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Dynamic Asymmetric Tail Dependence in Asian Developed Futures
Markets
Qing Xu† Xiaoming Li Abdullah Mamun
Department of Commerce, Massey University at Albany, Auckland,
New Zealand
June 2005
† Corresponding author. E-mail: [email protected] (Qing Xu),
[email protected] (Xiaoming Li), [email protected] (Abdullah
Mamun).
mailto:[email protected]:[email protected]:[email protected]
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Abstract
This paper employs three two-parameter Archimedean copulas (BB1,
BB4, and BB7) to
investigate dynamic asymmetric tail dependence in Asian
developed futures markets over
the post-crisis period. The estimation is consistent and
asymptotic with a careful
implementation of the two-stage method. Unlike previous
empirical research, we first let
each marginal model follow a conditional skewed-t distribution.
Based on robust
inference for dynamic marginal models, it is found that higher
moments of each filtered
index return series are significantly time-dependent. We then
extend those three two-
parameter copulas incorporating time-varying tail dependence to
capture dynamic
asymmetries. The estimation results of the copulas provide
strong evidence of
asymmetric tail dependence in Asian developed futures markets.
Moreover, based on the
goodness-of-fit tests, we find that the model BB7 is the optimal
one. The model’s results
suggest that the probability of dependence in bear markets is
higher than in bull markets
in the post-crisis period. This further confirms downside
dependent risk in Asian
developed futures markets. Our empirical findings provide a
basis for hedging downside
dependent risk, and thus make a contribution to the literature
of financial risk
management.
Keywords: Tail dependence, Time varying two-parameter copula,
Two-Stage Estimation, Threshold GARCH, Conditional skewed-t
distribution, Goodness-of-fit test.
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I Introduction
Asian developed futures markets, such as the Hong Kong Futures
Exchange, the
Osaka Stock Exchange, and the Singapore International Monetary
Exchange, have
attracted an extensive research interest. Previous studies have
focused on individual index
futures listed on these three futures markets. See, for example,
Fung, Cheng, and Chan
(1997), Chen, Duan, and Hung (1999), Cheng, Fung, and Chen
(2000), Duan and Zhang
(2001), Kim, Ko, and Noh (2002), Chung, Kang, and Rhee (2003),
Cheng, Jiang, and Ng
(2004), and So and Tse (2004). Little attention has been paid to
nonlinear dependence,
especially tail dependence caused by extreme events, between
these markets. The present
paper aims to fill this void using a technique known as the
copula method. Our study has
found strong evidence not only of asymmetric tail dependences
across the markets, but
also of downside dependent risks within the markets.
Tail dependence plays an increasingly important role in optimal
assets allocation and
asset pricing. A number of empirical studies have recently
uncovered that correlations
between international equity markets are higher during market
downturns than during
market upturns1. If all stock prices tend to fall together as
tail events occur, the value of
diversification might be overstated by those not taking the
increase in downside
dependence into account (Ang and Chen, 2002). As a consequence,
international
diversification is less beneficial than expected, and investors
have to reallocate more
assets into foreign markets with near-normal correlation
profiles to avoid the downside
risk. As far as asset pricing is concerned, asymmetric tail
dependence should also be
considered for valuing deep out-of-the-money puts and calls
since the structure of
dependence is different between left and right tails.2
Previous studies on dynamic dependence between markets are,
however, all based on
multivariate GARCH method (e.g. De Santis and Gerard (1997), and
Kroner and Ng
(1998)) and fail to capture tail dependence induced by rare
events. Until recently, there
has been a growing interest in applying copula theory in the
finance area. A copula is a
special multivariate distribution function which can fully
capture tail dependence among
1 e.g. Erb, Harvey and Viskanta (1994), King, Sentana and
Wadhwani (1994), De Santis and Gerard (1997), Longin and Solnik
(1995, 2001), Ang and Bekaert (2002) and, Ang and Chen (2002). 2
See Poon, Rockinger and Tawn (2004).
1
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two or more random variables. The major financial applications
can be found in Bouye et
al (2000), Bradley and Taqqu (2003), Embrechts, Lindskog, and
McNeil (2003), and
Cherubini, Luciano, and Vecchiato (2004), among others. Although
the copula model is a
new tool for evaluating multivariate dependence, empirical
fronts have been expanded in
many financial directions including asymmetric patterns of
financial market
comovements. See, for instance, Costinot et al (2000), Hu (2002,
2004), and de la Pena et
al (2004)).
Following Patton (2001, 2004), our methodologies are different
in three respects from
the above-cited studies. First, in estimating the model of a
marginal distribution, we
parameterize the dynamics of the conditional third and fourth
moments along with the
threshold heteroscedasticity process using Hansen’s (1994)
method. Compared to a large
body of previous researches which assumed that standardized
innovations are subject to
either a log-normal or a standard normal distribution, our
methods can better reflect the
characteristics of underlying returns. Second, another novelty
in our econometric
methodology is that we employ several dynamic two-parameter
Archimedean copulas to
trace dynamics of asymmetric dependence, rather than static
one-parameter Archimedean
copulas popularly used in the existing empirical literature. The
advantage of the dynamic
two-parameter Archimedean copula is that it can simultaneously
capture the time-varying
upper and lower tail dependences. This enables us to redress the
possible biasedness or
inaccuracy of the static one-parameter Archimedean copula that
assumes only one tail
dependence (either upper or lower) between bivariate random
variables. Third, to
enhance the consistence and efficiency of our two-stage maximum
likelihood estimates,
the asymptotic variance-covariance matrix are calculated by the
so called “sandwich
estimator” proposed by Newey and McFadden (1994) and White
(1994). The ensures
more robust statistic inference.
To our best knowledge, the present paper is the first to study
dynamic asymmetric
dependence using time-varying two-parameter copulas on Asian
developed futures
markets. It is organized as follows. In Sections 2 and 3, we
outline the models and the
estimation method. Section 4 presents and discusses empirical
results. Concluding
remarks are given in Section 5.
2
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II Models 2.1 Archimedean Copulas
A copula function C is defined as a cumulative distribution
function
(cdf) for a multivariate vector with support in [0, 1] :
),......,,( 21 mxxx
,( 21 XX ),......, mXm
C = Pr),......,,( 21 mxxx ),......,,( 2211 mm xXxXxX ≤≤≤ (2.1)
Let F be an m-dimensional distribution function with continuous
margins
. Then F has a unique copula for all x: mFFF ,......, 21
)](),......,(),([),......,,( 221121 mmm xFxFxFCxxxF = (2.2) If
each (i = 1, 2,……, m) and C are differentiable, the joint
density
is yielded as
iF ),......,,( 21 mxxxf
)(......)()(),......,,( 221121 mmm xfxfxfxxxf ×××=
)](),......,(),([ 2211 mm xFxFxFc× (2.3) where is the density
corresponding to and )( ii xf iF
)()......()()](),......,(),([
)](),......,(),([2211
22112211
mm
mmm
mm xFxFxFxFxFxFC
xFxFxFc∂∂∂
∂=
)(......)()(
),......,,(
2211
21
mm
m
xfxfxfxxxf×××
= (2.4)
is the density of copula.
