Does time varying risk premia exist in the international bond market? An empirical evidence from Australian and French bond market

A ab, Hi a; Beg, Rabiul Alam
A icle
Does ime a ying isk p emia exis in he in e na ional
bond ma ke ? An empi ical e idence om Aus alian and
F ench bond ma ke
In e na ional Jou nal o Financial S udies
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: A ab, Hi a; Beg, Rabiul Alam (2021) : Does ime a ying isk p emia exis in
he in e na ional bond ma ke ? An empi ical e idence om Aus alian and F ench bond ma ke ,
In e na ional Jou nal o Financial S udies, ISSN 2227-7072, MDPI, Basel, Vol. 9, Iss. 1, pp. 1-13,
h ps://doi.o g/10.3390/ij s9010003
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/257749
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
In e na ional Jou nal o
Financial S udies
A icle
Does Time Va ying Risk P emia Exis in he In e na ional Bond
Ma ke ? An Empi ical E idence om Aus alian and F ench
Bond Ma ke
Hi a A ab 1,* and A. B. M. Rabiul Alam Beg 2


Ci a ion: A ab, Hi a, and A. B. M.
Rabiul Alam Beg. 2021. Does Time
Va ying Risk P emia Exis in he
In e na ional Bond Ma ke ? An
Empi ical E idence om Aus alian
and F ench Bond Ma ke .
In e na ional Jou nal o Financial
S udies 9: 3. h ps://doi.o g/
10.3390/ij s9010003
Recei ed: 7 Sep embe 2020
Accep ed: 17 Decembe 2020
Published: 4 Janua y 2021
Publishe ’s No e: MDPI s ays neu-
al wi h ega d o ju isdic ional clai-
ms in published maps and ins i u io-
nal a ilia ions.
Copy igh : © 2021 by he au ho s. Li-
censee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and con-
di ions o he C ea i e Commons A -
ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Ins i u e o Business & In o ma ion Technology, Uni e si y o he Punjab, Laho e 54000, Pakis an
2College o Business, Law & Go e nance, James Cook Uni e si y, Towns ille 4811, Aus alia;
[email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +92-322-777-6498
Abs ac :
The p esence o isk p emium is an issue ha weakens he a ional expec a ion hypo hesis.
This pape in es iga es changing beha io o ime a ying isk p emium o holding 10 yea ma u i y
bond using a bi a ia e VARMA-DBEKK-AGARCH-M model. The model allows o asymme ic isk
p emia, causali y and co- ola ili y spillo e s join ly in he global bond ma ke s. Empi ical esul s
show signi ican asymme ic pa ial co- ola ili y spillo e s and isk p emium exis in he bond
ma ke s. The es ima es o he bi a ia e isk p emia show bi-di ec ional causali y exis be ween he
Aus alia and F ance Bond ma ke s. O e all esul s sugges nonexis ence o pu e a ional expec a ion
heo y in he isk p emium model. This in o ma ion is use ul o he agen s’ s a egic policy decision
making in global bond ma ke s.
Keywo ds:
asymme ic ola ili y; isk p emium; pa ial co- ola ili y spillo e s; bond ma ke ; G1;
C40; C13; C18
1. In oduc ion
Condi ional ola ili y models a e ou inely es ima ed wi hin he uni a ia e and mul i-
a ia e con ex s o ime a ying e u n ola ili y, isk-p emia, and ola ili y spillo e s in
he high- o-low equency inancial da a. Since ola ili y is unobse able, he esea che s
ha e a gued o model ola ili y u ilizing (i) ealized ola ili y, (ii) implied ola ili y, and
(iii) condi ional ola ili y in he inancial ma ke s, see McAlee e al. (2009) and Tsay
(2010).
