In eg a ion a a cos : E idence om ola ili y impulse
esponse unc ions
Eka e ini Panopoulou∗Theologos Pan elidis†
Feb ua y 25, 2005
Abs ac
We in es iga e he in e na ional in o ma ion ansmission be ween he U.S. and he es o he
G-7 coun ies using daily s ock ma ke e u n da a co e ing he las 20 yea s. A p e-1995 and pos -
1995 analysis e eals ha he linkages be ween he ma ke s ha e changed subs an ially in he mo e
ecen e a, sugges ing ha na ional ma ke s ha e become mo e in e dependen . In he majo i y o
he coun ies unde sc u iny, we p o ide e idence o di ec ola ili y spillo e s, unning mainly om
he US and poin ing o mo e apid in o ma ion ansmission du ing he ecen yea s. We u he
unco e he dynamics o he ola ili y spillo e s be ween he in e na ional s ock ma ke s by means
o a Vola ili y Impulse Response Analysis. Ou indings, based on h ee his o ical shocks ha ha e
caused u bulence in he s ock ma ke s, sugges ha he pe sis ence o ola ili y shocks has inc eased
subs an ially du ing he pos -1995 pe iod mainly due o inc eased pe sis ence and in e dependence in
he ola ili y o all ma ke s. As a esul , ola ili y shocks in he in e na ional s ock ma ke s nowadays
pe pe ua e o a signi ican longe pe iod compa ed o he p e-1995 e a.
JEL classi ica ion: G15, C32
Keywo ds: ola ili y spillo e s, ola ili y impulse esponse unc ions, s ock ma ke , ARCH-BEKK
Acknowledgmen s: Wea eg a e ul oT.Fla in,C.Ha ne andN.Pi is o help ulcommen s
and sugges ions. The au ho s hank he EU o inancial suppo unde he “PYTHAGORAS: Fund-
ing o esea ch g oups in he Uni e si y o Pi aeus” h ough he G eek Minis y o Na ional Educa ion
and Religious Affai s. The usual disclaime applies.
∗Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus, G eece and Depa men o
Economics, Na ional Uni e si y o I eland Maynoo h. Co espondence o: Eka e ini Panopoulou, Depa -
men o Economics, Na ional Uni e si y o I eland Maynoo h, Co.Kilda e, Republic o I eland. E-mail:
[email protected]. Tel: 00353 1 7083793. Fax: 00353 1 7083934.
†Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus, G eece.
1 In oduc ion
In he wake o he s ock ma ke c ash o Oc obe 1987, he s udy o he ansmission o
inancial shocks ac oss ma ke s o coun ies has eme ged as one o he mos in ensi e esea ch
opics in he in e na ional inance li e a u e in ecen yea s. The i s con ibu ions in he
so-called spillo e li e a u e came om Eun and Shim (1989) and Becke e al. (1990)
who, based on ei he a VAR model o a se o single linea equa ions, ied o cap u e he
dependence be ween in e na ional equi y e u ns. In a simila way, Koch and Koch (1991)
in es iga ed he e olu ion o con empo aneous and lead/lag ela ionships among eigh s ock
ma ke s and concluded ha egional in e dependence g ows o e ime and ha he in luence
o he Japanese ma ke inc eases a he expense o he US ma ke . Howe e , hese s udies
ocused on he e u n se ies and on how e u ns a e co ela ed ac oss ma ke s, i.e. hey
conside ed only in e dependence h ough he mean o he p ocess.
A second s and o he li e a u e, which is g owing apidly, explici ly ocuses on he
ola ili y o equi y e u ns, sugges ing he exis ence o highe -o de dependence s emming
om he second momen s. This amewo k is app op ia e when modeling high- equency
inancial ime se ies and i s impo ance has been ecognized e e since Engle (1982) in o-
duced he class o ARCH models. In his con ex , Hamao e al. (1990) employed uni a ia e
GARCH models o examine he s ock ma ke s a ound he 1987 US s ock ma ke c ash. They
ound e idence o signi ican p ice- ola ili y spillo e s om he US o he UK and Japan and
om he UK o Japan o he pos -c ash pe iod. In con as , no such spillo e s a e ound in
he p e-c ash pe iod. Thei esul s sugges ha shocks ha o igina e in he US a e la ge and
mo e pe sis en han he UK and Japanese ones. Lin e al. (1994), using a signal ex ac ion
model wi h GARCH p ocesses, ound a ecip ocal ela ionship be ween he p ice and ola il-
i y o he US and Japanese ma ke s. Susmel and Engle (1994) analyzed he in e ela ionship
be ween he US and he UK s ock ma ke s using hou ly e u ns and did no ind s ong
e idence o ei he mean o ola ili y spillo e s be ween he wo ma ke s. Ka olyi (1995)
examined he dynamic ela ionship be ween he US and Canadian s ock ma ke e u ns and
e u n ola ili ies using a bi a ia e GARCH model. He ound ha he effec so shockso ig-
ina ing in he US ma ke on he Canadian ma ke e u ns and ola ili y a e smalle and less
pe sis en han hose measu ed wi h adi ional ec o au o eg essi e models. Theodossiou
1
and Lee (1993) s udied he ela ionship be ween he US, he UK, Canadian, Ge man and
Japanese s ock ma ke s using a mul i a ia e GARCH-in-mean model and ound mean and
ola ili y spillo e s be ween some o hose ma ke s. The au ho s also documen ed ha he
US is he majo expo e o ola ili y. Mo e ecen ly, Kou mos and Boo h (1995) examined
p ice ola ili y spillo e s o he US, he UK and Japan in he con ex o an ex ensi e mul i-
a ia e Exponen ial GARCH model which can cap u e possible asymme ies in he ola ili y
ansmission mechanism. The au ho s ound apa om p ice spillo e s, ex ensi e and ecip-
ocal second momen in e ac ions, which a e asymme ic, i.e. nega i e inno a ions in a gi en
ma ke inc ease ola ili y in he nex ma ke o ade mo e han posi i e inno a ions. The
appea ance o he Asian inancial c isis in 1997 e i ed he in e es in he ma e and u ned
he ocus away om he majo s ock ma ke s owa ds he eme ging ones (see o example
Ng, 2000; Capo ale e al., 2001; He, 2001; Chen e al., 2002; Miyakoshi, 2003).
