large asymmetry and directional dependence by using copula modeling to currency exchange rates

7/27/2019 Large Asymmetry and Directional Dependence by Using Copula Modeling to Currency Exchange Rates

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Model Assisted Statistics and Applications 7 (2012) 327340 327DOI 10.3233/MAS-2012-00244IOS Press

Large asymmetry and directional dependence

by using copula modeling to currency

exchange rates

Daiho Uhma, Jong-Min Kimb and Yoon-Sung Jungc,

aDepartment of Statistics, Oklahoma State University, Stillwater, OK, USAbStatistics Discipline, Division of Science and Mathematics, University of Minnesota at Morris, Morris, MN, USAcCooperative Agricultural Research Center, College of Agriculture and Human Sciences, Prairie View A&M

University, Prairie View, TX, USA

Abstract.To examine the asymmetry of financial data in detail, we have considered both the tail dependence with diverse

copulas and Jung et al.s [8] directional dependence by copula. From the empirical study in this paper, we have found that

the tail dependence by Pattons [11] modified symmetrized Joe-Clayton copula function did not show the asymmetry property

sufficiently because there is no tail dependence for the currency exchange rates of Republic of Korea and Japan against the US

dollar after the 2008 financial crisis. However, by using the copula approach for directional dependence, we showed that there

exists the asymmetry of the currency exchange rates of Republic of Korea and Japan against the US dollar before and after the

2008 financial crisis because the directional dependence only when existing an asymmetry of data can be found. We conclude

that the copula approach for directional dependence can be supplemented to explain the asymmetry property in addition to the

tail dependence approach.

Keywords: Time copula, directional dependence, tail dependence, currency exchange rate, Farlie-Gumbel-Morgenstern copula

1. Introduction

Recently, a study of dependence by using copulas has been getting more attention in the areas of finance,

actuarial science, biomedical studies, and engineering. This is because a copula function does not require a normal

distribution and independent, identical distribution assumptions. Furthermore, the invariance property of copula

has been attractive in the finance area. Patton [11] proposed a modified symmetrized Joe-Clayton copula function

for checking an asymmetric property with currency exchange rate data. Rodriguez [12] modeled dependence with

switching-parameter copulas to study financial contagion by using daily returns from five East Asian stock indicesduring the Asian crisis, and from four Latin American stock indices during the Mexican crisis. Jung et al. [8]

proposed directional dependence of modified Farlie-Gumbel-Morgenstern(FGM) copula function by using currency

exchange rate data. In this paper, we have combined copula tail dependence and copula directional dependence for

analyzing the asymmetry of currency exchange rate data.

To investigate the asymmetry and directional dependence by using copula modeling, we have chosen the currency

exchange rates against US dollar for two countries: Republic of Korea (Developing country) and Japan (Developed

country) which Jung et al. [8] used. This provides a good example to see how two different countries relate to

each other before and after the 2008 financial crisis in terms of the asymmetry by tail dependence and directional

Corresponding author: Yoonsung Jung, Cooperative Agricultural Research Center, College of Agriculture and Human Sciences, Prairie ViewA&M University, Prairie View, TX 77446, USA. E-mail: [email protected].

ISSN 1574-1699/12/$27.50 2012 IOS Press and the authors. All rights reserved


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328 D. Uhm et al. / Large asymmetry and directional dependence by using copula modeling to currency exchange rates

dependence. The propertiesof copula, definitionof copula, and definitionof tail dependence by copula are introduced

in Section 2. Using a dataset of the daily exchange rates for the Korean won (KRW-USD) and the Japanese yenagainst the U.S. dollar (JPY-USD) from January 2, 2001 to December 31, 2010, we have fitted marginal distribution

of the daily exchange rates with an AR(1)-GARCH(1,1) model and then showed the tail dependence by using several

different copula models in Section 3. Section 4 reviewed the method of Jung et al.s [8] directional dependence

by FGM copula. Its application to a dataset of the daily exchange rates for the Korean won (KRW-USD) and the

Japanese yen against the U.S. dollar (JPY-USD) is given in Section 5. Section 6 gives the advantages of the methods

we propose in this paper.

