnitceles na etairporppa HCRA ,ledom we tsum evah a good aedi of what laciripme seitiraluger the ledom dluohs .erutpac ,suhT a feirb noissucsid of emos of the tnatropmi seitiraluger for asset snruter ytilitalov .swollof 1.2.1. Thick sliat tessA snruter dnet ot eb .citrukotpel ehT noitatnemucod of siht laciripme ytiraluger yb torblednaM (1963), amaF (1965) dna others del ot a egral erutaretil no gniledom stock snruter as .d.i.i draws morf deliat-kciht ;snoitubirtsid see, e.g. torblednaM (1963), amaF (1963,1965), kralC (1973) dna grebttalB dna sedenoG (1974). 1.2.2. Volatility clustering sA torblednaM (1963) wrote, . . . egral segnahc dnet ot eb dewollof yb egral ,segnahc of rehtie ,ngis dna llams segnahc dnet ot eb dewollof yb llams ..segnahc sihT ytilitalov gniretsulc nonemonehp si yletaidemmi tnerappa nehw asset snruter are dettolp hguorht .emit oT ,etartsulli erugiF 1 stolp the yliad latipac sniag no the dradnatS 90 etisopmoc stock xedni morf 2591-8291 denibmoc htiw dradnatS dna yliaD dradnatS dna s’rooP Capital sniaG 2 ’ 0291 0391 0491 0591 0691 0791 0891 0991 2000 Figure 1 Poor’s 500 xedni morf .0991-3591 ehT snruter are expressed ni ,tnecrep dna are ylsuounitnoc .dednuopmoc tI si raelc morf lausiv noitcepsni of the ,erugif dna yna elbanosaer lacitsitats test, that the snruter are ton .d.i.i hguorht .emit roF ,elpmaxe ytilitalov was ylraelc rehgih gnirud the 1930’s naht gnirud the 1960’s, as demrifnoc yb the noitamitse stluser reported ni hcnerF et .la (1987). A ralimis egassem si deniatnoc ni erugiF 2, hcihw stolp the yliad egatnecrep .S.U/kramhcstueD ralloD egnahcxe rate .noitaicerppa tcnitsiD sdoirep of egnahcxe tekram ecnelubrut dna ytiliuqnart are yletaidemmi .tnedive eW llahs nruter ot a lamrof sisylana of htob of these two emit seires ni noitceS 9 .woleb ytilitaloV gniretsulc dna kciht deliat snruter are yletamitni .detaler sA deton ni noitceS 1.1 ,evoba fi the lanoitidnocnu sisotruk of a, si ,etinif ~])$(E[/)~E(E 3 ,):z(E where the tsal ytilauqeni si tcirts sselnu to si .tnatsnoc Excess sisotruk ni E, nac therefore esira morf ssenmodnar ni ,lo morf excess sisotruk ni the lanoitidnoc noitubirtsid of ,Is ,.e.i ni ,lz or morf .htob 1.2.3. Leverage eflects ehT dellac-os egarevel“ effect,” tsrif deton yb kcalB (1976), refers ot the ycnednet for segnahc ni stock secirp ot eb ylevitagen detalerroc htiw segnahc ni stock .ytilitalov dexiF costs hcus as laicnanif dna gnitarepo egarevel edivorp a laitrap noitanalpxe for siht .nonemonehp A mrif htiw tbed dna ytiuqe gnidnatstuo yllacipyt .hC 49: ARCH Models 2965 yllaD .S.U kramhcstueD-ralloD noitaicerppA “p 0691 2691 4691 6691 1988 1990 1992 4991 erugiF 2 semoceb erom ylhgih degarevel nehw the eulav of the mrif .sllaf sihT sesiar ytiuqe snruter ytilitalov fi the snruter no the mrif as a elohw are .tnatsnoc kcalB (1976), ,revewoh deugra that the esnopser of stock ytilitalov ot the noitcerid of snruter si too egral ot eb denialpxe yb egarevel .enola sihT noisulcnoc si osla detroppus yb the laciripme work of eitsirhC (1982) dna Schwert .)b9891( 1.2.4. Non-trading