In his s udy, we ocus explici ly on unco e ing he ola ili y dynamics be ween he US
s ock ma ke and he emaining six o he G-7 coun ies using Vola ili y Impulse Response
Func ions o mul i a ia e GARCH models in oduced by Ha ne and He wa z (2006). In
his ein, we aim a es ablishing he pa e n o in o ma ion ansmission be ween hese
coun ies. As indica ed by Ross (1989), he ansmission o in o ma ion o a ma ke is
ela ed p ima ily o he ola ili y o an asse ’s p ice changes in an a bi age- ee economy,
i.e. he second momen is mo e impo an han he i s one in he low o in o ma ion. In
his ein, Engle e al. (1990) a ibu e mo emen s in ola ili y o he lag wi h which ma ke
pa icipan s p ocess new in o ma ion.
The p esen s udy makes a wo old con ibu ion. Fi s , we es ima e a bi a ia e GARCH
model, o which a BEKK ep esen a ion is adop ed (see Engle and K one , 1995), o each
o he six coun ies agains he US using daily e u ns o he las wen y yea s. This BEKK
o mula ion enables us o e eal he exis ence o any “me eo showe s”, i.e. ansmission o
ola ili y om one ma ke o ano he , as well as any “hea wa es”, i.e. inc eased pe sis ence
in ma ke ola ili y (see Engle e al., 1990). Spli ing ou sample in o wo non-o e lapping
sub-samples o equal leng h, we in es iga e whe he he effo s o mo e economic, mone a y
and inancial in eg a ion ha e undamen ally al e ed he sou ces and in ensi y o ola ili y
spillo e s o he indi idual s ock ma ke s. Second, by using a ecen ly de eloped echnique,
we es ima e he co esponding Vola ili y Impulse Response Func ions (VIRFs) implied by he
2
speci ica ion o each model. We hen assess he impac o h ee his o ically obse ed shocks,
i.e. he 1987 s ock ma ke c ash, he 1997 Asian Financial c isis and he 2001 e o is a ack
on he ola ili y and co- ola ili y o he ma ke s.
To he bes o ou knowledge, no o he s udy (excep o Ha ne and He wa z, 2006)
has employed his inno a i e echnique o VIRFs o s udy ola ili y dynamics in any ma ke .
Mo e impo an ly, he e a e se e al easons why VIRFs ep esen a con enien app oach o
analyse ola ili y spillo e s. Fi s , his echnique allows he esea che o de e mine p ecisely
how a shock o one ma ke in luences he dynamic adjus men o ola ili y o ano he ma ke
and he pe sis ence o hese spillo e effec s. Second, VIRFs depend on bo h he ola ili y
s a e and he unexpec ed e u ns ec o when he shock occu s. As a esul , he asymme ic
esponse o ola ili y on nega i e and posi i e “news” ypically documen ed in he li e a u e
(see e.g. Kou mos and Boo h, 1995) can easily be accommoda ed.1Thi d, con a y o ypical
Impulse Response Func ions, his speci ic me hodology a oids ypical o hogonaliza ion and
o de ing p oblems which would be ha dly easible in he case o highly in e ela ed and
obse ed a high equencies inancial ime se ies.
The only s udy ha is closely ela ed o ou s is he one by Leachman and F ancis (1996).
The au ho s use a wo-s age p ocedu e, i.e. hey i s es ima euni a ia eGARCHmodels
o he G-7 s ock ma ke e u ns and hen he es ima ed condi ional a iances a e used o
cons uc a VAR sys em. This me hodology enables hem o employ he s anda d Impulse
Response analysis and conduc Va iance Decomposi ions in o de o de e mine how a shock
o one ma ke in luences he dynamic adjus men o ola ili y in he emaining ma ke s and
he pe sis ence o hese ola ili y spillo e s. They can also quan i y he ela i e signi icance
o each ma ke in gene a ing and ansmi ing luc ua ions o o he ma ke s. In e es ingly,
he au ho s sugges ha a mul i a ia e GARCH app oach would gi e mo e efficien pa ame-
e es ima es han hei wo-s age app oach bu would no enable he esea che o ob ain
impulse esponse unc ions, as he la e a e no a ailable o GARCH p ocesses.
I is his gap in he li e a u e ha we in end o b idge by es ima ing he VIRFs o he
G-7 s ock ma ke e u ns accommoda ed by he a o emen ioned me hodology. Consis en
wi h he inc eased in eg a ion o capi al ma ke s al eady documen ed in he li e a u e (see,
1Nega i e “news”, i.e. unexpec ed e u ns, in one ma ke can esul in a diffe en ola ili y p o ile han
posi i e “news”, o he hings being equal.
3
o example, Ha ey, 1991; Bekae and Hod ick, 1992; Campbell and Hamao, 1992), ou
esul s sugges ha equi y ma ke s ha e become mo e in e dependen in he pos -1995 pe iod
compa ed wi h he p e-1995 pe iod. This g ea e in eg a ion esul ed in a signi ican inc ease
in he pe sis ence o ola ili y shocks o all he coun ies a hand. The exis ence o bo h
ele a ed “hea wa es” and “me eo showe s” effec s is depic ed in he pa e n and size o he
VIRFs.
The emainde o his s udy is o ganized as ollows. Sec ion 2 discusses he econome ic
me hodology and Sec ion 3 desc ibes he da a and p esen s he empi ical indings o bo h he
p e-1995 pe iod and he pos -1995 pe iod. Sec ion 4 offe s a summa y and some concluding
ema ks.
