ARCH modeling in finance

55 Pages·2001·4.05 MB·English

lanruoJ fo scirtemonocE 25 )2991( .95-5 dnalloH-htroN
HCRA gniledom ni *ecnanif
A weiver of eht yroeht dna laciripme ecnedive
miT velsrelloB
Northwestern ,ytiswrinU Er,anston, IL 60208, USA
yaR Y. uohC
Georgia Institute of Technoloa, Atlanta, GA 30332, USA
htenneK .F renorK
University of Arizona, Tucson, AZ 85721, USA
hguohtlA ytilitalov gniretsulc sah a gnol yrotsih sa a tneilas laciripme ytiraluger gniziretcarahc
ycneuqerf-hgih evitaluceps ,secirp ti saw ton litnu yltnecer taht deilppa srehcraeser ni ecnanif
evah dezingocer eht ecnatropmi fo ylticilpxe gniledom gniyrav-emit redro-dnoces .stnemom
latnemurtsnI ni tsom fo eseht laciripme seiduts sah neeb eht evissergerotuA lanoitidnoC
yticitsadeksoreteH )HCRA( ledom decudortni yb elgnE .)2891( sihT repap sniatnoc na weivrevo
fo emos fo eht stnempoleved ni eht noitalumrof fo HCRA sledom dna a yevrus fo eht suoremun
laciripme snoitacilppa gnisu laicnanif .atad lareveS snoitseggus rof erutuf ,hcraeser gnidulcni eht
noitatnemelpmi dna stset fo gnitepmoc tessa gnicirp ,seiroeht tekram erutcurtsorcim ,sledom
noitamrofni noissimsnart ,smsinahcem cimanyd gnigdeh ,seigetarts dna eht gnicirp fo evitavired
,stessa era osla .dessucsid
nA* reilrae noisrev fo siht repap yb T. ,velsrelloB .R ,uohC .N namarayaJ dna .K renorK saw
:deltitne HCRA‘ gniledoM ni :ecnaniF A evitceleS weiveR fo eht yroehT dna laciripmE
,ecnedivE htiw snoitsegguS rof erutuF .’hcraeseR eW dluow ekil ot knaht ruo seugaelloc ohw
depleh ylppus eht secnerefer detic ni siht .yevrus gnomA ynam ,srehto ew dluow yllaicepse ekil
ot knaht zuB ,kcorB nhoJ ,llebpmaC yaR ,oranneGeD knarF ,dlobeiD boR ,elgnE nitraM ,snavE
sakiG ivaR ,nahtannagaJ nanayaraN ,namarayaJ .J notsuH ,hcolluCcM moT
,ydruCcM naD ,nosleN nairdA .nagaP reteP ,nosniboR lliB ,trewhcS nehpetS ,rolyaT -icitrap
stnap ta eht ecnerefnoC no lacitsitatS sledoM fo laicnaniF ytilitaloV ta DSCU no lirpA ,7-6
,0991 dna na suomynona eerefer rof yrev lufpleh dna deliated stnemmoc no na reilrae .tfard miT
,velsrelloB yaR ,uohC dna neK renorK dluow ekil ot egdelwonkca laicnanif troppus morf FSN
,70822-09SES# eht aigroeG hceT ,noitadnuoF dna eht lraK rellE retneC ta eht ytisrevinU fo
,anozirA .ylevitcepser
000.50$/29/6704-4030 reiveslE-2991 ecneicS srehsilbuP .V.B llA sthgir devreser
6 T. Bollerslec et al., ARCH modeling in finance
1. noitcudortnI
ytniatrecnU si lartnec ot hcum of nredom ecnanif .yroeht gnidroccA ot
tsom asset gnicirp seiroeht the ksir muimerp si denimreted yb the ecnairavoc
neewteb the erutuf nruter no the asset dna eno or erom kramhcneb -oftrop
;soil e.g., the tekram oiloftrop or the growth rate ni .noitpmusnoc nI noitpo
gnicirp the ytniatrecnu detaicossa htiw the erutuf ecirp of the gniylrednu
asset si the tsom tnatropmi tnanimreted ni the gnicirp .noitcnuf ehT -noc
noitcurts of hedge soiloftrop si rehtona elpmaxe where the lanoitidnoc erutuf
secnairav dna secnairavoc gnoma the tnereffid assets devlovni yalp na -ropmi
tnat .elor
elihW ti has neeb dezingocer for etiuq emos emit that the ytniatrecnu of
evitaluceps ,secirp as derusaem yb the secnairav dna ,secnairavoc are -gnahc
gni hguorht emit [see, e.g., torblednaM (1963) dna amaF (1965)1, ti was ton
litnu yltnecer that deilppa researchers ni laicnanif dna yratenom scimonoce
evah started ylticilpxe gniledom emit noitairav ni -dnoces or redro-rehgih