Engle (1982) i s explici ly de eloped a condi ional ola ili y model known as
au o eg essi e condi ional he e oscedas ici y (o ARCH) model. Subsequen ly, Bolle sle
(1986) ex ended he ARCH o Engle (1982) o include dynamic ola ili y in he ARCH
speci ica ion, known as he gene alized ARCH (o GARCH) model. Tsay (1987) showed
ha he Engle’s (1982) ARCH can be de i ed om a i s o de andom coe icien au o e-
g essi e p ocess. The basic ARCH and GARCH models a e popula in applied uni a ia e
economics and inance, ye hey a e incapable o cap u e asymme ic “news” ha o en
a i es in he inancial ma ke s du ing he pe iods o asse ading and delayed ansac ions.
Since “news” a e unobse able and andom, a ious p oxies ha e been used in he inance
li e a u e o ackle he unobse able na u e o he news a iable, see Glos en e al. (1993),
Ding e al. (1993), Beg and Anwa (2014) among o he s. As he inancial ola ili y o e u ns
a e “news” dependen , i is o in e es o c ea e a a iable ha can be used as a p oxy
o he “news” o unde s and he e ec o he so-called “good” and “bad” news in he
inancial ma ke s in gene al. Wi hin he uni a ia e con ex , Glos en e al. (1993) de elop a
h eshold ype GARCH (synonym wi h TGARCH o GJR-GARCH o asymme ic GARCH)
and Nelson (1991) de eloped asymme ic ola ili y model known as Exponen ial GARCH
(EGARCH) model. Bo h he GJR and EGARCH cap u e news e ec o ola ili y bu hei
unc ional o ms a e di e en . Engle and Ng (1993) de elop nonpa ame ic diagnos ic es
ha emphasize he asymme y o ola ili y esponse o news. The condi ional ola ili y
In . J. Financial S ud. 2021,9, 3. h ps://doi.o g/10.3390/ij s9010003 h ps://www.mdpi.com/jou nal/ij s
In . J. Financial S ud. 2021,9, 3 2 o 13
speci ica ion o Ding e al. (1993) is called asymme ic powe ARCH (o APARCH) model.
The APARCH model nes s bo h asymme ic model o Glos en e al. (1993) and EGARCH
model o Nelson (1991). Ano he de elopmen o he uni a ia e ARCH/GARCH model
is he ime a ying ARCH-in-mean (o ARCH-M) model, i s in oduced by Engle e al.
(1987). This model cap u es isk p emium o holding isky asse s.
In his pape , we employ bo h uni a ia e and bi a ia e asymme ic GARCH-in mean
(AGARCH-M) model whe e a condi ional a iance is a de e minan o ime- a ying isk
p emia. Which en e s in he o ecas equa ion o he expec ed bond e u ns. Any inc ease
in he expec ed e u n will be iden i ied as isk p emium. The p esence o isk p emium is
an issue ha weakens he a ional expec a ion hypo hesis see, Shille (1978,1981), Shille
e al. (1983); Campbell (1986); Engle e al. (1987) among o he s o he uni a ia e case.
This pape , explo es linkages be ween Aus alia and F ance bond ma ke s connec ing wo
di e en con inen s using bi a ia e VARMA-DBEKK-AGARCH-M model. The model is
es ima ed by he quasi-maximum likelihood (in absence o mul i a ia e Gausini y) o he
Aus alian and F ench da a. The model in es iga es he di ec ion o causali y, co- ola ili y
spillo e s, and p esence o isk p emia join ly ac oss he wo ma ke s. The exis ence o
ime a ying isk p emium, and asymme ic co- ola ili y spillo e s a e he mos aluable
sou ces o in o ma ion h ough which e icien po olio alloca ion and di e si ica ion can
be unde s ood ac oss ma ke s bo h locally and globally. This in o ma ion is use ul o
measu ing and p edic ing ola ili y, p icing secu i ies, and isk managemen in gene al.
The s uc u e o he pape is as ollows. In Sec ion 2, Re iew o ela ed li e a u e and
in Sec ion 3, Da a, models and me hodology a e discussed. Sec ion 4p esen s empi ical
esul s. Finally, Sec ion 5concludes he pape .