2 Econome ic Me hodology
In his sec ion, we i s p esen he model we employ o in es iga e he ola ili y spillo e s
be ween he s ock ma ke s unde sc u iny and hen p o ide a b ie desc ip ion o he ola ili y
impulse esponse me hod employed o analyse he pe sis ence o he ola ili y shocks in he
in e na ional s ock ma ke s.
2.1 The BEKK Model
The analysis is based on a bi a ia e VAR(1)-GARCH(1,1) model. Le Y =(y1 ,y
2 )0be
he e u ns ec o , wi h y2 deno ing he US s ock ma ke and y1 one o he emaining G-7
coun ies. The condi ional mean o he p ocess is modeled as ollows:
Y =C+M∗Y −1+E (1)
whe e Cis a 2×1 ec o o cons an s and Mis a 2×2coefficien ma ix and E =(e1 ,e
2 )0
is he ec o o he ze o-mean e o e ms. We allow E o ha e a ime- a ying condi ional
a iance, ha is Va (E |F
−1)=H whe e F −1deno es he σ− ield gene a ed by all
in o ma ion a ailable a ime −1. We u he assume ha he condi ional a iance, H ,
ollows a bi a ia e GARCH(1,1) model and we, speci ically, conside he ollowing BEKK
4

ep esen a ion, in oduced by Engle and K one (1995):
E =H1/2
∗Z
H =Ω∗Ω0+A∗E −1∗E0
−1∗A0+B∗H −1∗B0(2)
whe e Ω=[ωij],i,j=1,2is a 2x2 lowe iangula ma ix o cons an s, A=[aij]and
B=[bij],i,j=1,2a e 2x2 coefficien ma ices and Z =(z1 ,z
2 )0∼iid(


0
0


,


10
01


).
Ma ix Ameasu es he ex en o which condi ional a iances a e co ela ed wi h pas squa ed
unexpec ed e u ns (i.e. de ia ions om he mean) and consequen ly cap u es he effec s o
shocks on ola ili y. On he o he hand, ma ix Bdepic s he ex en o which cu en le els
o condi ional a iances and co a iances a e ela ed o pas condi ional a iances and co a i-
ances. Apa om displaying sufficien gene ali y, his model ensu es ha he condi ional
a iance-co a iance ma ices, H =[hij, ],i,j=1,2,a eposi i ede ini e unde a he weak
assump ions.2
Compa ed o al e na i e mul i a ia e GARCH ep esen a ions, he BEKK model is mo e
con enien o es ima ion, because i in ol es ewe pa ame e s.Engle and K one (1995)
p o e ha he BEKK model in (2) is second-o de s a iona y i and only i all he eigen alues
o (A⊗A+B⊗B)a e less han uni y in modulus. In his case, he uncondi ional a iance o
E ,Va (E ), can easily be calcula ed by: ec[Va (E )] = [I4−(A⊗A)0−(B⊗B)0]−1∗ ec(Ω0Ω)
whe e ec is he ope a o ha s acks he columns o a squa e ma ix.3
Mo e in de ail, he condi ional a iance o each equa ion can be expanded o he bi-
a ia e GARCH(1,1) as ollows:
h11, =ω2
11 +a2
11e2
1 −1+2a11a12e1 −1e2 −1+a2
12e2
2 −1+(3)
+b2
11h11, −1+2b11b12h12, −1+b2
12h22, −1
h22, =ω2
21 +ω2
22 +a2
21e2
1 −1+2a21a22e1 −1e2 −1+a2
22e2
2 −1+(4)
+b2
21h11, −1+2b21b22h12, −1+b2
22h22, −1
2Engle and K one (1995) show ha H is posi i e de ini e i a leas one o Ωo Bis o ull ank.
3The momen p ope ies o mul i a ia e GARCH p ocesses a e also examined by Ha ne (2003).
5
h12, =ω11ω21 +a11a21e2
1 −1+(a11a22 +a12a21)e1 −1e2 −1+a12a22e2
2 −1+(5)
+b11b21h11, −1+(b11b22 +b12b21)h12, −1+b12b22h22, −1
Suppose ha we es ima e a bi a ia e sys em o Canada and he US based on equa ions
(1) - (2). In such a case, h11, and h22, deno e he condi ional a iance o Canada and he US
espec i ely, while h12, deno es he condi ional co a iance be ween he se ies. Signi icance o
any o bo h he elemen s a12,b
12 sugges s ha ola ili y in he Canadian ma ke is affec ed
by de elopmen s in he ola ili y o he US ma ke h ough ei he he pas ola ili y o he US
ma ke , h22, −1, o he pas squa ed inno a ions e2
2 −1(o e en he c oss p oduc s, e1 −1e2 −1,
o pas inno a ions). Fu he mo e, indi ec eedbacks may exis h ough he pas alue o he
condi ional co a iance h12, −1. When conside ing he e olu ion o he US ma ke ola ili y
and i s dependence on he Canadian one, he easoning is simila and ollows di ec ly om
equa ion (4). The con empo aneous co-mo emen in he ola ili y o he se ies is gi en by
equa ion (5) and is a unc ion o pas squa ed inno a ions, c oss p oduc s o inno a ions, pas
condi ional ola ili ies and na u ally pas condi ional co a iance. This ich pa ame e iza ion
sugges s ha e en in he case ha condi ional ola ili ies be ween he se ies a e no linked
di ec ly, i.e. b12 =b21 =0, he in e ac ions be ween he condi ional a iances is ensu ed by
pas e u n inno a ions.
To cope wi h he excess ku osis usually ound in he es ima ed s anda dised esiduals
unde he assump ion o Gaussian inno a ions, we ollow Bolle sle (1987) and e alua e
(and maximize) he sample log-likelihood unc ion unde he assump ion o (ν)−dis ibu ed
inno a ions. In such a case, gi en a sample o Tobse a ions, a ec o o unknown pa ame e s
θand a 2×1 ec o o e u ns Y , he bi a ia e BEKK model is es ima ed by maximizing
he ollowing likelihood unc ion:
L(θ)=
T
X
=1
ln(l (θ)) (6)
wi h
l =Γ((T+ )/2)
Γ( /2)[π( −2)]T/2|H |−1/2·1+ 1
−2E0
H−1
E ¸−(T+ )/2
(7)
whe e νdeno es he deg ees o eedom o he −dis ibu ion and Γ(·)is he gamma unc ion.
