ARCH MODELS” - Duke University

80 Pages·1999·4.69 MB·English

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