2. Copula method

A copula is a multivariate distribution function defined on the unit[0, 1]n, with uniformly distributed marginals.In this paper, we focus on a bivariate (two-dimensional) copula, where n = 2. Sklar [14] shows that any bivariatedistribution function,FXY(x, y), can be represented as a function of its marginal distribution ofXandY, FX (x)andFY(y), by using a two-dimensional copulaC(, ). More specifically, the copula may be written as

FXY(x, y) =C(FX (x), FY(y)) =C(u, v),

whereu andv are the continuous empirical marginal distribution functionFX (x) andFY(y), respectively. Notethatuandv have uniform distributionU(0, 1).

Therefore, the copula function represents how the function,FXY(x, y), is copuled with its marginal distributionfunctions, FX (x) and FY(y). It also describes the dependence mechanism between two random variables byeliminating the influence of the marginals or any monotone transformation of the marginals. In this section, we

review the basic concepts of copulas focusing on some preliminary propertiesof copulas. We start with the Definition

1 and 2 for subcopulas and coplua, and Theorem 1 (Nelsen [10]).

Definition 1 A subcopula is a function, C

, with the following properties: (i) Domain C

= D1D2, whereD1, D2 [0, 1] including 0 and 1; (ii) C(u, 0) = C(0, v) = 0 for all u D1, v D2; (iii) C(u, 1) = u,C(1, v) = v for all u D1, v D2; (iv) 0 C(u1, v1)C(u1, v2)C(u2, v1) +C(u2, v2), whenu1, u2D1,v1, v2D2, andu1 u2,v1 v2.Definition 2 A copula is a subcopula whose domain is the entire unit square.

The following theorems, which are due to Sklar [15], show that a bivariate distribution function can be represented

as a function of its marginals by using a two-dimensional copula.

Theorem 1 IfX andYare continuous random variables with distribution functions FX (x) andFY(y) and jointdistribution functionFXY(x, y), then there exists a unique subcopula,C, such that

FXY(x, y) =C(FX (x), FY(y)) (1)

for allx, y (DomainFX (x) DomainFY(y)), and there are couplas,C, such thatFXY(x, y) =C(FX (x), FY(y)) (2)

for all x, yR. IfFX (x)and FY(y)are continuous distribution function, then Cis unique; otherwise Cis uniquelydetermined on (DomainFX (x)DomainFY(y)).

LetX, Ybe random variables with continuous distribution functionsFX (x)and FY(y), respectively, letXandYbe continuous random variables with copula C and marginal distribution functions FX (x) and FY(y) so thatX FX (x), Y FY(y), and (X, Y) FXY(x, y), and let u = FX (x), v = FY(y), and (u, v) C. ThenSpearmansand Kendallsare given, respectively, by

C= 12 1

0 1

0 C(u, v) uvdudv, (3)


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D. Uhm et al. / Large asymmetry and directional dependence by using copula modeling to currency exchange rates 329

Table 1Copula tail dependence

Copula L U

Gaussian 0 0

Clayton 21/ 0Frank 0 0

Gumbel 0 2 21/

Student 2t+1

(+ 1)(1 )/(1 + )

2t+1

(+ 1)(1 )/(1 + )

L is the coefficient of lower (left) tail dependence.U is the coefficient of upper (right) tail dependence.is the coefficient of linear correlation of two random variables.L andU for the Gaussian Copula are zeroif and only if |V > ] = lim1

P[V > |U > ] (6)

ifU [0, 1]andL [0, 1]exist.Table 1 is a summary table for copula tail dependence. The bivariate Gaussian Copula is:

CG(u, v) =

1(u)

1(v)

1

2

1 2 exp(r2 2rs+s2)

2(1 2)

dr ds,

where1 is the inverse of the standard normal distribution, is the linear correlation coefficient betweenuandv,andu, v[0, 1].

The lower tail dependence and upper tail dependence for the Gaussian Copula is:

L =U = 0if and only if


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Table 2Summary statistics

20012005 20062010

N=1255 KRW-USD JPY-USD KRW-USD JPY-USD

Mean 1.87e-04 2.18e-05 8.87e-05 2.88e-04Std. Dev. 4.83e-03 6.07e-03 1.04e-02 7.30e-03Skewness 0.5115 0.2536 0.0522 0.4160Kurtosis 5.4128 4.3214 29.5625 6.4888Jarque-Bera statistic 358.86 104.68 36866.5 672.15

Correlation 0.3584 0.1695p < 0.001, Jarque-Bera test for normality.

Fig. 1. Daily exchange rates for Korean won and Japanese yen against U.S. dollar.

separately in two time-periods: one is from January 2, 2001 to December 31, 2005, and the other is from January 2,

2006.