This log-likelihood unc ion is maximized using he Be nd , Hall, Hall and Hausman (1974)
6
algo i hm (BHHH).4
2.2 Vola ili y Impulse Response Func ions
We now b ie ly desc ibe he Vola ili y Impulse Response Func ion (VIRF) in oduced by
Ha ne and He wa z (2006). The au ho s de i e he VIRF based on an al e na i e mul-
i a ia e GARCH ep esen a ion, namely he ec- ep esen a ion (in oduced by Engle and
K one , 1995), gi en by:
ech(H )=Q+R∗ ech(E −1∗E0
−1)+P∗ ech(H −1)(8)
whe e Qis a 3×1ma ix o cons an s, while Rand Pa e 3×3coefficien ma ices. ech is
he ope a o ha s acks he lowe iangula pa o a squa e ma ix. I is impo an o no e
ha he ec- ep esen a ion gi en in (8) equi es he es ima ion o 21 pa ame e s, while he
BEKK ep esen a ion gi en in (2) has only 11 pa ame e s. Thus, he BEKK model manages
o educe subs an ially he numbe o pa ame e s, acili a ing he es ima ion p ocedu e.
Ob iously, he ec-model is mo e gene al han he BEKK model, since he BEKK model
educes he numbe o pa ame e s by imposing some speci ic es ic ions on he ec-model.
In gene al, any gi en BEKK model has a unique equi alen ec- ep esen a ion (Engle and
K one 1995), while he con e se is no ue.5,6In summa y, by employing he BEKK model
we achie e he educ ion o he numbe o pa ame e s wi h i ually no cos in e ms o
he gene ali y o he model. The de i a ion o he unique equi alen ec- ep esen a ion o
a BEKK model is s aigh o wa d.7In he es o his pape , we assume ha (8) is he
equi alen ec- ep esen a ion o (2).
Assume ha a ime =0 he condi ional a iance is a an ini ial s a e H0and an ini ial
4Susmel and Engle (1994) sugges ha in he case o high- equency inancial da a using he -dis ibu ion
gene a es a mo e efficien es ima ion o condi ional e o s han he no mal dis ibu ion.
5Two GARCH ep esen a ions a e equi alen i e e y sequence o e o s {E }gene a es he same sequence
o condi ional ola ili ies {H } o bo h ep esen a ions.
6I is possible ha a ec-model has no equi alen BEKK ep esen a ion. Mo eo e , i he e is an equi alen
BEKK ep esen a ion o a ec-model, his BEKK ep esen a ion is no unique.
7The necessa y assump ions o he equi alence o he wo ep esen a ions a e he ollowing: Q=


ω2
11
ω11ω21
ω2
21 +ω2
22

,R=

a2
11 2a11a12 a2
12
a11a21 a11a22 +a12a21 a22a12
a2
21 2a21a22 a2
22

and P=

b2
11 2b11b12 b2
12
b11b21 b11b22 +b12b21 b22b12
b2
21 2b21b22 b2
22

.
7
shock Z0=(z1,0,z
2,0)0occu s. The VIRF, V (Z0),is hende ined as ollows:
V (Z0)=E[ ech(H )|F
−1,Z
0]−E[ ech(H )|F
−1]
The i s and hi d elemen s o V (Z0)(deno ed as 1, and 3, espec i ely) ep esen he
eac ion o he condi ional a iance o he i s and second a iable espec i ely o he shock,
Z0, ha occu ed pe iods ago. Simila ly, he second elemen o V (Z0)(deno ed as 2, )
ep esen s he eac ion o he condi ional co a iance o he shock, Z0, ha occu ed pe iods
ago. The VIRF can easily be compu ed ecu si ely based on he ollowing ela ions:
V1(Z0)=R∗{ ech(H1/2
0Z0Z0
0H1/2
0)− ech(H0)}(9)
V (Z0)=(R+P)∗V −1(Z0), >1
The VIRF has wo impo an diffe ences compa ed o he adi ional Impulse Response
Func ion (IRF) in he condi ional mean. Fi s , he VIRF is an e en unc ion o he ini ial
shock, ha is V (Z0)=V (−Z0), con a y o he IRF ha is an odd unc ion o he ini ial
shock. Second, he IRF is a linea unc ion, i.e. IRF(k∗Z0)=k∗IRF(Z0),while heVIRF
is no homogeneous o any deg ee.
Be o e p esen ing he empi ical esul s o his s udy, we b ie ly desc ibe he beha io o
he VIRF. Fi s o all, le Ψ=[ψi,1]:= ech(H1/2
0Z0Z0
0H1/2
0)− ech(H0)whe e i=1,2,3.
I is ob ious ha he elemen s o Ψa e unc ions o he elemen s o he ini ial s a e H0and
he elemen s o he shock Z0. The ollowing h ee cases a e o in e es :
Case I: Diagonal BEKK model (i.e. a12 =a21 =b12 =b21 =0)
In his case, bo h Rand P(and hus R+P) a e diagonal ma ices (see oo no e 7). I
is easy o show ha :
1,1=a2
11ψ1,1and 1, =(a2
11 +b2
11) −1 1,1 o >1
2,1=a11a22ψ2,1and 2, =(a11a22 +b11b22) −1 2,1 o >1
3,1=a2
22ψ3,1and 3, =(a2
22 +b2
22) −1 3,1 o >1
The e o e, in his pa icula case he e a e no ola ili y spillo e s, since bo h 1, and 3,
8
10−4,2.03 ×10−4)0 espec i ely o he Canada-US model, In his case, he ini ial shock is
es ima ed o be b
Z0=(b
H1/2
)−1b
E =(−9.74,−14.04)0. The co esponding ini ial shocks o
he emaining i e models a e calcula ed in a simila way.