A family of bivariate copulas known as Archimedean copulas
offers us an easy way
to capture asymmetry in joint skewness and joint kurtosis.
Although several popular one-
parameter Archimedean copulas such as Clayton, Frank, and Gumbel
copulas have been
extensively studied in previous researches, they cannot
distinguish between lower and
upper tail dependences. However, correlation between financial
markets may well be
asymmetric between market downturns and market upturns. For this
reason, we employ
three families of two-parameter Archimedean copulas referred to
as BB1, BB4 and BB7
in this study. From Fig. 1 and Fig. 2, it is clear that these
copulas can capture asymmetric
tail dependence simultaneously.
3
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2.2 Tail Dependence
According to Joe (1997), if a bivariate survival copula C of a
bivariate copula C,
U
uuuuC τ=−
→)1/(),(lim
1 (2.6)
exists, then C has upper tail dependence if (0, 1] and no upper
tail dependence if
= 0. Similarly, if
∈Uτ Uτ
L
uuuuC τ=
→/),(lim
0 (2.7)
exists, C has lower tail dependence if (0, 1) and no lower tail
dependence if = 0. ∈Lτ Lτ
Tail dependence may be understood as the joint probability of
large market
comovements, the probability of an extremely large negative
(positive) return on one
asset given that the other asset has yielded an extremely large
negative (positive) return.
The coefficients of upper and lower tail dependences are
therefore expressed as
uuuCuuUuUuUuU
uttuttu
U
−+−
=>>=>>=→→→ 1
),(21lim)|Pr(lim)|Pr(lim1,1,21,2,11
τ
uuuCuUuUuUuU
uttuttu
L ),(lim)|Pr(lim)|Pr(lim0,1,20,2,10 →→→
=≤≤=≤≤=τ (2.8)
The closed forms for the two-parameter copulas BB1, BB4, and
BB7, and the details
of tail dependence in these three models are provided in Table
1.3 Note that the two-
parameter copulas listed in Table 1 are characterized by
constant tail dependence. To
capture the dynamics of parameters, we follow Patton (2001,
2004) and specify the
dynamic two-parameter copulas as an ARMA (1, p) process:
−⋅++Λ=
−⋅++Λ=
∑
∑
=−−
−−
=−−
−−
p
jjtjt
LLt
LLLt
p
jjtjt
UUt
UUUt
up
up
1
11
1
11
||
||
υψτδωτ
υψτδωτ
(2.9)
where and tu tυ are marginal distributions, and are
autoregressive terms, Ut
U1−τδ
Lt
L1−τδ
∑=
−−⋅p
jtU up
1|| υ− jt
−
j
1ψ and |∑=
−−⋅
p
jjt
L p1
1 | − −jtu υψ are forcing variables, and
3 The functions of densities for these three two-parameter
copulas are very long. Matlab codes are available from the author
upon request.
4
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)]exp(1/[1)( kk −+=Λ
tt XX ,2,1 ,......,,(
∑∑= =
n
t
m
j1 1
ln1∑=
n
tc
is the logistic transformation to ensure (0, 1) at all time.
Regarding the number of lag for the forcing variables, there
does not exist a general rule
to follow. Here, we try p = 1.
∈LtUt ττ ,
]ˆM
,, 1,1,2 xx tt −
[INSERT TABLE 1 HERE] [INSERT FIGURE 1 AND FIGURE 2 HERE]
III Estimation Method 3.1 Two-Stage Maximum Likelihood
Estimator
Joe (1997, 2005) proposed a two-stage estimation procedure to
estimate the unknown
parameters of a copula. In the first step, for a sample size n
with m observed random
vectors , we can estimate the parameters of each margin {θ }
parametrically.
nttmX 1, ) = M
{θ M } = arg max (3.1) Mtjtj xf ,, );(ln θ
Next, based on the estimated parameters {θ } and a given density
of the copula, the
parameter estimates of each copula {θ } can be obtained via the
maximum likelihood
method in the second step.
M
C
{θ C } = arg max (3.2) ,);(),......,(),([ ,,,2,2,1,1 Ctmtmtttt
xFxFxF θθ
Joe (2005) shows that the two-stage estimation method generally
has good efficiency
properties.
3.2 Marginal Distribution
Based on the properties of classical copulas, Patton (2001)
suggests that the σ-algebra
ℱ generated by all previous joint observations ℱ t =
),,......,,( 1,21,11,2,1 xxxx tt −σ
must be considered. Thus, for the bivariate copula
ttt xxF ,2,1 ,( | ℱ t ) = C | ℱ t ), ℱ t ) | ℱ t ) 1− ttt xF
,1,1 (( 1− tt xF ,2,2 ( | 1− 1− ∈∀ tt xx ,2,1 ,
t = 1, 2, ……, n (3.3)
5
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Accordingly, the conditional mean and conditional
heteroscedasticity can be naturally
taken into account for modeling margins. However, the
distribution of each margin is
unknown and what we can do is to assume that each margin
approximately subjects to a
specified distribution. In this research, we use a parametric
method to estimate the
margins with data being assumed to follow a conditional skewed-t
distribution with a
threshold GARCH (1, 1) process.
The threshold GARCH (1, 1) (TGARCH) proposed by Glosten et al
(1993) and
Zakoian (1994) is
102
102
100 )()(
)0,max(
)0,max(
−−−
+−
−
+
+++=
−=
=
⋅=
+=
tttt
tt
tt
ttt
ttt
hdecebah
ee
ee
zhe
ekR
(3.4)
| ℱ t ~ N (0, 1) tz 1− where , is conditional mean, is
innovation, is conditional volatility, and is standardized
residual.
)/ln(100 1−⋅= ttt XXR
tztk te th
Black (1976) notes that movements of stock prices commonly
contain “leverage
effect” or “volatility feedback effect”. That is, when the value
of a stock falls due to bad
news, volatility of returns will increase as the debt-to-equity
ratio rises. The TGARCH
model allows bad news and good news (even extreme events) to
have a different impact
on volatility while the standard GARCH model does not. The
empirical result of Engle
and Ng (1993) suggests that the TGARCH model is the best to
model asymmetry,
compared to other nonlinear GARCH models such as EGARCH or
asymmetric GARCH
(AGARCH) models. A visual method to illustrate the advantage of
TGARCH model is
the so-called news impact curve (NIC) introduced by Pagan and
Schwert (1990) and
popularized by Engle and Ng (1993).