2. Re iew o Rela ed Li e a u e
Ma kowi z (1959) de eloped a es able o m o asse alloca ion u ilizing mean- a iance
app oach. The p inciples o he mean- a iance lies on he ollowing op imiza ion ules:
•Minimize he a iance o po olio e u n gi en expec ed e u n, and
•Maximize expec ed e u n, gi en a iance.
Mo i a ed by he wo k o Ma kowi z (1959); Sha pe (1964) and Lin ne (1965) in-
dependen ly de eloped a model o “dependence” be ween expec ed e u ns and isk in
dealing wi h isk– e u n nexus. Sha pe (1964) and Lin ne (1965), in oduce hei models
wi h addi ional wo key assump ions: (a) All in es o s a e assumed o ollow he mean
a iance ule, and (b) unlimi ed lending and bo owing a he isk– ee a e,
, which does
no depend on he amoun bo owed o len .
This model is known as capi al asse p icing model (CAPM). The heo y o CAPM
s a es ha he isk p emium on a secu i y is p opo ional o he isk p emium on ma ke
po olio. Tha is
i− ∝ m− 
, whe e
i
and
a e he e u ns on secu i y
i
and he
isk- ee a e, espec i ely,
m
is he e u n on he ma ke po olio, and he p opo ionali y
cons an o he model is deno ed by
βi
is he
i−
h secu i y’s “be a” alue. A s ock’s be a is
impo an o in es o s and policy make s since i e eals he s ock’s ola ili y. This model
has been ex ensi ely used in empi ical inance. Al hough heo e ically, he CAPM is sound
bu su e s om empi ical e idence.
Poo empi ical pe o mance o he adi ional s a ic CAPM gi es ise o modi ica ion
o he CAPM. Basu (1977,1983) ga e e idence ha when common s ocks a e so ed on
ea nings–p ice a ios (E/P), he u u e e u ns on high E/P s ocks a e highe han p edic ed
by he adi ional CAPM. Banz (1981) documen ed a size e ec when s ocks a e so ed by
ma ke capi aliza ion, in which a e age e u ns on small s ocks a e highe han p edic ed
by he CAPM. S a man (1980) and Rosenbe g e al. (1985) documen ed hose s ocks wi h
high book- o-ma ke equi y a ios ha e high a e age e u ns ha a e no cap u ed by hei
be as. Fama and F ench (1992) upda ed and syn hesized he e idence on he empi ical
ailu es o he CAPM. Using he c oss-sec ional eg ession app oach, Fama and F ench
con i m ha size, ea nings–p ice, deb –equi y, and book- o-ma ke a ios added o he
explana ion o expec ed s ock e u ns p o ided by ma ke be as. Fama and F ench (1996)
In . J. Financial S ud. 2021,9, 3 3 o 13
each he same conclusion using he ime-se ies eg ession applied o po olios o s ocks
so ed by p ice a ios.
Jaganna han and Wang (1996) included o he isk ac o s, di e en om Fama and
F ench (1992), in o he model and ound some suppo o he heo y and p ac ice o CAPM.
Speci ically, hey ound some imp o emen s o he model o mon hly da a a he han
annual da a. The Fama and F ench (1992) model is known as h ee ac o asse p icing
model in inance. They u he ex ended he model by including a ew o he exogenous
a iables in o he model. The Fama and F ench (2015) i e ac o asse p icing model
di ec ed a cap u ing he size, alue, p o i abili y, and in es men pa e ns in a e age s ock
e u ns and ound ha he i e- ac o model pe o ms be e han he h ee- ac o model.