.stnemom enO of the tsom tnenimorp sloot that has degreme for -ziretcarahc
gni hcus gnignahc secnairav si the evissergerotuA lanoitidnoC -sadeksoreteH
yticit )HCRA( ledom of elgnE (1982) dna sti suoirav .snoisnetxe ecniS the
noitcudortni of the HCRA ledom lareves derdnuh research papers gniylppa
siht gniledom ygetarts ot laicnanif emit seires data evah ydaerla appeared. nI
siht paper we yevrus those snoitubirtnoc that we redisnoc ot eb the tsom
tnatropmi dna gnisimorp ni the noitalumrof of epyt-HCRA sledom dna rieht
snoitacilppa ni the gniledom of evitaluceps .secirp lareveS gnitseretni scipot
ni laciripme ecnanif gnitiawa erutuf research are osla .dessucsid
ehT nalp of siht paper si as .swollof eW nigeb ni noitces 2 htiw a feirb
weivrevo of emos of the tnatropmi laciteroeht stnempoleved ni the -emarap
noitaziret dna noitatnemelpmi of epyt-HCRA ,sledom dna eunitnoc ni -ces
noit 3 htiw snoitacilppa of the HCRA ygolodohtem ot stock nruter data.
snoitceS 4 dna 5 revoc the gniledom of tseretni rates dna ngierof egnahcxe
rates, .ylevitcepser A deliated yhpargoilbib si nevig at the dne of the paper.
2. HCRA
gniwolloF the lanimes paper yb elgnE (1982) we llahs refer ot lla etercsid
emit citsahcots processes 1~~( of the mrof
)1(
,E = ;OtZ
7
z, ,.d.i.i E( )lz = 0, (rav zr) = 1,
htiw a, a ,gniyrav-emit ,evitisop dna elbarusaem noitcnuf of the emit t - 1
noitamrofni set, as na HCRA .ledom roF won E, si demussa ot eb a
.T Bollersle~ et ,.la ARCH modeling in finance 7
etairavinu process, tub snoisnetxe ot etairavitlum sgnittes are drawrofthgiarts
as dessucsid .woleb yB noitinifed E, si yllaires detalerrocnu htiw naem zero,
tub the lanoitidnoc ecnairav of ~F slauqe ,2tu hcihw yam eb gnignahc hguorht
.emit nI tsom snoitacilppa E, lliw dnopserroc ot the noitavonni ni the naem
for emos other citsahcots process, yas ,},y{ where
)b;,_,x(g=,y ,rs+ )3(
dna g(x,_ ;, )b setoned a noitcnuf of x,_ 1 dna the retemarap rotcev b, where
x,_ 1 si ni the emit t - 1 noitamrofni set. oT yfilpmis the ,noitisopxe ni tsom
of the noissucsid woleb we llahs emussa that t& si flesti .elbavresbo
Let f(z,) etoned the ytisned noitcnuf for z,, dna tel 0 eb the rotcev of lla
the nwonknu sretemarap ni the .ledom yB the noitciderp error -isopmoced
,noit the doohilekil-gol noitcnuf for the elpmas Ed, e7._,, . . . , F, ,semoceb
apart morf laitini ’,snoitidnoc
L(6) = :i )&+gol[ - .],tUgol )4(
l=t
ehT dnoces mret ni the noitammus si a naibocaJ mret gnisira morf the
noitamrofsnart morf z, ot .,F Note that (4) osla senifed the elpmas -ilekil-gol
hood for ,7y _,y , , . . . , ,y as nevig yb (3). neviG a cirtemarap noitatneserper
for f(z,), mumixam doohilekil setamitse for the sretemarap of tseretni nac eb
detupmoc yltcerid morf (4) yb a rebmun of tnereffid laciremun noitazimitpo
.seuqinhcet
ehT putes ni eqs. (1) dna (2) si ylemertxe lareneg dna swolla for a ediw
yteirav of .sledom tA the emas ,emit the cimonoce yroeht gninialpxe -opmet
lar noitairav ni lanoitidnoc secnairav si yrev .detimil ,yltneuqesnoC ni the
redniamer of siht noitces we llahs etartnecnoc no emos of the erom -sseccus
luf emit seires seuqinhcet that evah neeb depoleved for gniledom at2. esehT
sledom for the laropmet ecnedneped ni lanoitidnoc sdnoces stnemom raeb
hcum ecnalbmeser ot the emit seires seuqinhcet for lanoitidnoc tsrif stnemom
