dynamic asymmetric tail dependence in asian developed … · 2007. 7. 24. · iii estimation method...

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]

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.

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

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

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

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

)]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

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

)(

)(

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.

1−

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

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

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

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

[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

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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Deheuvels, P. (1981), “A non parametric test for independence,” Publications de l’Institut de Statistique de l’Université de Paris 26:29-50 de la Pena, V., L. Chollete, and C.C. Lu (2004), “Comovement of international financial markets,” Working Paper, Columbia University. De Santis, G. and B. Gerard (1997), “International asset pricing and portfolio diversification with time varying risk,” Journal of Finance, 52, 1881-1912. Duan, J.C. and H. Zhang (2001), “Pricing Hang Seng Index options around the Asian financial crisis – a GARCH approach,” Journal of Banking and Finance, 25, 1989-2014. Embrechts, P., F. Lindskog, and A.J. McNeil (2003), “Modeling dependence with copulas and applications to risk management,” in S. Rachev (ed.), Handbook of Heavy Tailed Distributions in Finance, 329-384, Elsevier. Embrechts, P., A.J. McNeil, and D. Straumann (2000), “Correlation: pitfalls and alternatives,” in P. Embrechts (ed.), Extremes and Integrated Risk Management, 71-76, Risk Books, Risk Waters Group Ltd. Embrechts, P., A.J. McNeil, and D. Straumann (2002), “Correlation and dependence in risk management: properties and pitfalls,” in M.A.H. Dempster (ed.), Risk Management: Value at Risk and Beyond, 176-223, Cambridge University Press, U.K. Engle, R.F. (1982), “Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation,” Econometrica, 50, 987-1008. Engle, R.F. and V.K. Ng (1993), “Measuring and testing the impact of news and volatility,” Journal of Finance, 48, 1749-1777. Erb, C.B., C.R. Harvey, and T.E. Viskanta (1994), “Forecasting international equity correlations,” Financial Analysts Journal, November-December, 32-45. Fung, J.K.W., L.T.W. Cheng, and K.C. Chan (1997), “The intraday pricing efficiency of Hong Kong Hang Seng index options and futures markets,” Journal of Futures Markets, 17, 797-815. Glosten, R.T., R. Jagannathan, and D.Runkle (1993), “On the relation between the expected value and the volatility of the nominal excess return on stocks,” Journal of Finance, 48, 1779-1801. Hansen, B.E. (1994), “Autoregressive conditional density estimation,” International Economic Review, 35, 705-730. Hu, L. (2002), “Dependence patterns across financial markets: methods and evidence,” Working Paper, Department of Economics, Ohio State University.

16

Hu, L. (2004), “Dependence patterns across financial markets: a mixed copula approach,” Working Paper, Department of Economics, Ohio State University. Joe, H. (1997), Multivariate Models and Dependence Concepts, Chapman and Hall, London. Joe, H. (2005), “Asymmetric efficiency of the two-stage estimation method for copula-based models,” Journal of Multivariate Analysis, 94, 401-419. Jondeau, E. and M. Rockinger (2003), “Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements,” Journal of Economic Dynamics and Control, 27, 1699-1737. Kim, I.J., K. Ko, and S.K. Noh (2002), “Time-varying bid-ask components of Nikkei 225 index futures on SIMEX,” Pacific-Basin Finance Journal, 10, 183-200. King, M., E. Sentana, and S. Wadhwani (1994), “Volatility and links between national stock markets,” Econometrica, 62, 901-933. Kroner, K. and V.K. Ng (1998), “Modeling asymmetric comovements of asset returns,” Review of Financial Studies, 11, 817-844. Longin, F. and B. Solnik (1995), “Is the correlation in international equity returns constant: 1960-1990?” Journal of International Money and Finance, 14, 3-26. Longin, F. and B. Solnik (2001), “Extreme correlation of international equity markets,” Journal of Finance, 56, 649-676. Mardia, K.V. (1970), “Measures of multivariate skewness and kurtosis with applications,” Biometrika, 57, 519-530. Nelsen, R.B. (1999), An Introduction to Copulas, Lecture Notes in Statistics, vol. 139, Springer, New York. Newey, W.K. and D. McFadden (1994), “Large sample estimation and hypothesis testing,” in R.F. Engle and D. McFadden (ed.) Handbook of Econometrics, Vol. 4, North Holland, Amsterdam. Pagan, A. and G.W. Schwert (1990), “Alternative models for conditional stock volatility,” Journal of Econometrics, 45, 267-290. Patton, A.J. (2001), “Modeling time-varying exchange rate dependence using the conditional copula,” Working Paper, Department of Economics, University of California, San Diego.

17

Patton, A.J. (2004), “Modeling asymmetric exchange rate dependence,” Working Paper, London School of Economics. Poon, S.H., M. Rockinger, and J. Tawn (2004), “Extreme value dependence in financial markets: diagnostics, models, and financial implications,” Review of Financial Studies, 17, 501-610. Sklar, A. (1959), “Functions de répartition à n dimensions et leurs marges,” Publications de l’Institut de Statistique de l’Université de Paris, 8, 229-231. So, R.W. and Y. Tse (2004), “Price discovery in the Hang Seng index markets: index, futures, and the tracker fund,” Journal of Futures Markets, 24, 887-907. White, H. (1982), “Maximum likelihood estimation of misspecified models,” Econometrica, 50, 1-25. White, H. (1994), Estimation, Inference and Specification Analysis, Econometric Society Monographs No. 22, Cambridge University Press, Cambridge, U.K. Zakoїan, J.M. (1994), “Threshold heteroskedastic model,” Journal of Economic Dynamic and Control, 18, 931-955.

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


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



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



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


Y

f(X,Y

)


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


Y

f(X,Y

)


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


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


v

c(u,

v)

u

v


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


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


f( ηt, λ

t)

0 500 1000 1500

1

1.5

2

2.5

3


0 500 1000 1500-6

-4

-2

0

2


0 500 1000 1500

-0.25

-0.2

-0.15

-0.1

-0.05


0 500 1000 15002

4

6

8


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


f( ηt, λ

t)

0 500 1000 1500

1

2

3


0 500 1000 1500

-4

-2

0

2

4


0 500 1000 15000

0.1

0.2

0.3

0.4


0 500 1000 15002

4

6

8

10

12


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


Hang Seng

MS

CI S

IN

38

-6 -4 -2 0 2

-4

-2

0

2

4


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

dynamic asymmetric tail dependence in asian developed … · 2007. 7. 24. · iii estimation method...

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