Howe e , he i e- ac o model
'
s main p oblem is i s ailu e o cap u e he low a e age
e u ns on small s ocks whose e u ns beha e like hose o i ms ha in es a lo despi e low
p o i abili y. Ra ios in ol ing s ock p ices ha e in o ma ion abou expec ed e u ns missed
by ma ke be as. Such a ios a e hus p ime candida es o expose sho comings o asse
p icing models in he case o he CAPM, sho comings o he p edic ion ha ma ke be as
su ice o explain expec ed e u ns. These obse a ions may be ega ded as misspeci ica ion
o he adi ional CAPM due o omi ed a iables. So, he consequence o he adi ional
CAPM migh su e om bias and inconsis ency.
Ano he impo an issue o he ailu e o empi ical suppo o CAPM migh be he
linea i y assump ion o expec ed e u ns. The linea model could pe o m badly in empi i-
cal applica ions i he linea i y assump ion is iola ed. I is well known ha mos o he
asse e u ns exhibi s ylized ac s, e.g., limi cycles, sudden jumps, ampli ude- equency
dependencies, and nonlinea i y. The inhe en nonlinea i y o he adi ional linea CAPM
could be a model speci ica ion p oblem. The e o e, he assump ion o linea i y o CAPM
needs o be es ed be o e adop ing such a model o policy decision analysis. One o he
sou ces o inhe en nonlinea i y may en e in o he model h ough he condi ional second
momen o he inancial e u n se ies. I linea i y does no hold hen he use o co ela ion
as a measu e o dependence be ween di e en inancial asse s is no app op ia e o op-
imal po olio selec ion by he CAPM. The e o e, he adi ional s a ic CAPM app oach
ounded on he assump ion o mul i a ia e no mali y could no be app op ia e. This may
be ega ded as unc ional misspeci ica ion o he adi ional CAPM.
The nonlinea i y ha may en e in o he e u ns se ies was i s cle e ly modelled by
he Nobel Lau ea e Robe Engle in 1982. This model is known as au o eg essi e condi ional
he e oskedas ic (ARCH) model, widely used in he inance and elsewhe e. Engle showed
ha i is possible o model he condi ional mean and condi ional a iance o a se ies o
obse a ions join ly. This heo y is a s onge addi ional con ibu ion o he adi ional s a ic
CAPM. The ARCH model cap u es a ious s ylized ac s ha exhibi ed by he inancial
asse e u ns, such as ola ili y clus e ing, asymme y, and a high deg ee o pe sis ence.
Bolle sle (1986) ex ended Engle’s (1982) ARCH by de eloping a echnique ha allows
he condi ional a iance o be an au o eg essi e mo ing a e age (ARMA) p ocess. This
expanded condi ional a iance is widely known as Gene alized Au o eg essi e Condi ional
He e oskedas ic (GARCH) model, Bolle sle (1986). These wo models a e widely used in
empi ical inance o ola ili y modelling.
Al hough he ARCH/GARCH models a e popula and ex ensi ely used in inance
li e a u e, howe e hey a e es ic ed only o symme ic in o ma ion. Va ious ex ensions
o ARCH/GARCH has appea ed in he li e a u e o o e come some inhe en nonlinea i y
p oblems wi hin GARCH class o models. Since ola ili y clus e ing is he likely cha ac-
e is ic o inancial e u ns which is nonlinea by na u e, can be modelled by asymme ic
-dis ibu ion, gene alized e o dis ibu ion, ex eme- alue heo y among o he s. A popu-
la nonlinea ex ension o ARCH/GARCH is he Nelson’s (1991) exponen ial gene alized
au o eg essi e condi ional he oscedas ici y (EGARCH) model. I a emp ed o include
asymme ic impac o shocks on ola ili y. In addi ion, his model does no equi e he
non-nega i i y es ic ions on he pa ame e s con adic o y o ARCH/GARCH condi ional
ola ili y models.