periods noitamrofnI that setalumucca nehw laicnanif stekram are desolc si detcelfer ni secirp after the stekram .nepoer ,fI for ,elpmaxe noitamrofni setalumucca at a tnatsnoc rate revo radnelac ,emit neht the ecnairav of snruter revo the doirep morf the yadirF esolc ot the yadnoM esolc dluohs eb three semit the ecnairav morf the yadnoM esolc ot the yadseuT .esolc amaF (1965) dna hcnerF dna lloR (1986) evah ,dnuof ,revewoh that noitamrofni setalumucca erom ylwols nehw the stekram are desolc naht nehw yeht are .nepo secnairaV are rehgih gniwollof sdnekeew dna syadiloh naht no other ,syad tub ton ylraen yb as hcum as dluow eb expected fi the swen lavirra rate were .tnatsnoc roF ,ecnatsni gnisu data no yliad snruter across lla NYSE dna XEMA stocks morf ,2891-3691 hcnerF dna lloR (1986) dnif that ytilitalov si 70 semit rehgih per ruoh no egareva nehw the tekram si nepo naht nehw ti si .desolc eilliaB dna velsrelloB (1989) report ylevitatilauq ralimis stluser for ngierof egnahcxe rates. 1.2.5. Forecastable events Not ,ylgnisirprus elbatsacerof sesaeler of tnatropmi noitamrofni are detaicossa htiw hgih ex etna .ytilitalov roF ,elpmaxe llenroC )8791( dna Pate11 dna nosfloW (1979, 6692 T Bollersleu te al. 1981) show that laudividni ’smrif stock snruter ytilitalov si hgih dnuora sgninrae .stnemecnuonna ,ylralimiS yevraH dna gnauH (1991,1992) dnif that dexif emocni dna ngierof egnahcxe ytilitalov si rehgih gnirud sdoirep of yvaeh gnidart yb lartnec sknab or nehw cimonoceorcam swen si gnieb .desaeler erehT are osla tnatropmi elbatciderp segnahc ni ytilitalov across the gnidart .yad roF ,elpmaxe ytilitalov si yllacipyt hcum rehgih at the nepo dna esolc of stock dna ngierof egnahcxe gnidart naht gnirud the elddim of the .yad sihT nrettap has neeb detnemucod yb sirraH (1986), ytireG dna nirehluM (1992) dna eilliaB dna velsrelloB (1991) gnoma others. ehT esaercni ni ytilitalov at the nepo at tsael yltrap stcelfer noitamrofni detalumucca elihw the tekram was .desolc ehT ytilitalov egrus at the esolc si ssel ylisae .deterpretni 1.2.6. Volatility and serial correlation noraBeL (1992) sdnif a gnorts esrevni noitaler neewteb ytilitalov dna laires -erroc noital for .S.U stock .secidni sihT gnidnif appears ylbakramer tsubor ot the eciohc of elpmas ,doirep tekram ,xedni tnemerusaem lavretni dna ytilitalov .erusaem miK (1989) stnemucod a ralimis pihsnoitaler ni ngierof egnahcxe rate data. 1.2.7. Co-movements ni volatilities kcalB (1976) devresbo that . . . there si a tol of ytilanommoc ni ytilitalov segnahc across :skcots a 1% tekram ytilitalov egnahc yllacipyt seilpmi a 1% ytilitalov egnahc for each stock. ,lleW perhaps the hgih ytilitalov stocks are tahwemos erom evitisnes ot tekram ytilitalov segnahc naht the wol ytilitalov stocks. nI lareneg ti smees riaf ot yas that nehw stock seitilitalov ,egnahc yeht lla dnet ot egnahc ni the emas .noitcerid dlobeiD dna evolreN (1989) dna yevraH et .la (1992) osla eugra for the ecnetsixe of a few nommoc factors gninialpxe egnahcxe rate ytilitalov .stnemevom elgnE et .la )b0991( show