The second shock we conside is he Asian inancial c isis in 1997. Mo e speci ically, we
calcula e he ini ial shock based on he es ima ed models (Table 3, Panel B) o Oc obe
27, 1997. The las shock we conside is he one associa ed wi h he e o is a ack o he
Twin Towe s in Sep embe 2001. Since he i s ading day o he US s ock ma ke a e
he a ack was he Sep embe 17, 2001, he ini ial shocks a e calcula ed based on his day.
The es ima ed ini ial shocks o hese h ee his o ical shocks wi h espec o ou es ima ed
models a e epo ed in Table 5.
Apa om he es ima ed pa ame e s o he BEKK models and he co esponding ini ial
shocks, he calcula ion o he VIRFs equi es he ini ial s a e o ola ili y, H0,assugges ed
by equa ion (9). To make ou indings in a ian o he choice o ini ial s a e, we selec he
las day o ou sample, i.e. Oc obe 8, 2004 and employ his es ima ed condi ional a iance-
co a iance ma ix in bo h sub—pe iods.15 This allows a di ec compa ison o he VIRFs
be ween he wo sub-pe iods unde conside a ion.
We i s in es iga e he size o he effec o each shock on he condi ional a iance o he
se ies unde examina ion. Table 7 epo s he maximum alue and he 1-s ep ahead alue
o he VIRF di ided by he ini ial condi ional a iance. As expec ed, he effec o he 1987
c ash on he condi ional ola ili y dynamics is subs an ially g ea e han he co esponding
effec s s emming om he o he wo shocks conside ed in his analysis. Fo example, in he
case o he US, he c ash o 1987 induces a ise in ola ili y ha is 6.57 and 17.18 imes
he ini ial ola ili y o he p e-1995 and pos -1995 pe iod espec i ely.16 The co esponding
igu es o he Asian inancial c isis a e 1.06 and 2.78 o he pe iods examined. E en milde
effec s a e p e alen when he hi d shock is conside ed.17
The inc eased in ensi y o ola ili y spillo e s du ing he ecen yea s is easily shown when
15The ini ial s a es employed o each pai o coun ies a e p esen ed in Table 6. Al e na i ely, he es ima ed
uncondi ional a iance ma ix o E could be employed as an ini ial s a e. In such a case, ou esul s would
be quali a i ely simila o he ones epo ed.
16The analysis o he US is based on he Canada-US model, al hough quan i a i ely simila esul s a e
d awn om he es o he bi a ia e models.
17In his case, i a simila shock occu ed in he p e-1995 pe iod, he inc ease in he US ola ili y would be
jus 0.53 imes he ini ial ola ili y, while in he case o he pos -1995 pe iod, he ola ili y inc ease would be
1.38 imes he ini ial ola ili y.
15

he pa h o he impulse esponses in he ola ili y o each coun y is conside ed. Figu es 2-4
plo he VIRFs o he h ee shocks espec i ely. In mos cases, he VIRF is maximised he
day a e he shock. Howe e , in some cases he effec o he ini ial shock g adually inc eases,
eaching i s maximum alue a e many days o e en weeks. In gene al, he impulses a e
declining s a ing om a high le el o ola ili y. In some cases, howe e , he e is e idence
o an ini ial shock ampli ica ion which inc eases u he he ini ial effec . E en ually, in all
cases he VIRFs esume hei declining pa h owa ds ze o.
We now ocus on he es ima ed hal -li e o he ola ili y shocks, which a e epo ed in
Table 8 (Panels A, B and C o he h ee shocks espec i ely) o all six pai s o coun ies and
he wo sub-pe iods.18 Appa en ly, he hal -li es o ola ili y shocks pain a simila pic u e
poin ing o mo e pe sis en shocks in he mo e ecen e a. Fo example, o he p e-1995
pe iod and o he 1987 c ash, he es ima ed hal -li es o he US, he coun y om which
he c ash o igina ed, ange om 48 o 91 days, while i he shock occu ed in he pos -1995
pe iod, he espec i e igu es would be 98 days o 219 days. The hal -li es o he es o he
coun ies a e lowe han he US in he i s sub-pe iod anging om 13 days (UK and F ance)
o 41 days (Japan). Su p isingly, ou indings om he ecen e a sugges ha a simila o he
“1987 c ash” shock would induce ola ili y spillo e s ha would las o signi ican ly longe
pe iod nowadays compa ed o he p e-1995 pe iod. Ou esul s o he o he wo shocks a e
quali a i e simila o hose o he i s shock, so we do no discuss hem sepa a ely o sa e
space.
In summa y, he empi ical indings o his ola ili y impulse esponse expe imen sugges
ha he inc eased in eg a ion o he in e na ional s ock ma ke s du ing he pos -1995 pe iod
has also caused an inc ease in he pe sis ence o ola ili y shocks. As a esul , simila
ola ili y shocks can pe pe ua e o a signi ican longe pe iod nowadays compa ed o he
p e-1995 e a.
4 Conclusions
The e is ex ensi e empi ical wo k in he li e a u e wi h espec o in e dependencies be ween
inancial ma ke s and mo e speci ically, na ional s ock ma ke s. This pape ocuses on second-
18No e ha he hal -li es in Table 8 a e calcula ed a e he Ini ial Shock Ampli ica ion has been deduc ed.
16
o de in e dependencies, i.e. linkages h ough he condi ional a iances o he se ies. The
analysis was pe o med using daily closing s ock index da a om he G-7 s ock ma ke s o
helas 20yea s. Byadop ingabi a ia eBEKK ep esen a ion and spli ing ou sample
in o wo 10-yea sub-samples, we i s examined whe he s ock ma ke linkages be ween he
US and he emaining o he G-7 coun ies ha e changed du ing he ecen yea s. As a second
s ep, we employed a new echnique de eloped by Ha ne and He wa z (2006) and es ima ed
he Vola ili y Impulse Response Func ions (VIRFs) ela ed o each pai o ou coun ies.