deziralupop ni the ylrae .seitneves tsuJ as the noitargetni of emit seires
seuqinhcet for the lanoitidnoc naem otni larutcurts cirtemonoce ledom
gnidliub has del ot a hcum deeper dna rehcir gnidnatsrednu of the gniylrednu
,scimanyd ralimis stluser evah ydaerla started ot egreme ni the gniledom of
lanoitidnoc secnairav dna .secnairavoc
‘Throughout this ,repap the ecnedneped of E( dna ,a on the retemarap rotcev 6 era
desserppus for notational .ecneinevnoc
8 T. Bollerslec et al., ARCH modeling in finance
2.1. The linear ARCH(q) model
sA elgnE (1982) stseggus ni sih lanimes paper, eno elbissop -aziretemarap
noit for at2 si ot express 2tu as a raenil noitcnuf of past derauqs seulav of the
process,
4
ta +w=2 ,:E)L(a+w=i_:&,a~
i=l
where w > 0 dna ~Y( 2 0, dna L setoned the gal operator. sihT ledom si
nwonk as the raenil )q(HCRA .ledom htiW laicnanif data ti serutpac the
ycnednet for ytilitalov ,gniretsulc ,.e.i for egral )llams( ecirp segnahc ot eb
dewollof yb other egral )llams( ecirp ,segnahc tub of elbatciderpnu .ngis nI
order ot ecuder the rebmun of sretemarap dna erusne a cinotonom gninilced
effect of erom tnatsid shocks, na ad hoc ylraenil gninilced gal erutcurts was
netfo desopmi ni ynam of the reilrae snoitacilppa of the ;ledom ,.e.i ~Y( =
a(q + 1 - i)/(q(q + 1)) as ni elgnE (1982, 1983).
roF z, yllamron ,detubirtsid the lanoitidnoc ytisned gniretne the doohilekil
noitcnuf ni (4) takes the mrof
gol f(8tVf1) = - 0.5 rr2gol - ?,U2,&5.0 (6)
mumixaM doohilekil (ML) desab ecnerefni serudecorp for the HCRA ssalc
of sledom rednu siht lanoitubirtsid noitpmussa are dessucsid ni elgnE (1982)
dna alutnaP (1985). hguohtlA the doohilekil noitcnuf si ylhgih raenilnon ni
the ,sretemarap a elpmis gnirocs mhtirogla si elbaliava for the raenil
)q(HCRA ledom denifed ni (5). ,eromrehtruF a egnargaL reilpitluM (LM)
test for (Y, = .. . =a4 = 0 nac eb detaluclac as TR* morf the noisserger of &f
no ,,-:F . . , ,~-:E where T setoned the elpmas .ezis sihT emas test si
yllareneg dilav gnisu yltnetsisnoc detamitse slaudiser morf the ledom nevig ni
(3). nA evitanretla tub yllacitotpmysa tnelaviuqe gnitset erudecorp si ot
tcejbus :& ot dradnats tests for laires noitalerroc desab no the -alerrocotua
noit ,erutcurts gnidulcni lanoitnevnoc uaetnamtrop tests as ni gnujL dna xoB
(1978). nI ,noitidda yrogerG (1989) stseggus a cirtemarapnon test for
)q(HCRA devired morf a etinif state vokraM niahc ,noitamixorppa elihw
nosniboR (1991) stneserp na LM test for yrev lareneg yllaires tnedneped
.yticitsadeksoreteh ehT llams elpmas ecnamrofrep of emos of these -amitse
tors dna test scitsitats evah neeb dezylana yb ,elgnE ,yrdneH dna elbmurT
(1985), dlobeiD dna yluaP (1989), velsrelloB dna egdirdlooW (19911, dna
yrogerG (1989). ,ylgnitseretnI the nwonk-llew llams elpmas drawnwod saib
for the retemarap setamitse ni evissergerotua sledom for the naem seirrac
revo ot the setamitse for (Y,, . . . , qa *.osla
tI‘ si osla htrow gniton taht ni eht ecneserp fo )q(HCRA ,stceffe dradnats stset rof laires
noitalerroc ni eht naem lliw dael ot ;snoitcejerrevo ees ssieW ,)4891( rolyaT ,)4891( johliM
,)5891( dlobeiD ,)7891( dna ztiwomoD dna oikkaH )7891( rof rehtruf .noissucsid
T. Bollerslec et al., ARCH modeling in finance 9
sA na evitanretla ot ML ,noitamitse epyt-HCRA sledom nac osla eb
detamitse yltcerid htiw dezilareneG Method of stnemoM .)MMG( sihT was