Letxtbe an exchange rate at timet, then the return or relative gain, yt, of the exchange rate at timetis defined as

yt = log (xt) log(xt1).In Table 2, the statistics of the returns are summarized into the two time periods from 2001 to 2005, and from 2006

to 2010. These include means of approximating zero, standard deviation, skewness, and kurtosis. The two time

periods have 1,255 daily returns each. For normality the skewness ranges between0.4160 and 0.5115, whichinclude little departure from symmetry, and the kurtosis is at least 4.3214. The returns of KRW-USD in 20062010

have the highest kurtosis of 29.5625 with a Jarque-Bera statistic of 36866.5 for normality test. It might be caused by

the volatility of rate changing over the time period. The Jarque-Bera test rejects normality for all returns of exchange

rates. The correlation coefficients between the returns of KRW-USD and JPY-USD are 0.3584 and 0.1695 in eachperiod. The negative correlation shows the opposite trends of the two exchange rates in 20062010. Based on the

fact of normality violation of the data, the dependence by copula function is a useful approach to better understand

the behavior of the data.

To see the behavior of the marginal distribution of the currency exchange rates, each exchange rate was modeled

with an autoregressive (AR) time series model. Table 3 shows estimates and statistics for AR(2) models for each

exchange rate. In comparing estimates and their standard errors all intercepts are not significant, and only AR1


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Table 3Estimates and statistics for AR model

20012005 20062010N=1255 KRW-USD JPY-USD KRW-USD JPY-USD

Intercept 2e-04 0e+00 1e-04 3e-04(1e-04) (2e-04) (3e-04) (2e-04)

AR1 0.0935 0.0430 0.0432 0.0428(0.0282) (0.0282) (0.0281) (0.0282)

AR2 0.0119 0.0175 0.0887 0.0025(0.0282) (0.0283) (0.0281) (0.0282)

Log-Likelihood 4915.53 4622.62 3951.84 4392.06AIC 9823.06 9237.24 7895.67 8776.12BIC 9818.67 9232.85 7891.29 8771.73

Standard errors are reported in parentheses.

Fig. 2. F statistics in Chow test.

estimate for KRW-USD in 2001205 and AR2 estimate for KRW-USD in 20062010 are significantly not zeros.

When they do have a serious violation in a constant variance, they might give unstable estimates. Since the AR(2)model is not suitable, using AR(1) model F test statistics are showed for the null hypothesisof no structural change

in Fig. 2 (Chow test, Chow [4]). In both exchange rates there are evidences for structural change in 20062010.

The Lagrange multiplier (LM) test was performed for autoregressive conditional heteroscedasticity (ARCH,

Engle [5]) based on the residuals of AR(1) using 12 lags. The ARCH model is used when the return,yt, does nothave constant variance. The LM test can help in determining the order of ARCH model and its appropriateness for

modeling the heteroscedasticity. In Table 4, the exchange rate of JPY-USD between 2001 and 2005 does not have a

significant ARCH effect by the LM test; however, the LM statistic of KRW-USD between 2006 and 2010 is 385.31.

This shows serious volatility in the Korean exchange rate from 2006 to 2010 (See Fig. 3). Table 4 also gives the

maximum likelihood estimates (MLE) and their standard errors in a generalized ARCH (GARCH, Bollerslev [3])

model. The AR(1)-GARCH(1,1) model is defined as:

yt= yt1+t,


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Table 4Estimates and statistics for AR(1)-GARCH(1,1) model

20012005 20062010KRW-USD JPY-USD KRW-USD JPY-USD

0.0947 0.0423 0.0427(7.90e-04) (7.96e-04) (7.96e-04)

ARCH LM statistic 70.25 5.45 385.31 85.74

2.92e-06 6.93e-07 2.92e-070 (3.32e-07) (1.01e-07) (1.07e-07)

2.46e-01 1.48e-01 3.63e-021 (3.54e-02) (1.36e-02) (5.56e-03)

6.75e-01 8.49e-01 9.59e-011 (2.92e-02) (1.19e-02) (5.58e-03)

Box-Ljung statistic 1.7484 1.0e-04 0.0273 0.3573

p


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Table 5Statistics for Copulas: 20012010