In . J. Financial S ud. 2021,9, 3 4 o 13
Ano he popula ex ension o GARCH is he Glos en e al. (1993) is known as GJR-
e u n ola ili y model in inancial econome ics. O he asymme ic models include “News
impac cu es” o Engle and Ng (1993), nonlinea asymme ic GARCH (NAGARCH) and
ec o AGARCH (VAGARCH) o Engle (1990) among o he s. These models ha e di e en
cen e s han he EGARCH and GJR. I is impo an o no e ha i a nega i e e u n shock
causes mo e ola ili y han a posi i e shock o he same size, he classical GARCH model
unde p edic s he amoun o ola ili y ollowing bad news and o e p edic s he amoun
o ola ili y ollowing good news.
The asse p icing heo ies ag ee ha a high isk has o be compensa ed by highe
expec ed e u ns. I is he e o e, easonable o include a iance in o he expec ed e u n
model o ake accoun o isk p emium. The esul ing model is known as ARCH-in-Mean
(ARCH-M) model and GARCH-in-Mean (GARCH-M) models wi hin he ARCH/GARCH
con ex . ARCH-M model was i s in oduced by Engle e al. (1987) in he uni a ia e
con ex . Signi icance o he ARCH-M could be ea ed as a ailu e o e icien ma ke
hypo hesis (EMH) ep esen ed by he adi ional CAPM. Bolle sle e al. (1988) add ess he
issue o isk p emia wi hin he mul i a ia e GARCH-M amewo k. Thei esul suppo
he ime- a ying condi ional a iance–co a iance o he asse e u ns. They also ound
signi ican isk p emia in luenced by he condi ional momen esul s. On he issue o isk
p emium, Ch is o e sen e al. (2012) de i ed he dis ibu ion o e u ns using a wo- ac o
ola ili y model, namely dynamic ola ili y and dynamic jump in ensi y. In hei model
each ac o ies own isk p emium. Using U.S. e u ns, hey ind s a is ically signi ican
esul s which ou pe o m he s anda d model wi hou he jumps. They ound signi ican
isk p emium on he dynamic jump in ensi y which has a much la ge impac on op ion
p ices. Combining he jump di usion and GARCH, A shanapalli e al. (2011) es ed he
isk– e u n ela ionship in he U.S. s ock e u ns. They ound signi ican ela ionship
be ween he isk and he e u n.
Campbell e al. (2020), gene a es ime- a ying isk p emia on bonds and s ocks
based on consump ion-based habi model o homoskedas ic mac oeconomic dynamics.
They ound co-mo emen o mac oeconomic dynamics o s ocks and bonds e u n. This
in o ma ion could help modelling ime- a ying isk p emia in a wide ange o consump ion-
based isk p emium. Coch ane and Piazzesi (2005) s udied ime a ia ion in expec ed
excess bond e u ns. They ocus on eal isk p emia in he eal e m s uc u e. Thei
mul iple eg ession o excess e u ns on all o wa d a es p o ide s onge e idence agains
expec a ions hypo hesis. Which indica es ha a single linea combina ion o o wa d
a es o ecas s e u ns o all ma u i ies. They do no include he ime- a ying p emium o
mac oeconomic o mone a y undamen s in he model.
3. The Da a, Models and Me hodology
3.1. The Da a
The 10 yea ma u i y bond p ice se ies o he Aus alia and F ance ma ke s a e
ex ac ed om The Bloombe g da abase. The se ies s a s a 4 Janua y 1990 and he sample
pe iod ends on 30 Decembe 2016. The daily bond e u n ha ing ma u i y o en yea is
cons uc ed using
= 100 * ln (p /p −1) (1)
whe e
p
is he bond p ice a ime and
p −1
is he one pe iod lag se ies. The
in (1) is
called he con inuously compounded e u n o log e u n in pe cen age.
Empi ical analysis begins wi h nume ical desc ip i e s a is ics and g aphical means
o obse ing he p ope ies o o he Aus alian and F ench bond e u ns da a. We hen
pe o m he uni oo es s ollowed by diagnos ic es s o he e u n se ies o examine
he s a is ical p ope ies in Sec ion 4. We epo he quasi-maximum likelihood es ima es
(QMLEs) in absence o Gaussini y o he s anda dized e u n shock o he uni a ia e au-
o eg essi e mo ing a e age asymme ic GARCH in mean (ARMA-AGARCH-M) models
ollowed by he mul i a ia e es ima ion o he wo coun y’s ec o au o eg essi e mo ing
a e age diagonal BEKK—asymme ic GARCH-M (VARMA-DBEKK-AGARCH-M) model.