This echnique enabled us o quan i y he size and he pe sis ence o h ee his o ical shocks
ha ha e caused u bulence in he s ock ma ke s. Fu he mo e, he signi ican ly diffe en
s uc u e o s ock ma ke s in he p e- and pos -1995 pe iods allowed compa isons ha shed
some ligh in o he cu en beha io o s ock ma ke s.
Ou empi ical indings can be summa ised as ollows. We con i med he es ablished iew
ha he US s ock ma ke is he majo ola ili y expo e coun y. Speci ically, he e is
e idence o signi ican ola ili y spillo e s om he US o Canada, F ance and Ge many
du ing he p e-1995 pe iod. Fo he same pe iod, he es o he G-7 coun ies, i.e. I aly,
Japan and he UK appea secluded and in ulne able o shocks o igina ing in he US. On he
o he hand, ou indings o he mo e ecen pe iod poin o inc eased in eg a ion be ween
he ma ke s. Speci ically, he smalle o he G-7 coun ies, i.e. Canada, F ance, Ge many
and I aly mainly impo ola ili y om he US. A mo e impo an inding, howe e , is he
e idence in a o o bidi ec ional ola ili y spillo e s be ween he US and Japan, as well as
he US and he UK. Ou esul s sugges ha shocks o igina ing in he UK affec posi i ely
he US s ock ma ke while he Japanese ones in luence he US ma ke nega i ely, inducing
lowe le els o ola ili y. Ou VIRFs analysis o h ee his o ical shocks, namely he 1987
c ash, he 1997 Asian inancial c ash and he 2001 e o is a ack p o ided use ul insigh s
wi h espec o he size and pe sis ence o ola ili y shocks. We speci ically ound e idence
in a o o inc eased ampli ude and du a ion o ola ili y spillo e s in he pos -1995 sample
compa ed o he p e-1995 one. This in ensi y o shocks mainly s ems om he inc eased
in e dependence and pe sis ence o he equi y ma ke ola ili ies documen ed in he ecen
e a. Consequen ly, had a shock simila o he one o he 1987 c ash occu ed in he mo e
ecen yea s, he ime equi ed o his shock o die ou would ha e been ex emely longe
nowadays compa ed o he p e-1995 pe iod.
17
The me hod employed he e can also be applied o o he cases ha in ol e high equency
da a, mainly inancial da a, o examine linkages and unco e he ola ili y dynamics be ween
he se ies unde examina ion. Vola ili y spillo e s be ween exchange a e ma ke s o be ween
s ock ma ke s and exchange a es can be de ec ed and quan i ied h ough he VIRFs. An-
o he p omising ou e o u he in es iga ion may be he ex ension o his bi a ia e analysis
o a highe o de one, allowing o in e ac ions among h ee o mo e coun ies. Bo h hese
ex ensions will be he objec o ou u u e wo k.
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20

21
Table 1: Summa y Desc ip i e S a is ics
Panel A: Full Sample (31/12/84-8/10/04)
Canada F ance Ge many I aly Japan UK US
Mean 0.00025 0.00035 0.00028 0.00029 9.59E-05 0.00044 0.00034
Median 0.00049 0.00046 0.00032 0.00037 0.00000 0.00055 0.00025
Maximum 0.08874 0.08289 0.08769 0.07099 0.12883 0.07231 0.09095
Minimum -0.12111 -0.08430 -0.11494 -0.10678 -0.13823 -0.14047 -0.22899
S d. De . 0.00960 0.01355 0.01453 0.01304 0.01602 0.01150 0.01093
Skewness -1.17403 -0.22157 -0.29519 -0.36396 0.11682 -0.59935 -2.07463
Ku osis 17.7069 5.88741 7.16838 6.90276 7.63023 10.6913 47.1517
Panel B: Fi s subsample (31/12/84-31/12/94)
Canada F ance Ge many I aly Japan UK US
Mean 0.00016 0.00044 0.00031 0.00027 0.00048 0.00054 0.00033
Median 0.00037 0.00049 0.00000 0.00034 0.00053 0.00054 0.00029
Maximum 0.08874 0.08289 0.08769 0.07099 0.12883 0.07231 0.09095
Minimum -0.12111 -0.08430 -0.11494 -0.10678 -0.13823 -0.14047 -0.22899
S d. De . 0.00800 0.01317 0.01351 0.01397 0.01560 0.01179 0.01045
Skewness -2.07996 -0.36377 -0.50435 -0.39156 -0.01657 -1.01165 -4.80774
Ku osis 43.4971 7.11751 10.3723 7.59009 10.1313 15.5090 108.379
Panel C: Second subsample (1/1/95-8/10/04)
Canada F ance Ge many I aly Japan UK US
Mean 0.00034 0.00028 0.00027 0.00029 -0.00025 0.00035 0.00035
Median 0.00064 0.00041 0.00053 0.00045 -0.00043 0.00056 0.00017
Maximum 0.04690 0.06198 0.06837 0.05592 0.12354 0.05797 0.05574
Minimum -0.09033 -0.07362 -0.08559 -0.07543 -0.06592 -0.05886 -0.07114
S d. De . 0.01088 0.01389 0.01542 0.01213 0.01639 0.01124 0.01136
Skewness -0.80306 -0.10832 -0.16263 -0.31626 0.22749 -0.16387 -0.10826
Ku osis 8.96719 4.94645 5.24439 5.42687 5.72985 5.29419 6.25458
22
Table 2: Un es ic ed Es ima ed GARCH(1,1)-BEKK Models
Panel A: 1s subsample (31/12/84-31/12/94) Panel B: 2nd subsample (1/1/95-8/10/04)
c
11 α11 α12 b11 b
12 d. . c11 α11 α12 b11 b
12 d. .
c
21 c
22 α21 α22 b21 b
22
Eigen-
alues (s.e.) c21 c
22 α21 α22 b21 b
22
Eigen-
alues (s.e.)