Log-Likelihood AIC BIC L U

Gaussian copula 8.1283 16.2558 16.2535 0 0Claytons copula 12.6170 25.2331 25.2308 0.0026 0Rotated Clayton copula 9.4539 18.9070 18.9046 0 0.0011(with tail dep. in upper tail instead of lower)Plackett copula 18.7311 37.4615 37.4591 0 0Frank copula 16.9707 33.9407 33.9384 0 0Gumbel copula 14.1408 28.2807 28.2784 0 0.1221Rotated Gumbel copula 19.7077 39.4147 39.4124 0.1281 0Students t copula 66.0887 132.1766 132.1743 0.0898 0.0898Symmetrised Joe-Clayton 16.7929 33.5842 33.5795 0.0373 0Time-varying normal copula 79.1476 158.2929 158.2859Time-varying rotated Gumbel 80.1311 160.2597 160.2527Time-varying SJC copula 59.8096 119.6144 119.6004

CJC(u, |U, L) = 1 (1 {[1 (1 u)] + [1 (1 )] 1}1/)1/,where= 1/log2(2 U),=1/log2(L),U (0, 1), andL (0, 1).

To investigate asymmetry of dependence by using both tail dependencies, Patton [11] proposed the symmetrized

Joe-Clayton (SJC) Copula:

CSJ C(u, |U, L) = 0.5 [CJC(u, |U, L) +CJC(1 u, 1 |U, L) +u+ 1]. (7)A copula model usedin econometrics is the time-varying normal copula:

CG(u, v|) = 1(u)

1(v)

1

2

(1 2) exp(r2 2rs+s2)

2(1 2)

drds ,

where1 is the inverse of the standard normal distribution function and

t =

+t1+ 110

10j=1

1(utj ) 1(tj )

,

where = (1 ex)(1 +ex)1 =tanh(x/2)is the modified logistic transformation as Patton [11].The purpose of using tail dependence in financial data is to investigate the behavior of the random variables during

extreme events. In this paper, we want to examine the probability of the extreme events (before and after the 2008

financial crisis) of an extremely large depreciation (appreciation) of both the KRW against the USD and the JPYagainst the USD, by using tail dependence. Table 5 shows the lower and the upper tail dependencies of several

copulas for the daily exchange rates for KRW-USD and JPY-USD from January 2, 2001 to December 31, 2010.Among twelve copula models in Table 5, the time-varying rotated Gumbel copula is the best model in terms of AICand BIC criteria.

Symmetrized Joe-Clayton copula show thatL =0.0373 is larger thanU =0, which means that the probability

of both extremely large depreciations of the KRW-USD and the JPY-USD is greater than the extremely largeappreciations of both the KRW-USD and the JPY-USD from January 2, 2001 to December 31, 2010. Furthermore,there exists no probability occurring of the extremely large appreciations of the currency exchange rates of the two

countries.In Fig. 4, the conditional correlation estimates from the time-varying normal copula defined in Patton [11] during

20012010 show that there exists estimates of positive correlation of the KRW-USD and the JPY-USD from 2001 to

late 2007, and that there exists estimates of negative correlation of the KRW-USD and the JPY-USD from late 2007to 2010. This result corresponds to Fig. 1, which is Daily exchange rates for the Korean won and the Japanese yenagainst the U.S. dollar.

To look at the effect of the financial crisis, we performed the tail dependence test for the two time period groups(group 1: 20012005 and group 2: 20062010).

Table 6 shows the lower and upper tail dependenciesof several copulas for the daily exchange rates for KRW-USD

and JPY-USD from January 2, 2001 to December 31, 2005. Among twelve copula models in Table 6, Time-varying


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Fig. 4. Conditional Correlation Estimates from Time-varying Normal Copula 20012010.

SJC copula is the best model in terms of AIC and BIC criteria. Symmetrized Joe-Clayton copula shows that L =0.1574 is smaller thanU =0.1842, which means that the probability of both extremely large appreciations of theKRW-USD and the JPY-USD is greater than one or both extremely large depreciations of the KRW-USD and the

JPY-USD from January 2, 2001 to December 31, 2005. This reflects that there is a more positive relationship of the

currency exchange rates of the two countries during this time period.Table 7 shows the lower and upper tail dependencies of several copulas of the daily exchange rates for KRW-USD

and JPY-USD from January 2, 2006 to December 31, 2010. Among the twelve copula models in Table 7, Students t

copula is the best model in terms of AIC and BIC criteria. Symmetrized Joe-Clayton copula shows that L =U =0, which means that there exists no probability occurring of the extremely large appreciations (depreciations) of the

currency exchange rates of the two countries from January 2, 2006 to December 31, 2010. From these results, we

found that the tail dependence by a copula function could detect the asymmetry in data properly. To overcome this

problem, we suggest a copula directional dependence approach to analyzing data.