In . J. Financial S ud. 2021,9, 3 5 o 13
The G ange causali y o he bond e u ns, and he pa ial co- ola ili y spillo e s o he
bi a ia e bond e u ns in es iga ed.
3.2. Speci ica ion o he Model
To unde s and he dynamic in e dependence o bond e u ns, ime- a ying isk-
p emium, causali y, and co- ola ili y spillo e s, we u ilize VARMA-DBEKK-AGARCH-M
model. This model nes s a wide ange o mul i a ia e ola ili y models and conside s
a ious issues o modelling eal inancial se ies. This model is capable o ex ac asymme y,
G ange - ype causali y and Chang e al. (2018) ype co- ola ili y spillo e s be ween asse s
ac oss coun ies.
3.2.1. Uni a ia e Models o Condi ional Mean and Condi ional Vola ili y
Fo he condi ional mean o secu i y e u n,
, we use Box–Jenkin’s au o eg essi e
mo ing a e age (ARMA) model as ollows.
|F −1=φ0+∑k
j=1φj −j+∑m
l=1ψlε −l+ε (2)
whe e
φo
,
φj
,
ψl
a e scala pa ame e s o he ARMA(
k
,
m
) p ocess,
ε
is he inno a ion
o e u n shock,
F −1
is he se o in o ma ion a ailable a ime
. The AIC and BIC a e
ou inely used in empi ical applica ions o ARMA o de selec ion. A benchma k model
o ola ili y p oposed by Bolle sle (1986) called gene alized au o eg essi e condi ional
he e oskedas ici y (GARCH) model, which akes he ollowing o m.
GARCH(p,q):
E(ε2
|F −1) = h =w+∑q
j=1αjε2
−j+∑p
l=1βlh −l(3)
The o de o GARCH q= 1 and p= 1 has been ound app op ia e in eal applica ions,
Bolle sle (1986,1987).
The GARCH is a gene aliza ion o Engle (1982) au o eg essi e condi ional he e oskedas-
ic (ARCH) model. The basic ARCH/GARCH model canno dis inguish be ween he
asymme ic shocks on ola ili y, which is a common phenomenon o inancial e u n se ies.
Glos en e al. (1993) (o GJR) de elop a model which accoun s o asymme ic ola ili y
called AGARCH, akes he ollowing o m.
h |F −1=w+∑p
l=1βlh −l+∑q
j=1αjlε2
−l+∑p
l=jγjd −jε2
−j, whe e
d −j=1i ε −j<0
0o he wise
(4)
Engle e al. (1987), allows he condi ional mean e u ns o depend on i s own con-
di ional a iance. This model is sui able o analysis o he asse ma ke s’ ime a ying
isk p emiums wi h in en o conside si ua ions whe e he isk-a e se agen s equi e
compensa ion o holding isky asse s. This model is gene ally known as isk p emium
model exp essed as ollows.
=µ +δg(h )+ε (5)
whe e
µ
is he condi ional mean gene a ed by model (2),
h
is as de ined in (3). In inance,
δg(h )
ep esen he isk p emium, see Be a and Higgins (1993). In mos applica ions
g(h )=√h
has been used, o example, Domowi z and Hakkio (1985) and Bolle sle
e al. (1988). The GJR speci ica ion o condi ional ola ili y when added in he asse e u n
equa ion, he esul ing model (5) becomes GJR-GARCH-M.