Canada 0.0013* 0.2596* -0.0173 0.9387* 0.0149* 0.9885 4.8333*
Canada 0.0007* 0.2125* 0.0314 0.9747* -0.0074 0.9963 7.3849*
(0.0001) (0.0291) (0.0183) (0.0116) (0.0060) 0.9710 (0.3160) (0.0001) (0.0183) (0.0185) (0.0043) (0.0050) 0.9961 (0.5860)
0.0004* 0.0006* 0.0656* 0.1355* -0.029* 0.9938* 0.9692 LL 0.0005* 0.0006* 0.0344 0.2301* -0.0074 0.9705* 0.9940 LL
(0.0001) (0.0002) (0.0331) (0.0193) (0.0128) (0.0062) 0.9625 17066.3 (0.0002) (0.0001) (0.0208) (0.0213) (0.0054) (0.0055) 0.9939 17029.1
F ance 0.0036* 0.2614* 0.0545* 0.9213* -0.0012 0.9890 5.6514*
F ance 0.0013* 0.2167* -0.0271 0.9692* 0.0110 0.9977 8.4343*
(0.0004) (0.0280) (0.0221) (0.0160) (0.0078) 0.9458 (0.4033) (0.0002) (0.0189) (0.0235) (0.0057) (0.0071) 0.9886 (0.8219)
0.0004 0.0007* -0.0110 0.1533* -0.0019 0.9830* 0.9425 LL -0.0003 0.0008* 0.0106 0.2315* 0.0018 0.9682* 0.9883 LL
(0.0002) (0.0001) (0.0146) (0.0130) (0.0073) (0.0026) 0.9214 15138.7 (0.0002) (0.0001) (0.0164) (0.0186) (0.0051) (0.0053) 0.9798 15977.8
Ge many 0.0028* 0.2738* 0.0423* 0.9339* 0.0044 0.9881 5.2204*
Ge many 0.0007* 0.2205* 0.0384 0.9733* -0.0055 0.9997 8.4185*
(0.0003) (0.0257) (0.0207) (0.0109) (0.0067) 0.9593 (0.3353) (0.0002) (0.0165) (0.0227) (0.0043) (0.0069) 0.9932 (0.8173)
-0.0001 0.0008* -0.0064 0.1415* 0.0041 0.9828* 0.9566 LL -0.0003 0.0009* 0.0017 0.2415* 0.0021 0.9656* 0.9931 LL
(0.0001) (0.0001) (0.0137) (0.0123) (0.0055) (0.0027) 0.9423 15197.8 (0.0002) (0.0001) (0.0122) (0.0162) (0.0034) (0.0047) 0.9869 15921.2
I aly 0.0021* 0.2436* 0.0132 0.9584* -0.0044 0.9896 5.6541*
I aly 0.0017* 0.2409* 0.0068 0.9572* 0.0027 0.9964 8.6927*
(0.0003) (0.0223) (0.0163) (0.0076) (0.0046) 0.9786 (0.4018) (0.0002) (0.0205) (0.0180) (0.0074) (0.0053) 0.9863 (0.8834)
0.0002 0.0007* 0.0082 0.1477* -0.0027 0.9836* 0.9784 LL 0.0001 0.0007* 0.0236 0.2293* -0.0067 0.9718* 0.9855 LL
(0.0001) (0.0001) (0.0107) (0.0121) (0.0036) (0.0022) 0.9780 14978.5 (0.0002) (0.0001) (0.0180) (0.0165) (0.0064) (0.0042) 0.9745 16219.0
Japan 0.0025* 0.3437* 0.0593* 0.9281* -0.0104 0.9898 5.2258*
Japan 0.0020* 0.2072* 0.0069 0.9696* 0.0041 0.9946 8.3958*
(0.0002) (0.0235) (0.0279) (0.0089) (0.0074) 0.9808 (0.3588) (0.0003) (0.0170) (0.0249) (0.0050) (0.0063) 0.9902 (0.8265)
0.0002 0.0007* 0.0243* 0.1537* -0.008* 0.9827* 0.9648 LL -0.0002 0.0007* -0.026* 0.2308* 0.0049 0.9709* 0.9893 LL
(0.0002) (0.0001) (0.0100) (0.0122) (0.0038) (0.0024) 0.9634 14890.7 (0.0002) (0.0001) (0.0132) (0.0153) (0.0039) (0.0038) 0.9837 15242.6
UK 0.0031* 0.2693* 0.0061 0.9239* 0.0057 0.9864 6.0113* UK 0.0013* 0.2186* -0.095* 0.9574* 0.0320* 0.9976 9.0356*
(0.0004) (0.0304) (0.0250) (0.0164) (0.0073) 0.9497 (0.4041) (0.0001) (0.0170) (0.0191) (0.0065) (0.0073) 0.9776 (0.8716)
0.0004 0.0008* -0.0049 0.1656* -0.0020 0.9796* 0.9485 LL -0.000* 0.0008* 0.0565* 0.2343* 0.0003 0.9623* 0.9649 LL
(0.0002) (0.0001) (0.0188) (0.0135) (0.0090) (0.0033) 0.9281 15438.0 (0.0002) (0.0002) (0.0182) (0.0209) (0.0075) (0.0072) 0.9506 16512.5
No es: S anda d e o s a e epo ed in pa en heses. An as e isk indica es signi icance a he 5% le el. d. . e e s o deg ees o eedom o he -dis ibu ion. LL
e e s o he alue o he log-likelihood unc ion.
23
Table 3: Res ic ed Es ima ed GARCH(1,1)-BEKK Models
Panel A: 1s subsample (31/12/84-31/12/94) Panel B: 2nd subsample(1/1/95-8/10/04)
c
11 α11 α12 b11 b
12 d. . c11 α11 α12 b11 b
12 d. .
c
21 c
22 α21 α22 b21 b
22
Eigen-
alues (s.e.) c21 c
22 α21 α22 b21 b
22
Eigen-
alues (s.e.)