++=−−
+−
210
2102
00 )(
)(
t
t
ec
ebdaNIC σ if (3.5)
<>
00
t
t
ee
(σ is the unconditional standard deviation).
6
-
As we will see from Fig. 5 in Section 4.3, NIC clearly shows the
asymmetric relationship
between the lagged shocks and conditional volatility, holding
constant all past and
current information.
Although the TGARCH model can fully capture the asymmetry of
conditional
volatility, the assumption that the standardized innovations
subject to standard normal
distribution is unrealistic. In view of this, Hansen (1994)
introduced a skewed-t (skt)
distribution. The original density of this unconditional
distribution is defined as
tz
++⋅
−+⋅
−+⋅
−+⋅
=+−
+−
2/)1(2
2/)1(2
1211
1211
),|(η
η
λη
ληλη
AzBCB
AzBCB
zf skt if BAzBAz
//
−≥−< (3.6)
where
124
−−
≡ηηλCA , , 222 31 AB −+≡ λ
)2/()2()2/)1((
ηηπη
Γ−+Γ
≡C .
The degree of freedom η and the skewness parameter λ are
restricted within (2, ∞ ) and (-
1, 1) respectively. Figure 4 exhibits different patterns of
skewed-t pdf with various
parameters.
[INSERT FIGURE 3 HERE]
Further, given a cdf of the traditional student-t distribution
with η degree, the cdf of
a skewed-t distribution is
η,tF
−
−++⋅
⋅⋅−
−−+⋅
⋅⋅−
=
ληη
λλ
ηη
λλ
λη
η
η
21)1(
21)1(
),|(
,
,
AzBF
AzBF
zF
t
t
skt if z
BAzBA
//
−≥−< (3.7)
For the proof of Eq. (3.7), the reader may refer to Jondeau and
Rockinger (2003). We use
the TGARCH (1, 1) model with conditional skewed-t innovations as
proposed in Jondeau
and Rockinger (2003)4 as follows
4 There are six constructed variations in Jondeau and Rockinger
(2003). In this research, we only consider model M2.
7
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)(
)(
12122
11111−−
+−
−−
+−
++Ξ=
++Ξ=
ttt
ttt
eceba
eceba
λ
η (3.8)
Note that (k) = L + ((U – L)/(1 + exp (-k))) is a logistic
function forcing Ξ ),2( ∞∈tη ,
and ,1(−∈tλ 1) where L and U are lower and upper bounds
respectively. Accordingly,
|~tz ,t( tskt λη ℱ t ). Then the marginal distributions of the
bivariate copula in Eq. (3.3)
can be obtained via
1−
),|( ,,, tititiskt zF λη , t = 1, 2,……, n, i = 1, 2.
3.3 Consistent Asymptotic Estimation
Consistency and asymptotic normality are two major desirable
properties of
maximum likelihood estimators. However, the asymptotic property
holds only when the
model is correctly specified, but it is by no means a necessary
condition for the consistent
estimation of particular parameters of interest (White (1982)).
In practice, investigators
often strongly rely on explicit distributional assumptions but
can not completely extract
information from finite samples. Thus, the deviation of the
principal asymptotic
properties of maximum likelihood estimators might induce serious
model
misspecification. Following propositions of Newey and McFadden
(1994) and White
(1994), our asymptotic covariance matrix for the two-stage
procedure is therefore
consistently estimated by the so-called “sandwich estimator”:
1−n ·H ·OPG ·H = {θ , θ } (3.9) 1− )ˆ(θ )ˆ(θ 1− )ˆ(θ θ̂ M C
where H is inverse Hessian matrix and OPG represents the
outer-product-of-the-
gradient or BHHH estimator advocated by Bernt, Hall, Hall and
Hausman (1974). As
such, the standard error of each parameter is replaced by the
robust standard error for
statistical inference.
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IV Empirical Results 4.1 Data
We choose Hang Seng index futures traded on the Hong Kong
Futures Exchange,
Nikkei 225 index futures traded on the Osaka Stock Exchange, and
Morgan Stanley
Capital International (MSCI SIN) index futures traded on the
Singapore International
Monetary Exchange, as proxies of Asian developed futures
markets. All daily data are
8
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collected from Datastream and cover the post-crisis period from
07 September 1998 to 28
February 2005. Index futures returns are defined as )/log(100
1−⋅= ttt XXR where (t
= 1, 2, ……, n) is the underlying price of an index futures.
tX
4.2 Preliminary Analysis
Table 2 presents a range of descriptive statistics for all
return series. The statistics of
first and second moments of each return series indicate that
empirical distributions are
not standard normal. Daily mean returns of Hang Seng and MSCI
SIN are positive
around 0.04% in contrast to Nikkei 225 with negative mean
–0.01%. The average of all
unconditional standard deviations is about 1.58%. The values of
skewness which ranges
between –0.0477 and 0.2352, and the values of kurtosis which
ranges between 4.8551
and 6.3724, further reveal that each return series is
asymmetrically distributed with fat
tails. Although the Ljung-Box statistics (Q ) for up to 10, 20
and 50 lags calculated for
each raw return show the absence of linear autocorrelation, the
results for squared returns
( ) strongly suggest the presence of nonlinear autocorrelation.
Meanwhile, the LM
tests of Engle (1982) also significantly exhibit heteroscedastic
effect in the data. These
descriptive statistics provide evidence that the three return
series non-normally
distributed with heteroscedasticity.
x
xxQ
[INSERT TABLE 2 HERE] 4.3 Dynamic Marginal Distributions
Table 3 presents the results of the margin models in which
asymmetries on
conditional heteroscedasticity are allowed for and higher
moments are time varying.
Based on robust standard errors, all parameters are highly
significant. The autoregressive
effect in the volatility specification is strong as is around
0.9265 suggesting extreme
clustering effects. The parameter of negative returns c is
positive and greater than the
parameter of positive returns b indicating the presence of the
leverage-effect for these
three index futures returns. Not surprisingly, the TGARCH NIC is
steeper than the
GARCH NIC for negative news, and less steep for positive news,
as shown in Fig. 5. The
asymmetric conditional variance of the TGARCH (1, 1) model
represented by circles is
0d
0
0
9
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clearly in contrast to the symmetric pattern of GARCH (1, 1)
model represented by dots.