detseggus dna detnemelpmi yb Mark (1988) ahtrudoB dna Mark (1991)
,netsolG ,nahtannagaJ dna elknuR (1991), nomiS (1989) dna ,hciR ,dnomyaR
dna reltuB (1990a, ,)b dna ni a ylesolc detaler txetnoc yb yevraH (1989) dna
nosreF (1989). A nosirapmoc of the ycneiciffe of exact ML, isauQ mumixaM
doohilekiL (QML), dna MMG setamitse gnisu tnereffid tnemurtsni sets
dluow eb .gnitseretni naiseyaB ecnerefni serudecorp nihtiw the HCRA ssalc
of sledom are depoleved ni a seires of papers yb ekeweG (1988, 1989a, ,)b
who sesu etnoM olraC sdohtem ot enimreted the exact roiretsop -ubirtsid
.snoit
nA yllanoitavresbo tnelaviuqe noitatneserper for the ledom ni ,)lc (2) dna
(5) si nevig yb the gniyrav-emit retemarap )q(AM ,ledom
4
,i-q;,z&
&,=W,+
l=r
where .,,,a,,~ . qta,. are .d.i.i htiw naem zero dna ecnairav w, (Y,, . . , ,qra
.ylevitcepser sihT pihsnoitaler neewteb the gniyrav-emit retemarap ssalc of
sledom dna the raenil )q(HCRA ledom has neeb rehtruf deiduts yb yasT
(1987) Bera dna Lee (1989,1991), miK dna nosleN (1989) ffloW (1989)
gnuehC dna yluaP (1990) dna Bera, ,sniggiH dna Lee (1991). ,ylralimiS ni
ssieW )b6891( dna sniggiH dna Bera (1989a) snosirapmoc ot the raenilib emit
seires ssalc of sledom are .deredisnoc
2.2. The linear GARCH(p, q) model
nI ynam of the snoitacilppa htiw the raenil )q(HCRA ledom a gnol gal
htgnel q si dellac for. nA evitanretla dna erom elbixelf gal erutcurts si netfo
dedivorp yb the dezilareneG ,HCRA or ,p(HCRAG q), ledom ni velsrelloB
(1986)”
;a;e:_,+
.2~u)L(B+:a)L(a+w=,‘~u;B~
,a +w=2 )7(
=i
1 l=i
oT erusne a denifed-llew process lla the sretemarap ni the redro-etinifni RA
noitatneserper a,* = :EM = (1 - :E)L(~‘-))L(P tsum eb ,evitagennon
where ti si demussa that the roots of the laimonylop P(h) = 1 eil edistuo the
tinu ;elcric see nosleN dna Cao (1991) dna Drost dna namjiN (1991). roF a
)l,l(HCRAG process siht stnuoma ot gnirusne that htob iy~ dna p, are
.evitagennon tI swollof osla that E, si ecnairavoc yranoitats fi dna ylno fi
ehT3 elpmis GARCH(1, 1) ledom saw yltnednepedni detseggus yb rolyaT .)6891(
01 T. Bollersleue t al., ARCH modeling in finance
)l(a + )l(p < 1.4 Of ,esruoc ni that noitautis the ,p(HCRAG q) ledom -erroc
sdnops yltcaxe ot na redro-etinifni raenil HCRA ledom htiw yllacirtemoeg
gninilced .sretemarap
nA gnilaeppa erutaef of the ,p(HCRAG q) ledom snrecnoc the emit seires
ecnedneped ni .:E gnignarraeR ,smret (7) si ylidaer deterpretni as na AMRA
ledom for E,” htiw evissergerotua sretemarap c4L) + p(L), gnivom egareva
sretemarap ,)L(p-- dna yllaires detalerrocnu noitavonni ecneuqes ”,E{ - .}:a
gniwolloF velsrelloB (1988), siht aedi nac eb desu ni the noitacifitnedi of the
orders p dna q, hguohtla ni tsom snoitacilppa p = q = 1 si dnuof ot 5.eciffus
hcuM of nredom ecnanif yroeht si cast ni smret of suounitnoc emit
citsahcots laitnereffid ,snoitauqe elihw yllautriv lla laicnanif emit seires are
elbaliava at etercsid emit slavretni .ylno sihT tnerappa gap neewteb the
yllaciripme detavitom HCRA sledom dna the gniylrednu cimonoce yroeht si
the sucof of nosleN ,)b0991( who shows that the etercsid emit GARCH(1, 1)
ledom segrevnoc ot a suounitnoc emit noisuffid ledom as the gnilpmas
lavretni gets ylirartibra 6Llams gnolA ralimis ,senil nosleN (1992) shows that fi
the eurt ledom si a noisuffid ledom htiw on ,spmuj neht the etercsid emit
