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HenrykGurgul,RobertSyrek
specificationwhereconditionaldistributionsareskewt-distributionofFernandezand
Steel(1998).Thesedistributionsareparameterizedwithskewξ
mandshapev
mparam-
eters(ξ
sandv
sforsubindexreturns).Inordertoobtainuniformlydistributedvalues
weapplythetransformation:
u
it
=
F
mt
(
r
mt
|
μ
mt
ł
h
mt
ł
ξ
m
ł
v
m
)
and
u
st
=
Fr
st
(
st
|
μ
mt
ł
h
mt
ł
ξ
m
ł
v
m
)
.
Todescribethedependencestructureofreturnsweapply(inthecalculationswithR
packagermgarchbyAlexiosGhalanos)bivariatetcopulawithtime-varyingcorrela-
tionR
tandconstantparameterυ.Thedensityofbivariatetdistributionisgivenby:
cu
t
(
mt
ł
u
st
|
R
t
ł)
υ
=
R
t
12
/
Γ
(
|
k
Γ
υ
k
|
(
υ
+
2
2
+
2
1
N(N
|||
)k)k
N
|
Γ
)k
(
|
1
υ
2
+
(
z
|
υ
1
mt
2
+
N
|
)
zRz
-
υ
′
2
+
1
υ
t
(
|
k
-
1
1
+
N
|
)
-
z
υ
υ
2
st
+
2
2
N
|
)
-
υ
+
2
1
,
(2)
where
z
=
(
t
υ
-
1
(
u
mt
)ł
t
υ
-
1
(
u
st
)
)
andt
υ(·)isunivariatetdistributionwithυdegreesof
freedom.
InthispaperweusethedynamicconditionalcorrelationDCC(1,1)modelofEngle
(2002)whichcanbysummarizedas:
H
t
=
DRD
t
t
t
,
D
t
=
diag
(
h
mt
ł
h
st
)
,
R
t
=
diag(
Q
t
)
-
12
/
Q
t
diag(
Q
t
)
-
12
/
,
Q
t
=
(
1
--
ab
)
Q
+
a
ćć
t
-
1
t
′
-
1
+
b
Q
t
-
1
,
(3)
where
Q
isunconditionalcovariancematrixofstandardizedresiduals
ć
t
.Itispossible
totakeintoaccounttheasymmetriceffectinthissetting.Cappielloetal.(2006)intro-
ducedasymetricDCCmodelinwhichnegativeshocksandpositiveshockshavedifferent
impactonfuturecorrelations.
3.2.SystemicriskmeasurementwithΔCoVaR
ThesystemicriskmeasurewhichisbasedontheconceptofValue-at-Risk(Adrian
andBrunnermeier2011,2016)isΔCoVaR.Let
(
()
r
st
besomeeventforreturnsofsub-
indexs.ThenCoVaRatconfidencelevelαcorrespondstoconditionalVaRofthemarket
return(system):
Pr
(
mt
≤
CoVaR
m
t
|(
C
r
st
)
|()
C
r
st
)
=
α
.
(4)