Meanwhile, the condition for covariance-stationarity is
satisfied since
for all the three series. To capture the dynamic nonnormality of
the
residuals, we let the conditional skewness
12/)( 000
-
Straumann (2002) propose to use rank correlation such as
Spearman’s Sρ or Kendall’s
Kτ instead of Pearson’s Pρ as a measure of nonlinear
dependence.5 From panels B and C
of Table 5, we can see that the statistics of Spearman’s Sρ and
Kendall’s Kτ are all
highly significant indicating strong nonlinear dependence
between standardized residuals.
Nevertheless, these rank correlations do not tell us anything
about co-skewness and co-
kurtosis. We thus employ ellipticity test proposed by Mardia
(1970) to detect multivariate
skewness and kurtosis. Again, the results of Table 6 for both
the bivariate skewness test
and the bivariate kurtosis test allow us to reject the null
hypothesis of bivariate normality,
implying the possibility of asymmetric comovements caused by
tail dependence.
[INSERT TABLE 3 AND TABLE 4 HERE] [INSERT FIGURE 4 AND FIGURE 5
HERE] [INSERT TABLE 5 HERE] [INSERT FIGURE 6] [INSERT TABLE 6 HERE]
4.4 Time varying two-parameter copulas
We begin investigation of tail dependence with the static
two-parameter Archimedean
copulas. Table 7 shows that unconditional tail dependence
exhibits constant asymmetry.
All lower tail dependences are greater than upper tail
dependences except for the Hang
Seng-Nikkei 225 pair in model BB7. In spite of the significant
statistic values, however,
information provided in Table 7 is not good enough to trace
dynamic dependence. Table
8 sets out the results of the two-stage maximum likelihood
estimator for the time varying
two-parameter copulas. Although the intercept of conditional
lower tail dependence
for the Hang Seng-MSCI SIN pair is insignificant, all other
parameters for both models
BB1 and BB7 as shown in Table 8 are highly significant based on
robust standard errors,
Lω
5 At this stage, the Spearman’s Sρ and the Kendall’s Kτ are
calculated by nonparametric method. The parametric calculations of
the Spearman’s Sρ and Kendall’s Kτ by copula method can be found in
Embrechts, Lindskog and McNeil (2003).
11
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suggesting that tail dependence is time varying in all case.
Meanwhile, there are common
features between these two models in conditional upper tail
dependence but different
ones in conditional lower tail dependence. First, all of the
intercept are negative.
Second, the dynamic upper tail dependence parameters have a
positive relationship with
autoregressive terms ( > 0) and a negative relationship with
forcing variables (
Uω
Uδ Uψ < 0).
For the lower tails in models BB1 and BB7, the parameters of the
forcing variables Lψ are all negative except for the Nikkei
225-MSCI SIN pair in model BB1. As Patton
(2001, 2004) mentioned, the implication of the negative values
of Lψ and Uψ for the
forcing variables is that a smaller mean absolute difference ||
jt−jtu − −υ will lead to an
increase in tail dependence. Therefore, the tail dependences
within these three futures
markets are very sensitive to the distance between margins since
we only choose one lag
for the mean absolute difference. This reflects how fast the
propagation of unexpected
shocks between markets is.
The computation of the dynamic BB4 model is somewhat difficult.
Each parameter
varies sensitively with different starting value selection.
Results in the Table 8 thus are
obtained with our best effort. Among mixed results, the best fit
is only for the pair Nikkei
225-MSCI SIN while parameters are significant only in
conditional lower tail
dependences for the remaining pairs.
[INSERT TABLE 7 AND TABLE 8 HERE]
[INSERT FIGURE 7 HERE]
Turning to summaries of results for the time varying
two-parameter copulas displayed
in Table 9, the different average values of time varying lower
and upper tail dependences
provide information about dynamic asymmetries. The strongest and
weakest asymmetric
tail dependences are found in the Hang Seng-MSCI SIN and the
Nikkei 225-MSCI SIN
pairs respectively. Meanwhile, five of the nine pairs show that
the average lower tail
dependences are greater than the average upper tail dependences.
Interestingly, model
BB4 yields the results that all lower tail dependences are
averagely less than upper tail
dependences, in contrast with the results in Table 7.
12
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[INSERT TABLE 9 HERE]
4.5 Determining the Optimal Dynamic Two-parameter Copulas
Based on our estimates of the parametric copulas, we perform a
set of goodness-of-fit
test and use two information criteria for selecting the optimal
copula model. The
goodness-of-fit tests include the Kolmogorov-Smirnov test, the
Anderson-Darling test,
and the Integrated Anderson-Darling test. The basic idea of
these tests is based on a
distance measure between the empirical and theoretical
distribution function. The
empirical copula was introduced by Deheuvels (1979, 1981) and
formally defined by
Nelsen (1999). Let ( denote a sample of size n from a continuous
m-
variate distribution and ( denote the rank statistic of the
sample. The
empirical copula is the function
nttmtt xxx 1,,2,1 ),......,, =
,......,, ,2,1 tt www ),tm
∑∑= =
=
n
t
m
i
mE nn
tnt
nt
C1 1
21 1,......,, 1 [ (4.1) ], iti tw ≤
If represents the theoretical copula, then the
Kolmogorov-Smirnov test is TC
KS = max | |TE CC − (4.2)
the Anderson-Darling test is
AD = max)1(||
TT
TE
CCCC−
− (4.3)
and the Integrated Anderson-Darling test is
IAD = ∑∑= = −
−n
t
n
t TT
TE
CCCC
1 1
2
1 2)1(
)( (4.4)
Moreover, two additional tests based on the maximized
log-likelihood function (Loglik)
are the Akaike information criterion
AIC = -2Loglik + 2q (4.5)
and the Bayesian information criterion
BIC = -2Loglik + qlnn (4.6)
13
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where q is the number of coefficients. The most informative
copula is thereby selected
according to the minimum values of the above tests and
criteria.
The goodness-of-fit tests results reported in Table 10 indicate
that the time varying
model BB7 is the superior time varying two-parameter copulas for
assessing conditional
tail dependences. In addition, both the minimum values of AIC
and BIC are also
consistent in ranking the model BB7 number one. The plots of
conditional tail
dependences and time varying parameters of model BB7 for the
three pairs are given in
Fig. 7. In the light of summary in Table 9, model BB7 is the one
where the mean values
of conditional lower tail dependence are higher than the mean
values of conditional upper
tail dependence. This suggests that the probability of downside
market comovements is
greater than the probability of upside market comovements
between Asia developed
futures markets during the post crisis period. This finding is
similar to previous ones by
Login and Solnik (2001) and Ang and Chen (2002). The two studies
find that correlation
between international equity markets is higher during bear
markets than during bull
markets.
[INSERT TABLE 10 HERE]
V. Concluding Remarks We employ three two-parameter Archimedean
copulas to investigate dynamic
asymmetric tail dependence in Asian developed futures markets.