secnairav are yltnetsisnoc detamitse yb a dethgiew egareva of past slaudiser
as ni the GARCH(1, 1) .noitalumrof rehtonA elbissop nosaer for the sseccus
of the ,p(HCRAG q) sledom ni gnitamitse lanoitidnoc secnairav si dessucsid
ni Brock, ,heisH dna noraBeL (1991). yehT show that fi :E si raenil ni the
esnes of yeltseirP (1981), the ,p(HCRAG q) noitatneserper yam eb nees as a
suoinomisrap noitamixorppa ot the ylbissop etinifni dloW noitatneserper
for .:E
elihW noitagergga ni lanoitnevnoc AMRA sledom for the lanoitidnoc
naem si ,drawrofthgiarts laropmet noitagergga nihtiw the HCRA ssalc of
sledom si ton .suoivbo ,revewoH ni na lufthgisni tnecer paper, Drost dna
namjiN (1991) show that the ssalc of ,p(HCRAG q) sledom si desolc rednu
laropmet ,noitagergga yletairporppa denifed ni smret of tseb raenil -cejorp
.snoit ,oslA dlobeiD ,b6891( ,>S891 gnisu a dradnats lartnec timil meroeht
epyt ,tnemugra shows ecnegrevnoc towards ytilamron of a elagnitram process
htiw HCRA errors rednu laropmet .noitagergga
2.3. Nonnormal conditional densities
tA the emas emit that ycneuqerf-hgih laicnanif data tibihxe ytilitalov
,gniretsulc ti si osla ylediw dezingocer that the lanoitidnocnu ecirp or nruter
n14 na gnitseretni tnecer ,repap nesnaH )0991( sevired tneiciffus snoitidnoc rof raen hcope
ecnedneped dna eht noitacilppa fo dradnats citotpmysa yroeht ni a GARCH(1, 1) .ledom
sA5 detniop tuo ni johliM )0991( nihtiw eht txetnoc fo eht )l(HCRA ,ledom eht citotpmysa
dradnats rorre rof eht snoitalerrocotua dna eht laitrap snoitalerrocotua rof e: sdeecxe /l if ni
eht ecneserp fo ,HCRA suht gnidael ot yllaitnetop rewol rewop fo hcus .stset
%ee osla eht nosirapmoc ni rolyaT )a0991( fo eht lacitsitats seitreporp fo eht GARCH(1, 1)
dna evissergerotua modnar elbairav )VRA( sledom detavitom yb noisuffid .snoitalumrof
T. BollersleLle t al., ARCH modeling in finance 11
snoitubirtsid dnet ot evah fatter sliat naht the lamron ;noitubirtsid for emos
of the tseilrae ecnedive see torblednaM (1963) dna amaF (1965). hguohtlA
the lanoitidnocnu noitubirtsid for E, ni the ,p(HCRAG >q ledom htiw -idnoc
lanoit lamron errors as nevig yb ,)lc (2), (6), dna (7) evah fatter sliat naht the
lamron noitubirtsid [see jlchliM (198.5) dna velsrelloB (1986)1, for ynam
laicnanif emit seires ti does ton yletauqeda tnuocca for the .sisotrukotpel
tahT ,si the dezidradnats slaudiser morf the detamitse ,sledom ,i = ,‘-tGflE
netfo appear ot eb ’.citrukotpel
gniwolloF etihW (1982), citotpmysa dradnats errors for the sretemarap ni
the lanoitidnoc naem dna ecnairav snoitcnuf that are tsubor ot serutraped
morf ytilamron evah neeb devired yb ssieW (1984, 1986a). velsrelloB dna
egdirdlooW (1991) tneserp a tnetsisnoc rotamitse for siht tsubor -oc-ecnairav
ecnairav xirtam ni na HCRA krowemarf that seriuqer ylno tsrif ,sevitavired
together htiw na noitartsulli of the llams elpmas ecnamrofrep of the -amitse
tor dna the seitreporp of the tsubor 'RT egnargaL reilpitluM tests ni
egdirdlooW (1988,199O). tI si dnuof that the lanoitnevnoc dradnats errors
desab no the retuo tcudorp of the tneidarg-isauq deniatbo rednu the -pmussa
noit of lanoitidnoc ytilamron dnet ot etatsrednu the eurt dradnats errors for
the sretemarap ni the lanoitidnoc ecnairav noitauqe nehw lanoitidnoc -pel
sisotrukot si .tneserp esehT saedi are osla detartsulli yllaciripme ni eilliaB
dna velsrelloB (1991).”