that .S.U dnob ytilitalov segnahc are ylesolc deknil across .seitirutam sihT ytilanommoc of ytilitalov segnahc sdloh ton ylno across assets nihtiw a ,tekram tub osla across tnereffid .stekram roF ,elpmaxe Schwert (1989a) sdnif that .S.U stock dna dnob seitilitalov evom together, elihw elgnE dna lemsuS (1993) dna oamaH et .la (1990) revocsid esolc sknil neewteb ytilitalov segnahc across lanoitanretni stock .stekram ehT ecnatropmi of lanoitanretni segaknil has neeb rehtruf derolpxe yb gniK et .la (1994), elgnE et .la (1990a), dna niL et .la (1994). tahT seitilitalov evom together dluohs eb gnigaruocne ot ledom ,sredliub ecnis ti setacidni that a few nommoc factors yam nialpxe hcum of the laropmet noitairav ni the lanoitidnoc secnairav dna secnairavoc of asset .snruter sihT smrof the sisab for the factor HCRA sledom dessucsid ni noitceS 6.2 .woleb Ch. 49: ARCH Models 2961 I .2.8. Macroeconomic variables and volatility ecniS stock seulav are ylesolc deit ot the htlaeh of the ,ymonoce ti si larutan ot expect that serusaem of cimonoceorcam ytniatrecnu hcus as the lanoitidnoc secnairav of lairtsudni ,noitcudorp tseretni rates, yenom growth, etc. dluohs pleh nialpxe segnahc ni stock tekram .ytilitalov Schwert (1989a, )b sdnif that hguohtla stock ytilitalov sesir ylprahs gnirud snoissecer dna laicnanif sesirc dna drops gnirud ,snoisnapxe the noitaler neewteb cimonoceorcam ytniatrecnu dna stock ytilitalov si ylgnisirprus weak. netsolG et .la (1993), no the other ,dnah revocnu a gnorts evitisop pihsnoitaler neewteb stock nruter ytilitalov dna tseretni rates. 1.3. Univariate parametric models 1.3.1. GARCH suoremuN cirtemarap snoitacificeps for the emit gniyrav lanoitidnoc ecnairav evah neeb proposed ni the .erutaretil nI the raenil )q(HCRA ledom yllanigiro decudortni yb elgnE (1982), the lanoitidnoc ecnairav si detalutsop ot eb a raenil noitcnuf of the past q derauqs ,snoitavonni +w=;o 1 CLiE~_i-W+C((L)E:_l, )6.1( i=l,q where L setoned the gal or tfihskcab operator, ,y’L = .~_,Y Of ,esruoc for siht ledom ot eb llew denifed dna the lanoitidnoc ecnairav ot eb ,evitisop tsomla ylerus the sretemarap tsum yfsitas w > 0 dna ~(c 3 0,. . . , a, > 0. gninifeD ,v = :E - ,:a the )q(HCRA ledom ni (1.6) yam eb nettirw-er as Ef = w + C((L)&f_ 1 + .,v (1.7) ecniS E, _ I ),v( = 0, the ledom sdnopserroc yltcerid ot na )q(RA ledom for the derauqs ,snoitavonni .:E ehT process si ecnairavoc yranoitats fi dna ylno fi the mus of the evitisop evissergerotua sretemarap si ssel naht ,eno ni hcihw case the lanoitidnocnu ecnairav slauqe Var(s,) = a2 = o/( 1 - 1U - ... - .)qU nevE hguoht the E,)S are yllaires ,detalerrocnu yeht are ylraelc ton tnednepedni hguorht .emit nI ecnadrocca htiw the dezilyts facts for asset snruter dessucsid ,evoba there si a ycnednet for egral )llams( etulosba seulav of the process ot eb dewollof yb other egral )llams( seulav of elbatciderpnu .ngis ,oslA as deton ,evoba fi the -irtsid noitub for the dezidradnats snoitavonni ni noitauqe (1.5) si demussa ot eb emit ,tnairavni the lanoitidnocnu noitubirtsid for E, lliw evah fatter sliat naht the -ubirtsid noit for z,. roF ,ecnatsni for the )l(HCRA ledom htiw yllanoitidnoc yllamron detubirtsid errors, E(sp)/E($)’ = 3( 1 - (/):a 1 - 34) fi T(c3 < 1, dna ~):E(E/)$.