Canada 0.0010* 0.2161* 0.9567* 0.0099* 0.9923 5.0970*
Canada 0.0007* 0.1896* 0.0495* 0.9796* -0.0116* 0.9968 7.3827*
(0.0001) (0.0170) (0.0073) (0.0026) 0.975 (0.3232) (0.0001) (0.0132) (0.0164) (0.0028) (0.0042) 0.9956 (0.5843)
0.0007* 0.1580* 0.9835* 0.975 LL
0.0006* 0.0007* 0.2555* 0.9652* 0.9939 LL
(0.0001) (0.0125) (0.0022) 0.9619 17065.9
(0.0002) (0.0001) (0.0169) (0.0044) 0.9939 17026.8
F ance 0.0036* 0.2692* 0.0424* 0.9218* 0.9886 5.6243*
F ance 0.0015* 0.2108* 0.9696* 0.0064* 0.9962 8.4791*
(0.0004) (0.0263) (0.0197) (0.0143) 0.9459 (0.3983) (0.0002) (0.0162) (0.0047) (0.0020) 0.9900 (0.8214)
0.0002* 0.0008* 0.1465* 0.9834* 0.9459 LL
0.0008* 0.2375* 0.9694* 0.9900 LL
(0.0001) (0.0001) (0.0122) (0.0023) 0.9221 15137.2
(0.0001) (0.0148) (0.0036) 0.9846 15975.4
Ge many 0.0028* 0.2667* 0.0545* 0.9373* 0.9881 5.2190*
Ge many 0.0009* 0.2233* 0.0187* 0.9726* 0.9964 8.4130*
(0.0003) (0.0221) (0.0152) (0.0089) 0.9603 (0.3334) (0.0002) (0.0145) (0.0075) (0.0033) 0.9959 (0.8164)
0.0009* 0.1445* 0.9835* 0.9603 LL
0.0009* 0.2358* 0.9700* 0.9959 LL
(0.0001) (0.0109) (0.0022) 0.9496 15196.3
(0.0001) (0.0146) (0.0036) 0.9957 15919.6
I aly 0.0021* 0.2426* 0.9596* 0.9898 5.6196*
I aly 0.0017* 0.2335* 0.9597* 0.0044* 0.9964 8.6295*
(0.0003) (0.0216) (0.0072) 0.9796 (0.3938) (0.0002) (0.0193) (0.0068) (0.0017) 0.9859 (0.8451)
0.0008* 0.1435* 0.9845* 0.9795 LL
0.0008* 0.2336* 0.9705* 0.9859 LL
(0.0001) (0.0121) (0.0022) 0.9795 14977.0
(0.0001) (0.0154) (0.0037) 0.9755 16217.8
Japan 0.0025* 0.3386* 0.9317* 0.9891 5.1422*
Japan 0.0021* 0.2112* 0.9682* 0.0060* 0.9936 8.4674*
(0.0003) (0.0231) (0.0085) 0.9827 (0.3418) (0.0003) (0.0171) (0.0052) (0.0025) 0.9893 (0.8384)
0.0008* 0.1569* 0.9821* 0.9681 LL
0.0008* -0.0115* 0.2323* 0.9712* 0.9874 LL
(0.0001) (0.0128) (0.0025) 0.9681 14885.1
(0.0001) (0.0058) (0.0151) (0.0036) 0.9874 15240.8
UK 0.0028* 0.2601* 0.9355* 0.9851 6.0111*
UK 0.0014* 0.2192* -0.0943* 0.9575* 0.0317* 0.9976 9.0840*
(0.0004) (0.0270) (0.0130) 0.9583 (0.3949) (0.0002) (0.0168) (0.0190) (0.0063) (0.0070) 0.9781 (0.8752)
0.0003* 0.0009* 0.1631* 0.9791* 0.9583 LL
-0.0006* 0.0008* 0.0561* 0.2347* 0.9626* 0.9658 LL
(0.0001) (0.0001) (0.0130) (0.0028) 0.9428 15437.0
(0.0002) (0.0002) (0.0145) (0.0160) (0.0043) 0.9513 16511.2
No es: See Table 2.
24
Table 4: Likelihood Ra io Tes s
Canada F ance Ge many I aly Japan UK
1s subsample
LR-s a . 0.8529 3.1034 3.0371 2.8405 11.2143 2.0793
d. . 6 6 7 8 8 7
p- alue 0.9906 0.7958 0.8815 0.9440 0.1898 0.9553
2nd subsample
LR-s a . 4.4864 4.8151 3.1584 2.4012 3.6086 2.6975
d. . 5 5 5 6 6 2
p- alue 0.4817 0.4389 0.3969 0.8794 0.7295 0.2596
No es: The null hypo hesis es ed is: Res ic ed Model p e e ed o Un es ic ed Model.
Table 5: His o ical Shocks
Canada F ance Ge many I aly Japan UK
C ash 1987
Z10 -9.74 -3.99 -2.46 -3.63 -7.28 -8.63
Z20 -14.04 -16.85 -16.97 -17.39 -17.32 -15.74
Asian C isis
Z10 -8.83 0.68 -0.41 -0.82 -1.37 0.96
Z20 -4.24 -7.47 -7.07 -6.82 -6.85 -7.05
Twin Towe s
Z10 1.16 0.68 -0.41 -2.37 -1.97 2.79
Z20 -5.27 -7.47 -7.07 -4.65 -4.88 -5.76
Table 6: Ini ial S a e H0
Canada F ance Ge many I aly Japan UK
h11,0 0.7390 1.0380 1.0517 0.5925 1.4174 0.5321
h12,0 0.3397 0.4019 0.4491 0.1299 0.2504 0.1942
h22,0 0.5724 0.5587 0.5553 0.5486 0.5272 0.5993
No es: Figu es a e exp essed in 10-4.