elihW the QML desab ecnerefni serudecorp are drawrofthgiarts ot -elpmi
,tnem ylluf tneiciffe mumixam doohilekil setamitse yam eb preferred ni emos
.snoitautis nI noitidda ot the laitnetop sniag ni ,ycneiciffe the exact mrof of
the error noitubirtsid osla syalp na tnatropmi elor ni lareves tnatropmi
snoitacilppa of the HCRA ,ledom hcus as noitpo gnicirp dna the noitcurtsnoc
of lamitpo forecast error ;slavretni see elgnE dna afatsuM (1992) dna eilliaB
dna velsrelloB (1992). velsrelloB (1987) stseggus gnisu the dezidradnats
t-tnedutS noitubirtsid htiw the degrees of modeerf gnieb ”.detamitse Other
cirtemarap seitisned that evah neeb deredisnoc ni the noitamitse of HCRA
sledom edulcni the nossioP-lamron erutxim noitubirtsid ni noiroJ (1988), the
power laitnenopxe noitubirtsid ni eilliaB dna velsrelloB (1989), the
lamrongol-lamron erutxim noitubirtsid ni heisH (1989a), dna the dezilareneg
laitnenopxe noitubirtsid ni nosleN .)~0991( nI a detaler ,txetnoc hcolluCcM
(1985) stseggus the esu of na etinifni ecnairav citrukotpel elbats naiteraP
tI‘ swollof morf s’nesneJ ytilauqeni taht htiw a yltcerroc deificeps lanoitidnoc .ecnairav eht
ssecxe sisotruk ni ’~,u,E tonnac deecxe eht ssecxe sisotruk ni ;,E ees heisH .)a9891(
tA‘ eht emas ,emit gnitcartsba morf yna ,ecnerefni nosleN )d0991( sah nwohs taht eht lamron
doohilekil-isauq sesaercni htiw erom esicerp ytilitalov setamitse yletairporppa( ,)denifed elihw
siht si ton yllareneg eurt rof lamronnon doohilekil .snoitcnuf
nI“ eht suounitnoc emit yllanoitidnoc lamron ,ICHCRAG )1 noisuffid noitamixorppa dessucsid
ni nosleN ,)b0991( eht snoitavonni devresbo revo trohs emit slavretni era yletamixorppa -birtsid-t
.detu
21 ?i Bollerslec et al., ARCH modeling in finance
noitubirtsid ni eht mumixam doohilekil noitamitse fo eht dellac-os evitpadA
lanoitidnoC ,yticitsadeksoreteH ro ,HCA .ledom
As an evitanretla ot mumixam ,doohilekil a cirtemarapimes ytisned -amitse
noit euqinhcet dluoc eb desu ni gnitamixorppa .),t(f gniwolloF tnallaG and
akhcyN ,17891( ni tnallaG and nehcuaT ,)9891( ,tnallaG ,heisH and nehcuaT
,19891( and ,tnallaG Rossi, and nehcuaT ,10991( ),z(f is decalper yb a
laimonylop ,noisnapxe saerehw elgnE and areviR-zelaznoG )1991( suggest a
ytisned rotamitse yllanigiro depoleved yb aipaT and nospmohT ”.18791( yB
gnidiova yna cificeps lanoitubirtsid ,noitpmussa cirtemarapimes ytisned -itse
noitam sevig an dedda ytilibixelf ni eht .noitacificeps fO ,esruoc derapmoc ot
lluf noitamrofni mumixam doohilekil htiw a yltcerroc deificeps ,ytisned eht
cirtemarapimes hcaorppa ylbairavni sevlovni a ssol ni citotpmysa .ycneiciffe
,revewoH htiw yldekram deweks snoitubirtsid eht ycneiciffe fo eht -arapimes
cirtem rotamitse serapmoc ylbarovaf htiw eht LMQ setamitse deniatbo rednu
eht noitpmussa fo lanoitidnoc ;ytilamron ees elgnE and areviR-zelaznoG
.)1991(
2.4.