With a careful
implementation of two-stage estimation, we found that higher
moments of each filtered
index futures return are time dependent. This is indicative of
conditional skewness and
leptokurtosis in each index futures return series. We then
extend a class of two-parameter
copulas incorporating time varying tail dependences to capture
the dynamic asymmetries.
The estimated results provide strong evidence of asymmetric
dependence across all Asia
developed futures markets. Moreover, based on the
goodness-of-fit tests, we found the
model BB7 is the optimal one which demonstrates that the
probability of dependence in
bear markets is higher than in bull markets during the post
crisis period further exposing
downside dependent risk. Therefore, our consistent parameter
estimates successfully
14
-
approximate the dynamic asymmetric tail dependences and
constitute a precise basis for
the purpose of hedging downside risk.
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18
-
Table 1. Bivariate Two-Parameter Archimedean Copulas Tail
Dependence Model C (u,υ ) ∈βα , L Uττ
BB1 αββαβα υ /1/1 }])1()1[(1{ −−− −+−+ u
α > 0, β ≥ 1 2 )/(1 αβ− 2-2 β/1
BB4 )/11(/1 }])1()1[(1{ αββαααα υυ +−−−−−−−− −+−−−+ uu β
α ≥ 0, β > 0 (2-2 )β/1− α/1− 2 β/1−
BB7 αββαβα υ /1/1 }]1)()1[(1{1 −−− −−+−−− u 1 uu −= 1 , υυ −=
1
α ≥ 1, β > 0 2 β/1− 2-2 α/1
Note that α and β are estimated parameters for copulas. and are
lower and upper tail dependences respectively. More details see Joe
(1997) and Nelsen (1999).Lτ Uτ
-
Table 2. Summary Statistics of Index Futures Returns Index
Futures Hang Seng Nikkei 225 MSCI SIN Observations 1690 1690 1690
Maximum 8.7515 8.0043 11.1124 Minimum -8.7116 -7.5986 -7.0921 Mean
0.0352 -0.0137 0.0365 Standard Deviation 1.7239 1.4910 1.5324
Skewness 0.0508 -0.0477 0.2352 Kurtosis 5.6085 4.8551 6.3724
Ljung-Box
xQ (10) 9.2167 13.5722 11.4151
xQ (20) 24.4408 19.6409 21.1178
xQ (50) 58.9410 47.8588 56.7702
xxQ (10) 178.2818* 128.2048* 173.8839*
xxQ (20) 312.4551* 229.2490* 288.0914*
xxQ (50) 550.2255* 249.6927* 433.9073* Engle (10) 108.7451*
81.3262* 94.5516* Engle (20) 134.8376* 118.6739* 125.3843* Engle
(50) 155.4204* 127.7343* 165.9947* The daily percentage index
futures return series on the three leading Asian futures markets
Hang Seng, Nikkei 225 and, MSCI SIN over the post crisis period 07
September 1998 to 28 February 2005 are measured as )log( 1−× tt
XX100 . The Ljung-Box statistics provides tests for the presence of
autocorrelations of raw returns and squared returns as well as the
LM test of Engle (1982) for the presence of ARCH effects. The
critical values of Ljung-Box test and LM test of Engle (1982) are
18.307 (lag 10), 31.410 (lag 20) and, 67.5048 (lag 50) at 5%. *
indicates significance at the 5% level.
20
-
Table 3. Parameter Estimates of the TGARCH (1,1) Model with
Conditional Skewness and Kurtosis Index Futures
Hang Seng Nikkei 225 MSCI SIN Parameter Estimate Robust Std.
Error
Estimate Robust Std. Error
Estimate Robust Std. Error
0a 0.0080 0.0001 0.0401 0.0015 0.0215 0.0007
0b 0.0069 0.0003 0.0386 0.0011 0.0644 0.0020
0c 0.0660 0.0006 0.1270 0.0033 0.1082 0.0027
0d 0.9623 0.0001 0.9084 0.0025 0.9088 0.0025
1a 1.8329 0.0287 1.0621 0.0331 1.6901 0.0418 1b 0.2312 0.0045
0.7091 0.0073 0.2151 0.0104 1c 0.0122 0.0045 0.4305 0.0084 -0.1542
0.0094 2a 0.0398 0.0002 0.0032 0.0003 0.0729 0.0011 2b -0.0182
0.0006 0.0717 0.0001 -0.0974 0.0006 2c -0.0400
0.0005
0.0234
0.0007
-0.0526
0.0010
Loglike
3130.5328 2967.2045 2918.8259
Elapsed Time (Sec.)
24.79 13.65 12.45
This table contains results of maximum likelihood estimator for
margin models with threshold GARCH (1, 1) and conditional skewness
tλ and kurtosis tη . The specified model is
102
102
100 )()(),0,max(),0,max(,, −−−
+−
−+ +++=−==⋅=+= tttttttttttttt hdecebaheeeezheekR , |,(~ ttt sktz
λη ℱ ) 1−t )(),( 1212211111 −−+−−−+− ++Ξ=++Ξ= tttttt ecebaeceba
ληRobust standard errors are calculated according to Newey and
McFadden (1994) and White (1994). Loglike is log likelihood.
Figures in bold indicate significance at 1% level. All results are
calculated by Matlab 7 with Pentium 4 2.80 GHz machine.
21
-
Table 4. Summary Statistics of Conditional Skewness and Kurtosis
Parameters and of Standardized Residuals Index Futures
Hang Seng Nikkei 225 MSCI SIN
tη tλ tz
tη tλ tz
tη tλ tz
Maximum 16.0000 0.1540 3.5737 16.0000 0.0000 3.8723 16.0000
0.4643 4.7751 Minimum 2.5847 -0.0199 -6.1508 2.0328 -0.2812 -6.8158
2.4689 -0.0364 -5.4601 Mean 5.4580 -0.0021 -0.0119 6.7664 -0.0276
-0.0062 6.3444 0.0044 -0.0138 Std. 0.9895 0.9761 1.0038 Skewness
-0.2658 -0.3296 -0.0709 Kurtosis Q
5.1048 5.1222 5.1143 x (10) 2.9781
8.1648 8.0995
xQ (20) 16.3434 11.9987 16.0964
xxQ (10) 6.5841 7.5002 4.7434
xxQ (20) 28.4008 20.1376 9.9770 Engle (10) 7.4916 8.1320 5.6728
Engle (20)
29.1952
21.0907
10.8469
Q
The summary statistics are based on the results of Table 3. and
are Ljung-Box statistics for standardized residuals and squared
standardized residuals respectively. Engle’s (1982) LM test for
squared standardized residuals is represented by Engle ( · ) for
different lags.
x xxQ tz2tz
2tz
22
-
Table 5. Correlation Measures for Bivariate Standardized
Residuals Index Futures
Hang Seng Nikkei 225 MSCI SIN
Panel A: Pearson’s Pρ
Hang Seng — 0.4480 (0.0000) 0.5449 (0.0000) Nikkei 225 — 0.3956
(0.0000) MSCI SIN —
Panel B: Spearman’s Sρ
Hang Seng — 0.4173 (0.0000) 0.5021 (0.0000) Nikkei 225 — 0.3696
(0.0000) MSCI SIN —
Panel C: Kendall’s Kτ
Hang Seng — 0.2870 (0.0000) 0.3523 (0.0000) Nikkei 225 — 0.2551
(0.0000) MSCI SIN — Panel A reports linear dependence Pearson’s Pρ
which is calculated as
yx
xyP σσ
σρ = .