In . J. Financial S ud. 2021,9, 3 6 o 13
3.2.2. Mul i a ia e Models o Condi ional Mean and Condi ional Vola ili y
Le
=( 1 , 2 , . . . . . . ., N )0
be an (
N×
1) ec o o
N
-dimensional asse e u ns o
log e u ns a he ime index =1, 2, . . . . . . , Twi h he ollowing s uc u e.
|F −1=µ +ε ,ε =H0.5e (6)
whe e
µ =E( |F −1)
is he condi ional expec a ion o he ec o
gi en he pas in o ma-
ion
F −1
and
ε =(ε1 ,ε2 , . . . . . . ., εN )0
is an
(N×1)
ec o o shocks, o inno a ion a ime
. Each componen o
ec o is a uni a ia e e u n o an asse . The
e
=
e1
,
e2
,
. . . . . .
,
eN
is an (
N×
1) ec o o
i.i.d.
andom ec o wi h p obabili y dis ibu ion, say,
G(0, IN)
,
whe e
G
is assumed o be a con inuous,
IN
is he iden i y co a iance ma ix, and 0 is
an
N×
1 mean ec o . The mul i a ia e condi ional e u n can be exp essed as a ec o
au o eg essi e mo ing a e age (VARMA) model as ollows.
=Φ0+∑k
i=1Φi −i+∑m
l=1Ψlε −l+ε (7)
whe e
Φ0
is a
(N×1)
ec o o in e cep ,
Φi
and
Ψl
a e bo h (
N×N
) ma ices o each iand
lo a ious lags. The ec o o e u ns can be es ed o s a iona i y. The
(N×N)
co a iance
ma ix
H
wi h componen
hij
,
(i,j=1, 2, . . . . . . N)
, need o be speci ied. Va ious o ms o
H
ha e been p oposed, o example, Sil ennoinen and Te äs i a
(2009)
,Bauwens e al.
(2006)
,Tsay (2006). The wo mos popula mul i a ia e condi ional ola ili y speci ica ions
a e he Bolle sle e al. (1988) VEC and Engle and K one (1995) BEKK speci ica ions. In
he p esen pape we ocus on a diagonal a ian o BEKK.
3.2.3. VARMA-DBEKK-AGARCH-in Mean Model
We conside he ollowing o m o he mul i a ia e isk p emia model.
Re u n:
|F −1=Φ0+∑k
i=1Φi −i+∑m
l=1Ψlε −l+δH1/2
+ε (8)
DBEKK-AGARCH:
H |F −1=C0C+∑q
j=1Ajε −jε0 −jA0+∑p
l=1BlH −lB0
l+∑q
j=1ΓjDj ×ε −jε0 −jΓ0
j(9)
In model (8) he condi ional expec ed e u n is augmen ed by he unc ion o condi-
ional ola ili y model (9). In he abo e model (9), he ma ices
A
,
B
, and
Γ
a e assumed
o be diagonal. The ma ix
C
is a lowe iangula ma ix. A simple e sion o (9) wi h
p=1 and q=1, akes he ollowing o m.
H |F −1=C0C+Aε −1ε0 −1A0+BlH −1B0+Γ(D −1×ε −1ε0 −1Γ0(10)
whe e
D −1=1i ε −1<0
0i ε −1≤0
he o he a iables a e as de ined abo e. Le us assume ha
ε =H η
, whe e
η
is a ec o
o iden ically and independen ly dis ibu ed ec o o andom a iables wi h mean ze o
ec o and uni a iance co a iance ma ix.
3.3. Es ima ion
The es ima o s o he pa ame e s o he models (8) and (9) a e ob ained by maximizing
he log likelihood unc ion
L(θ)=1
T∑T
=1l (θ)(11)
whe e l (θ)=−0.5 ∑ ln|H |+ε H−1
ε0 
In . J. Financial S ud. 2021,9, 3 7 o 13
whe e
θ
is he se o pa ame e s o he models (8) and (9),
l (θ)
is he log o he a gumen , |.|
is he de e minan o he a gumen . Equa ion (11) akes he o m o he Gaussian likelihood.