Nonlinear and nonparametric ARCH
nI eht ,p(HCRAG )q ledom )7( eht ecnairav ylno sdneped no eht -ingam
edut and ton eht sign fo .*E As dessucsid ni noitces 3.3 ,woleb this is
tahwemos at sddo htiw eht laciripme roivaheb fo kcots tekram secirp erehw
egarevel stceffe yam eb .tneserp nI eht laitnenopxE ,)q,p(HCRAG ro
,p(HCRAGE ledom decudortni yb nosleN ,)cO991( a: is an cirtemmysa
q),
noitcnuf fo past S‘,E as denifed yb ,11( ,12( and
~a,(9z,i+r[lz,~,I ~P,lOg~~;.
+o=Z,rcgol )]I,-,zIE- +
i=l i=l
)8(
ekilnU eht raenil (HCRAG ledom ni ,)7( ereht era on snoitcirtser no eht
p, q)
sretemarap (Y~ and ip ot erusne ytivitagennon fo eht lanoitidnoc .secnairav
,suhT eht noitatneserper ni 18( selbmeser an detcirtsernu ARMA(p,
q)
ledom rof gol .2ru fI 4iyc < 0, eht ecnairav sdnet ot esir )llaf( nehw .Y_~ is
evitagen )evitisop( ni ecnadrocca htiw eht laciripme ecnedive rof kcots
snruter dessucsid .woleb Assuming ,z is .d.i.i ,lamron ti swollof that (E is
ecnairavoc yranoitats dedivorp all eht stoor fo eht evissergerotua laimonylop
)A(P = 1 eil edistuo eht tinu .elcric ehT HCRAGE ledom is ylesolc detaler ot
ehT“‘ tcaf taht eht noitamrofni xirtam si ton kcolb lanogaid neewteb eht ytisned‘ ’sretemarap
dna eht HCRA sretemarap setacilpmoc eht evitpada noitamitse fo eht HCRA ,sretemarap sa
detseggus yb elgnE dna areviR-zelaznoG .)1991(
T. Bollerslec et al., ARCH modeling in finance 13
the cimhtiragol noitaziretemarap dessucsid yb ekeweG (19861 dna alutnaP
(1986) dna the evitacilpitluM HCRA ledom detseggus yb johliM ,b7891( c),
gol 2,u = w + t ,a gol ;_“,z + 5 gol(;P i-T - gol .>&
l=i l=i
ynaM other evitanretla cirtemarap HCRA snoitalumrof evah neeb -disnoc
ered ni the ,erutaretil gnidulcni power snoitamrofsnart of &f as ni the
raenilnoN HCRA ledom of sniggiH dna Bera )b9891( dna Bera dna sniggiH
(1991) dna a dlohserht HCRA ledom as ni naiokaZ ;10991( see osla elgnE
dna velsrelloB (1986). nI the dlohserht ,ledom the at2 si a raenil esiweceip
,noitcnuf ybereht gniwolla tnereffid snoitcaer of ytilitalov ot tnereffid sngis
dna sedutingam of the shocks. A detaler ,ledom desab no a vokraM niahc
approach, si proposed yb xuoreiruoG dna trofnoM (1992). ,oslA ,yevraH
,ziuR dna anatneS (1992) evah yltnecer proposed na devresbonu stnenopmoc
emit seires ledom ni hcihw HCRA secnabrutsid are decalp no htob the state
dna gnitadpu .snoitauqe A erom lamrof nosirapmoc of these ynam evitanretla
snoitalumrof dluow eb .evitamrofni
daetsnI of gniyler no a cirtemarap noitatneserper for a,‘, a cirtemarapnon
noitamitse euqinhcet dluoc eb desu ni gnitamixorppa the lanoitidnoc -irav
.ecna gniwolloF nosniboR (1987a, ,)b lareves ,srohtua gnidulcni nagaP dna
hallU (1988), nosniboR (19881, reltsihW (19881, nagaP dna gnoH (19901, dna
anatneS dna inawhdaW (1989), evah detacovda lenrek sdohtem ni hcihw 2Ta
si detamitse as a dethgiew egareva of ,:E t = 1,2,. . . , T, hcus that E,‘S for
hcihw the gninoitidnoc set denifed( ni smret of deggal )noitamrofni ’esolc‘ ot
that of 7& seviecer the tsehgih ”.thgiew lareveS tnereffid gnithgiew semehcs
are ,elbissop tub the tsom tneuqerf ni the noitamitse of HCRA sledom evah
neeb naissuaG .slenrek yreV elttil si nwonk tuoba the etinifni elpmas -reporp
seit of these seuqinhcet ni the tneserp .txetnoc
nA evitanretla cirtemarapnon noitamitse ygetarts sevlovni gnitamixorppa