(E = co :esiwrehto htob of hcihw exceed the lamron eulav of three. 2968 T uelsrelloB et .la ylevitanretlA the )q(HCRA ledom yam osla eb detneserper as a emit gniyrav retemarap )q(AM ledom for a,, ,E = 0 + cc(J!& _,El 1, )8.1( where },i{ setoned a ralacs .d.i.i citsahcots process htiw naem zero dna ecnairav .eno emiT gniyrav retemarap sledom evah a gnol yrotsih ni scirtemonoce dna .scitsitats ehT laeppa of the lanoitavresbo tnelaviuqe noitalumrof ni noitauqe (1.6) smets morf the ticilpxe sucof no the emit gniyrav lanoitidnoc ecnairav of the process. roF noissucsid of siht noitaterpretni of HCRA ,sledom see, e.g., yasT (1987), Bera et .la (1993) dna Bera dna Lee (1993). nI laciripme snoitacilppa of )q(HCRA sledom a gnol gal htgnel dna a egral rebmun of sretemarap are netfo dellac for. oT tnevmucric siht melborp velsrelloB (1986) proposed the dezilareneg ,HCRA or ,p(HCRAG ,)q ,ledom :~ = 0 + C C(iEf_i + C W + _:&)L(a _:a)L(B (1.9) ~jjb:_j ~ 1 + 1. i=l,q j= 1.~ roF the lanoitidnoc ecnairav ni the ,p(HCRAG q) ledom ot eb llew denifed lla the stneiciffeoc ni the gnidnopserroc etinifni order raenil HCRA ledom tsum eb .evitisop dedivorP that a(L) dna /J(L) evah on nommoc roots dna that the roots of the laimonylop )x(I/ = 1 eil edistuo the tinu ,elcric siht ytivitisop tniartsnoc si deifsitas fi dna ylno fi lla the stneiciffeoc ni the etinifni power seires noisnapxe for (/)~(a 1 - ))x(i/ are .evitagen-non yrasseceN dna tneiciffus snoitidnoc for siht are nevig ni nosleN dna Cao (1992). roF the elpmis ,l(HCRAG 1) ledom tsomla erus ytivitisop of :0 seriuqer that w 3 0, i(c B 0 dna 1I/ > 0. gnignarraeR the ,p(HCRAG q) ledom as ni noitauqe (1.7), ti swollof that ;a = 0 + [a(L) + B(L)l&:1_ - P(L)v,-1 + v,, (1.10) hcihw senifed na ]p,)q,p(xam[AMRA ledom for .:E yB dradnats ,stnemugra the ledom si ecnairavoc yranoitats fi dna ylno fi lla the roots of a(x) + )x(b = 1 eil edistuo the tinu ;elcric see velsrelloB (1986) for a lamrof proof. nI ynam snoitacilppa htiw hgih ycneuqerf laicnanif data the etamitse for (IC 1) + (I/ 1) snrut tuo ot eb yrev esolc ot .ytinu sihT sedivorp na laciripme noitavitom for the dellac-os detargetni HCRAG (p, ,)q or ,)q,p(HCRAGI ledom decudortni yb elgnE dna velsrelloB (1986). nI the HCRAGI ssalc of sledom the evissergerotua laimonylop ni noitauqe (1.10) has a tinu root, dna yltneuqesnoc a shock ot the lanoitidnoc ecnairav si tnetsisrep ni the esnes that ti sniamer tnatropmi for erutuf forecasts of lla .snoziroh rehtruF noissucsid of ytiranoitats snoitidnoc dna seussi of ecnetsisrep are deniatnoc ni noitceS 3 .woleb tsuJ as na AMRA ledom netfo sdael ot a erom suoinomisrap noitatneserper of the laropmet seicnedneped ni the lanoitidnoc naem naht na RA ,ledom the HCRAG (p, q) noitalumrof ni noitauqe (1.9) sedivorp a ralimis added ytilibixelf revo the raenil
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