Because we do no assume mul i a ia e no mali y o he s anda dized e u n shock
η
,
es ima o s o he pa ame e s om (11) a e he quasi-maximum likelihood es ima o s
(QMLEs). The QMLEs a e consis en and asymp o ically no mal, see Ling and McAlee
(2003), Chang e al. (2018). The e o e, he classical asymp o ic es s a e alid o s a is ical
in e ence.
3.4. Co-Vola ili y Spillo e s E ec
Vola ili y spillo e e ec s can be es ima ed u ilizing he de ini ions gi en in he pape
by Chang e al. (2017,2018). In his pape , we apply he pa ial co- ola ili y spillo e s as
ollows: ∂Hij
∂εk, −1
,i6=j,k=ei he io j(12)
4. Empi ical Resul s
4.1. P elimina y Da a Analysis
In his sec ion we p o ide bo h nume ical and g aphical desc ip i e analysis o he 10
yea bond a es o Aus alia and F ance, and ime se ies p ope ies o he se ies o 6144
daily obse a ions.
Table 1below p o ides he basic desc ip i e s a is ics o he Aus alia and F ance
bond e u n se ies each comp ising 6144 obse a ions.
Table 1. Desc ip i e s a is ics o he Daily 10-yea bond e u ns.
S a is ics Aus alia F ance
Mean −0.015006 (0.4072) −0.0031 (0.8938)
S anda d de ia ion (s .de ) 1.4191 1.8012
Minimum −10.7556 −15.7247
Maximum 12.9630 20.8273
Skewness 0.295699 (0.0000) 0.3952 (0.0000)
Excess Ku osis 5.2828 (0.0000) 7.5076 (0.0000)
Ja que and Be a (JB) 7232.9308 (0.0000) 14,586.7843 (0.0000)
Sample size 6144 6144
No e: 1. The e u ns a e in pe cen ages and he sample pe iod s a s 4 Janua y 1990 and ends 30 Decembe 2016.
The da a sou ce is gi en in he ex . 2. The p- alue o he es is in pa en heses.
The basic s a is ics o he wo se ies show excess ku osis, implying ha he se ies
ha e a ails. Bo h he se ies a e non-no mal by he Ja que and Be a (1987) es . Time plo
o he p ice and e u n ola ili y o each se ies shown in Figu e 1a,b below.
F om he Figu e 1a,b abo e, we obse e ha he pa e n o mo emen o he bond
p ices sloping downwa d o bo h Aus alia and F ance. Howe e , he ola ili y o e u ns
changes wi h a ying deg ee o clus e ing ac oss he wo bond ma ke s. The Aus alian
bond ma ke expe ience anquil pe iod om 2005 o 2007. Howe e , bond e u n ola ili y
s a ed luc ua ing om la e 2007 un il he beginning o 2010. Bond ma ke o F ance on he
o he hand exhibi anquili y be o e 2007. The bond ma ke o F ance inc eased sligh ly
du ing 2008 and con inue o luc ua e a a as e a e and peaked qui e high du ing 2015 o
2016 compa ed wi h 2010. This could be due o he global and Eu opean inancial c ises
and Russian inancial c isis. The Aus alian bond e u n ola ili y was compa a i ely high
du ing 1996 o 2004 han he p e ious yea s. This could be due o he Asian c ises. While
du ing 2008 o 2016 he ola ili y clus e ing was ela i ely high compa ed wi h he pe iods
2007. Bo h he ma ke s peaked up high ola ili y du ing he global inancial c isis.
In . J. Financial S ud. 2021,9, 3 8 o 13
In . J. Financial S ud. 2021, 9, x FOR PEER REVIEW 8 o 14
(a)
(b)
Figu e 1.
(
a
) Plo o Bond p ice, e u n ola ili y and squa ed e u n se ies o Aus alia; (
b
) Plo o Bond p ice, e u n
ola ili y and squa ed e u n se ies o F ance.