the nwonknu lanoitidnoc ecnairav noitcnuf yb a seires noisnapxe where the
rebmun of smret ni the noisnapxe si na gnisaercni noitcnuf of the elpmas
.ezis roF ,ecnatsni ni the elbixelF reiruoF mroF decudortni yb tnallaG (19811,
gf2 si detamixorppa yb a noitcnuf of laimonylop dna cirtemonogirt smret ni
deggal seulav of E,. nI the cirtemarapnonimes approach ni tnallaG dna
akhcyN (1987) the cisab aedi si ot ylpitlum the lamron ytisned ni (6) yb a
laimonylop .noisnapxe roF gnisaercni orders of siht ,noisnapxe rednu weak
snoitidnoc the mumixam doohilekil setamitse dnuof gnisu siht dohtem evah
the emas lacissalc seitreporp as cirtemarapnon .setamitse nI snoitacilppa of
siht aedi redro-wol-yrev laimonylop snoitamixorppa evah yllacipyt neeb ;desu
oT“ draug tsniaga eht ecneulfni fo ,sreiltuo eht ’tuo-eno-evael‘ rotamitse si netfo ;detpoda
,.e.i 7 # .t
14 T. Bollerslev et al., ARCH modeling in finance
see, e.g., nagaP dna gnoH (1988), tnallaG dna nehcuaT (1989), dna ,tnallaG
,heisH dna nehcuaT (1989).
A good noitartsulli dna nosirapmoc of emos of the tnereffid cirtemarap
dna cirtemarapnon sdohtem dessucsid ni siht noitcesbus si nevig ni nagaP
dna Schwert (1990).
2.5. ARCH-in-Mean models
nI the ,naeM-ni-HCRA or ,M-HCRA ledom decudortni yb ,elgnE ,neiliL
dna sniboR (19871, the lanoitidnoc naem si na ticilpxe noitcnuf of the
lanoitidnoc ecnairav of the process, as nevig yb (11, (21, dna
nI siht ,ledom na esaercni ni the lanoitidnoc ecnairav lliw eb detaicossa htiw
na esaercni or a decrease ni the lanoitidnoc naem of ty gnidneped no the
ngis of the laitrap evitavired of g(x,_,, ;‘(v b) htiw respect ot a,‘. ynaM
seiroeht ni ecnanif evlovni na ticilpxe tradeoff neewteb the ksir dna the
expected .nruter ehT M-HCRA ledom si yllaedi detius ot gnildnah hcus
snoitseuq ni a emit seires txetnoc where the lanoitidnoc ecnairav yam eb
.gniyrav-emit ehT tsom nommoc seciohc for the lanoitcnuf mrof of
g(x,_ ,i ;*lo b) evah devlovni raenil or cimhtiragol snoitcnuf of at2 or cf.
,yllamroF noitamitse of the M-HCRA ledom poses on added .seitluciffid
,revewoH ni the ecnesba of M-HCRA effects, the noitamrofni xirtam -bo
deniat rednu the yrailixua noitpmussa of lanoitidnoc ytilamron si kcolb
lanogaid neewteb the sretemarap ni the lanoitidnoc naem dna ecnairav
snoitcnuf of the .ledom sihT si on regnol eurt for the M-HCRA .ledom ,suhT
ekilnu the raenil HCRAG ledom ni (7) where tnetsisnoc setamitse of the
sretemarap ni the noitcnuf ;,_,x(g b) nac eb deniatbo neve ni the ecneserp
of noitacificepssim ni a,‘, tnetsisnoc noitamitse ni the M-HCRA ledom
seriuqer that the lluf ledom eb yltcerroc .deificeps sihT slellarap the stluser
for cirtemmysa ecnairav snoitalumrof hcus as the HCRAGE ledom ni (8),
where correct noitacificeps of the lluf ledom si yllareneg deriuqer ni order ot
eetnaraug ;ycnetsisnoc see nagaP dna uabaS (1987a) dna nagaP dna gnoH
(1990). citsongaiD tests for the ecnairav noitacificeps therefore emoceb yrev
tnatropmi erofeb snoitaterpretni are edam tuoba the retemarap .setamitse
ehT ycnetsisnoc tests deniltuo ni nagaP dna uabaS )b7891( mrof the sisab for
eno hcus set of .scitsongaid
2.6. Persistence in variance
A nommoc gnidnif ni hcum of the laciripme erutaretil gnisu ycneuqerf-hgih
laicnanif data snrecnoc the tnerappa ecnetsisrep deilpmi yb the setamitse for