Panel B reports nonlinear dependence Spearman’s Sρ which is
calculated as
∑ ∑∑= ==
−⋅−−−=n
i
n
i
Ri
Ri
Ri
Ri
n
i
Ri
Ri
Ri
RiS yyxxyyxx
1 1
22
1)()())((ρ
where and are rank statistics for random variables and
respectively. RixRiy ix iy
Panel C reports nonlinear dependence Kendall’s Kτ which is
calculated as
∑≤≤≤
−
−−
=
njijijiK yyxxsign
n
1
1
)])([(2
τ .
The associated p-values are provided in parentheses.
23
-
Table 6. Multivariate Normality Test for Bivariate Standardized
Residuals
Index Futures Hang Seng Nikkei 225 MSCI SIN
Panel A: Statistics of Bivariate Skewness Test
2,1b A 2,1b A 2,1b A Hang Seng — — 0.2276 64.1145* 0.1627
45.8379* Nikkei 225 — — 0.2150 60.5505* MSCI SIN — —
Panel B: Statistics of Bivariate Kurtosis Test 2,2b B 2,2b B
2,2b B
Hang Seng — — 12.3730 22.5199* 12.2820 22.0527* Nikkei 225 — —
13.0347 25.9203* MSCI SIN — — The purpose of Mardia’s (1970)
multivariate normality test is to see whether multivariate skewness
or kurtosis (or both) is significant at a certain level. Panel A
reports the statistics of bivariate skewness test A = n x b /6 ~
(4), with a critical value 9.49 at the 95% confidence interval (n
is the sample size) based on the calculation of b which is a basic
point for testing 2-demensional multivariate skewness. Panel B
reports the statistics of bivariate kurtosis test B =
2,12χ
2,1
n
nnb
/64
)1/()1(82,2 +−− ~ N (0,1), with a critical value 1.96
at the 95% confidence interval (n is the sample size) based on
the calculation of b which is a basic point for testing
2-demensional multivariate kurtosis. * indicates significance at
the 5% level.
2,2
24
-
Table 7. Parameter Estimates of the Static Two-Parameter
Archimedean Copulas Hang Seng – Nikkei 225 Hang Seng – MSCI SIN
Nikkei 225 – MSCI SIN
Model Parameter Estimate Robust Std. Error
Estimate Robust Std. Error
Estimate Robust Std. Error
α 0.3391 0.0039 0.4517 0.0039 0.3496 0.0038 β 1.1868 0.0041
1.2484 0.0043 1.1399 0.0038 BB1 Lτ 0.8204 — 0.7782 — 0.8085
—
Uτ 0.2068 —
0.2576 —
0.1631 — Loglike
-187.3493 -295.7946 -155.2558
α 0.6023 0.0012 0.8200 0.0017 0.5488 0.0011 β
L
0.0316
0.0591 0.0313 0.0626 0.0315 0.0634BB4 τ
U 0.3164
10−×— 0.4294
10−×— 0.2828
10−×—
τ 2.9767 10 —
2.4123 10 — 2.7765
10 — Loglike
-160.9679 -252.5507 -138.3157 -1
α 1.2343 0.0024 1.3168 0.0026 1.1763 0.0025 β 0.4830 0.0003
0.6666 0.0004 0.4581 0.00004 BB7 Lτ
U 0.2381 — 0.3535 — 0.2203 —
τ 0.2466 — 0.3072
— 0.1973
—Loglike
-184.9232 -292.8262 -153.8232
This table shows estimated parameters of static two-parameter
copulas via two-stage maximum likelihood estimator. The specified
model is provided in Table 1. Lτ and Uτ are unconditional lower and
upper tail dependences respectively. Robust standard errors are
calculated as propositions of Newey and McFadden (1994)
and White (1994). Loglike represents log likelihood. Figures in
bold indicate significant at 1% level. All results are computed by
Matlab 7 with Pentium 4 2.80 GHz machine.
25
-
Table 8. Parameter Estimates of the Time Varying Two-Parameter
Archimedean Copulas Hang Seng – Nikkei 225 Hang Seng – MSCI SIN
Nikkei 225 – MSCI SIN
Model Parameter Estimate Robust Std. Error
Estimate Robust Std. Error
Estimate Robust Std. Error
Lω 0.1691 0.0173
-0.0086 0.0175 -2.2256 0.1002 Lδ 0.8676 0.0125 0.8727 0.0399
-0.2853 0.0679 Lψ -1.5320 0.0024
-0.4759 0.0782 0.9406 0.0123 BB1 Uω -0.5871 0.0299 -0.5755
0.0043 -0.8652 0.0516
Uδ 0.4714 0.0177 0.1426 0.0036 0.1168 0.0292 Uψ -0.5642
0.0244
-1.5895 0.0431 -2.7935 0.0091 Loglike -190.8177 -299.9736
-159.1086
Elapsed Time (Sec.)
22.36 12.00 13.59
Lω -0.9998 0.0164 -4.7465 0.0781 -1.2881 0.0509 Lδ
L 0.9984 0.0164 0.9918 0.0163 0.9984 0.0110
ψ -0.9741 0.0160
-4.4554 0.0733 0.2964 0.0308 BB4 Uω
U 0.9392 0.8713 1.5398 1.6326 0.6420 0.0018
δU
0.3500 0.5635 -0.9537 0.1115 -0.8627 0.0142 ψ 0.8512 0.3691
0.9933 1.2673 1.6269 0.0089 Loglike -9729.1119 -10973.8094
-9507.0763
Elapsed Time (Sec.)
15.99 16.64 23.00
26
-
Lω 0.1487 0.0131
-0.2260 0.00003 -2.2458 0.0094 Lδ 0.8758 0.0108 0.1394 0.0005
-0.8142 0.0071 Lψ -1.2031 0.0087
-1.2675 0.0023 -0.0641 0.0094 BB7 Uω -0.3658 0.0100 -0.4062
0.0064 -0.2008 0.0283
Uδ 0.5224 0.0149 0.1717 0.0062 0.3919 0.0249 Uψ -0.7457
0.0268
-1.2619 0.0046 -3.0209 0.0447 Loglike -189.3858 -296.6299
-158.7936
Elapsed Time (Sec.)
27.18 14.45 17.83
This table reports the estimated parameters of the time varying
two-parameter Archimedean copulas via two-stage maximum likelihood
estimator. The conditional tail dependences are defined as pp where
L and U are
conditional lower and upper tail dependences respectively,
−⋅++Λ=
−⋅++Λ= ∑∑=
−−−
−=
−−−
−j
jtjtUU
tUUU
tj
jtjtLL
tLLL
t upup1
11
1
11 ||,|| υψτδωτυψτδωτ
)]exp(1/[1)( kk
tτ tτ
−+=Λ is the logistic transformation to ensure (0, 1) at all
time. Robust standard errors are calculated according to Newey and
McFadden (1994) and White (1994). Loglike represents log
likelihood. Figures in bold indicate significance at the 1% level.
All results are computed by Matlab 7 with Pentium 4 2.80 GHz
machine.
∈LtUt ττ ,
27
-
Table 9. A Summary of Conditional Tail Dependence and Time
Varying Parameters for the Two-Parameter Archimedean Copulas Hang
Seng – Nikkei 225 Hang Seng – MSCI SIN Nikkei 225 – MSCI SIN
Ltτ
Uτ t tα tβ
Ltτ
Uτ t tα tβ
Ltτ
Uτ t tα tβ
Maximum
0.5793 0.5479 0.9442 1.8582 0.5793 0.5479 0.8663 1.8582 0.5793
0.5479 0.6832 1.8582BB1 Minimum
0.0239 0.1412 0.1634 1,1181 0.1708 0.1029 0.3348 1.0825 0.0961
0.0229 0.2375 1.0169
Mean 0.1989 0.2037 0.3689 1.1841 0.2958 0.2587 0.4560 1.2545
0.1774 0.1590 0.3549 1.1407 – < + – > + – > +
Maximum 0.5793 0.9122 0.7426 7.5431 0.5793 0.9591 0.7426 16.5905
0.5793 0.9434 0.7426 11.8892BB4 Minimum
0.0000 0.5000 0.0001 1.0000 0.0000 0.2061 0.0001 0.4389 0.0000
0.2123 0.0001 0.4472
Mean 0.0006 0.8520 0.0011 4.4450 0.0003 0.6964 0.0010 2.4038
0.0006 0.6262 0.0019 1.8108 – < + – < + – < +
Maximum 0.5638 0.5793 1.9737 1.2094 0.5479 0.5793 1.9737 1.1520
0.5479 0.5793 1.9737 1.1520BB7 Minimum
0.0504 0.1478 1.1246 0.2319 0.1897 0.1552 1.1319 0.4169 0.0826
0.0210 1.0155 0.2780
Mean 0.2535 0.2430 1.2316 0.5197 0.3602 0.3079 1.3223 0.6853
0.2235 0.1918 1.1811 0.4629 – > + – > + – > +
This table provides summaries of conditional lower and upper
tail dependences and time varying parameters based on results of
Table 8. Symbol “–“ and “+” indicate average values of conditional
lower and upper tail dependences respectively.
28
-
Table 10. Goodness of Fit Tests and Information Criteria for
Dynamic Objective Dependence Functions
Hang Seng-Nikkei 225 Hang Seng-MSCI SIN Nikkei 225- MSCI SIN K-S
0.0718 0.0812 0.0589 AD 0.1721 0.1977 0.1722 BB1 IAD 0.0117 0.0168
0.0100 AIC 393.6354 611.9473 330.2173 BIC 426.2303 644.5422
362.8122 K-S 0.1533 0.1292 0.1136 AD 0.3174 0.2633 0.2288 BB4 IAD
0.0473 0.0362 0.0279 AIC 19470.2238 21959.6188 19026.1527 BIC
19502.8187 21992.2137 19058.7476 K-S 0.0701 0.0796 0.0578 AD 0.1718
0.1930 0.1690 BB7 IAD 0.0112 0.0160 0.0096 AIC 390.7717 605.2598
329.5873 BIC 423.3666 637.8547 362.1822 This table reports results
of goodness-of-fit tests and information criteria for dynamic
objective dependence functions. The basic idea of goodness-of-fit
tests is based on distance measures between the empirical copula
and theoretical copula C . K-S indicates Kolmogorov-Smirnov test:
KS = max | , AD
indicates Anderson-Darling test: AD = maxEC T |TE CC −
)1(||
TT
TE
CCCC−
− , and IAD indicates IAD = ∑∑ . The
information criteria are Akaike information criteria AIC =
-2Loglik + 2q and Bayesian information criteria BIC = -2Loglik +
qlnn respectively where n is sample size and q is number of
parameters. Figures in bold indicate the most informative dynamic
objective dependence function.
= =
n
t
n
t C1 11 2
(−−
TT
TE
CCC 2
)1()
29
-
-3-2
-10
12
-2
0
2
0.05
0.1
0.15
0.2
0.25
X
Joint Density with BB1 Copula α = 0.82 β = 1.67
Y
f(X,Y
)
Contour of Joint Density with BB1 Copula α = 0.82 β = 1.67
X
Y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-3-2
-10
12
-2
0
2
0.05
0.1
0.15
0.2
0.25
X
Joint Density with BB4 Copula α = 0.34 β = 1.05
Y
f(X,Y
)
Contour of Joint Density with BB4 Copula α = 0.34 β = 1.05
X
Y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
-3-2
-10
12
-2
0
2
0.05
0.1
0.15
0.2
X
Joint Density with BB7 Copula α = 1.34 β = 0.95
Y
f(X,Y
)
Contour of Joint Density with BB7 Copula α = 1.34 β = 0.95
X
Y
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
Figure 1. Surface and Contour of Joint Density with
Two-Parameter Copula
The surfaces and contours are plotted based on Eq. (2.4). For
the sake of convenience, all margins are subject to standard normal
distribution where X, Y ∈ [-3, 3].
30
-
u
v
Contour of BB1 Copula (α = 0.82, β = 1.67)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
00.2
0.40.6
0.81
0
0.5
10
5
10
15
20
u
Density of BB1 Copula (α = 0.82, β = 1.67)
v
c(u,
v)
u
v
Contour of BB4 Copula (α = 0.34, β = 1.05)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
00.2
0.40.6
0.81
0
0.5
10
5
10
15
20
u
Density of BB4 Copula (α = 0.34, β = 1.05)
v
c(u,
v)
u
v
Contour of BB7 Copula (α = 1.34, β = 0.95)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
00.2
0.40.6
0.81
0
0.5
10
2
4
6
8
10
12
u
Density of BB7 Copula (α = 1.34, β = 0.95)
v
c(u,
v)
Figure 2. Contour and Surface of Density of Two-parameter Copula
The contours and surfaces are plotted based on panel B of Table
1and Eq. (2.4) respectively. For the sake of convenience, all
margins are subject to standard normal distribution where ∈υ,u [0,
1].
31
-
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7Shapes of Skewed-t Distribution with Various Parameters
Quantile
PD
Fη = 4, λ = 0.2η = 4, λ = 0η = 4, λ = -0.3
Figure 3. Plot of Unconditional Skewed – t Distribution with
Various Parameters
The negative skewed-t density is represented by solid line, the
positive skewed-t density is represented by dash-dot line, and
standard Student-t density is represented by dashed line.
32
-
-10 -8 -6 -4 -2 0 2 4 6 8 102
3
4
5
6
7
8News Impact Curves for Hang Seng
Lagged Shock
Con
ditio
nal V
aria
nce
GARCHTGARCH
-8 -6 -4 -2 0 2 4 6 8 102
3
4
5
6
7
8
9
10News Impact Curves for Nikkei 225
Lagged Shock
Con
ditio
nal V
aria
nce
GARCHTGARCH
33
-
-8 -6 -4 -2 0 2 4 6 8 10 122
4
6
8
10
12
14News Impact Curves for MSCI SIN
Lagged Shock
Con
ditio
nal V
aria
nce
GARCHTGARCH
Figure 4. News Impact Curve (NIC) for Index Futures Returns All
NIC are plotted according to Eq. (3.5) where each return series is
subject to a conditional skewed-t distribution. The asymmetric
pattern of TGARCH (1, 1) is plotted by circles and the symmetric
pattern of GARCH (1, 1) by dots.
34
-
-6 -5 -4 -3 -2 -1 0 1 2 3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
PDF of Conditional Skewed-t Distribution with TGARCH(1,1) for
Hang Seng
Standardized Residuals
f( ηt, λ
t)
0 500 1000 15001
1.5
2
2.5
3
Conditional Volatilities of TGARCH (1,1)
0 500 1000 1500-6
-4
-2
0
2
Standardized Residuals of TGARCH (1,1)
0 500 1000 1500
0
0.05
0.1
0.15
Conditional Skew ness λt for Satndardized Residuals
0 500 1000 15002
3
4
5
Conditional Kurtosis ηt for Satndardized Residuals
Figure 5(a). Plots of Standardized Residuals and Conditional
Parameters for Hang Seng
35
-
-6 -5 -4 -3 -2 -1 0 1 2 3
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
PDF of Conditional Skewed-t Distribution with TGARCH(1,1) for
Nikkei 225
Standardized Residuals
f( ηt, λ
t)
0 500 1000 1500
1
1.5
2
2.5
3
Conditional Volatilities of TGARCH (1,1)
0 500 1000 1500-6
-4
-2
0
2
Standardized Residuals of TGARCH (1,1)
0 500 1000 1500
-0.25
-0.2
-0.15
-0.1
-0.05
Conditional Skew ness λt for Satndardized Residuals
0 500 1000 15002
4
6
8
Conditional Kurtosis ηt for Satndardized Residuals
Figure 5(b). Plots of Standardized Residuals and Conditional
Parameters for Nikkei 225
36
-
-5 -4 -3 -2 -1 0 1 2 3 4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
PDF of Conditional Skewed-t Distribution with TGARCH(1,1) for
MSCI SIN
Standardized Residuals
f( ηt, λ
t)
0 500 1000 1500
1
2
3
Conditional Volatilities of TGARCH (1,1)
0 500 1000 1500
-4
-2
0
2
4
Standardized Residuals of TGARCH (1,1)
0 500 1000 15000
0.1
0.2
0.3
0.4
Conditional Skew ness λt for Satndardized Residuals
0 500 1000 15002
4
6
8
10
12
Conditional Kurtosis ηt for Satndardized Residuals
Figure 5(c). Plots of Standardized Residuals and Conditional
Parameters for MSCI SIN
37
-
-6 -4 -2 0 2
-6
-4
-2
0
2
Scatter Plot of Bivariate Standardized Residuals
Hang Seng
Nik
kei 2
25
-6 -4 -2 0 2
-4
-2
0
2
4
Scatter Plot of Bivariate Standardized Residuals
Hang Seng
MS
CI S
IN
38
-
-6 -4 -2 0 2
-4
-2
0
2
4
Scatter Plot of Bivariate Standardized Residuals
Nikkei 225
MS
CI S
IN
Figure 6. Scatter Plot of Bivariate Standardized Residuals
Each plot clearly shows the absence of linear relationship
between filtered returns.
39
-
0 500 1000 15000
0.5
1Time Varying Low er Tail Dependence τL
0 500 1000 15000.15
0.2
0.25
0.3
Time Varying Upper Tail Dependence τU
0 500 1000 1500
1.15
1.2
1.25
1.3
1.35Time Varying Parameter αt
0 500 1000 1500
0.4
0.6
0.8
1
1.2Time Varying Parameter β t
Figure 7(a) Plots of Time Varying Tail Dependences and
Parameters of Model BB7
for Hang Seng-Nikkei 225
40
-
0 500 1000 15000
0.2
0.4
Time Varying Low er Tail Dependence τL
0 500 1000 15000
0.2
0.4
Time Varying Upper Tail Dependence τU
0 500 1000 1500
1.2
1.4
1.6
1.8
Time Varying Parameter αt
0 500 1000 1500
0.6
0.8
1
Time Varying Parameter β t
Figure 7(b) Plots of Time Varying Tail Dependences and
Parameters of Model BB7
for Hang Seng-MSCI SIN
41
-
0 500 1000 15000.21
0.22
0.23
0.24Time Varying Low er Tail Dependence τL
0 500 1000 15000
0.5
1Time Varying Upper Tail Dependence τU
0 500 1000 1500
1.2
1.4
1.6
1.8
Time Varying Parameter αt
0 500 1000 15000.44
0.45
0.46
0.47
Time Varying Parameter β t
Figure 7(c)
Plots of Time Varying Tail Dependences and Parameters of Model
BB7 for Nikkei 225-MSCI SIN
42
I IntroductionII ModelsIII Estimation MethodIV Empirical
ResultsV. Concluding Remarks
Figure 1. Surface and Contour of Joint Density with
Two-Parameter CopulaPaper Cover of Qing Xu et al 2005.pdfQing
Xu†Xiaoming LiAbdullah MamunJune 2005Abstract