4. Copula approach directional dependence

Sungur [16]proposed a directional dependence using copulafunction, so we will briefly introduce this. Sungur [16]

considered two types of directional dependence between two random variables,UandV, in regression: rV|U(u) =

E[V|U = u] and rU|V(v) = E[U|V = v] for the Rodriguez-Lallena, and Ubeda-Flores family of copula in theform:

C(u, v) =uv +f(u)g(v), (8)

whereE[V|U =u]is the conditional expectation ofVgiven thatU=u (Rodriguez et al. [13]).Note that the specific functional forms off andg determine the corresponding family of bivariate distributions

of(U, V). Iff and g are different, then the copula is not symmetric, in which case the form of the regressionfunctions forV andU will be different. Hence, one might consider two types of directional dependence, i.e. onein the direction fromU toV, and the other in the direction fromV toU. Since directional dependence can arisefrom marginal behavior, joint behavior, or both, one may consider the following general measure of directional

dependence:


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Gaussian copula 85.9645 171.9275 171.9234 0 0Claytons copula 64.3495 128.6975 128.6934 0.1964 0Rotated Clayton copula 69.4510 138.9003 138.8962 0 0.2106(with tail dep. in upper tail instead of lower)Plackett copula 73.6523 147.3031 147.2990 0 0Frank copula 72.8289 145.6562 145.6521 0 0Gumbel copula 80.0435 160.0854 160.0813 0 0.2720Rotated Gumbel copula 75.7503 151.4990 151.4949 0.2659 0Students t copula 87.4405 174.8794 174.8753 0.0066 0.0066Symmetrized Joe-Clayton 88.9110 177.8188 177.8106 0.1574 0.1842Time-varying normal copula 87.0603 174.1157 174.1034Time-varying rotated Gumbel 82.2794 164.5541 164.5418Time-varying SJC copula 96.3267 192.6438 192.6193



Gaussian copula 7.6087 15.2158 15.2117 0 0Claytons copula 0.0057 0.0129 0.0170 0 0Rotated Clayton copula 0.0093 0.0203 0.0243 0 0(with tail dep. in upper tail instead of lower)Plackett copula 4.1505 8.2994 8.2953 0 0Frank copula 0.0015 0.0047 0.0088 0 0Gumbel copula 26.5381 53.0779 53.0820 0 0.1221Rotated Gumbel copula 21.4873 42.9761 42.9802 0 01221 0Students t copula 32.7375 65.4733 65.4692 0.0536 0.0536Symmetrized Joe-Clayton 4.8732 9.7496 9.7578 0 0

Time-varying normal copula

12.1757

24.3466

24.3344

Time-varying rotated Gumbel 8.4967 16.9886 16.9763Time-varying SJC copula 10.8184 21.6463 21.6709

(k)XY =

E

rY|X (X) E[Y]k

k(Y) ifk(Y) =E[Y E[Y]]k = 0;

(9)

(k)YX =

E

rX|Y(Y) E[X]k

k(X) ifk(X)= 0,

where (k)XY is the proportion of the k-th central moment ofY explained by the regression ofY on X. For example,

(2)XYcan be interpreted as the proportion of variation explained by the regression ofY onXwith respect to totalvariation ofY.

Jung et al. [8] considered the following type of FGM distributions in the form of the Rodriguez-Lallena andUbeda-Flores copula family in (8):

C(u, v) =uv +ubvb(1 u)(1 v) for0 u, v 1and, 1, (10)where, andare parameters. C(u, v)defined in (10) is a copula function for satisfying

0 min

+ 1

11

,

+ 1

11

, (11)

see Bairamov et al. [2].

Let Xiand Yibe i.i.d. random variables ofXand Y for i= 1, . . . , n. Then uiand viare the marginal distributionfunctions ofFX (xi)andFY(yi). Note thatui andvihave uniform distributions on(0, 1). The empirical likelihoodis


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L(; u, v) =n

i=1

c(ui, vi), (12)

whereu = (u1, . . . , un), v= (v1, . . . , vn)

andc(ui, vi) = 2C(ui,vi)

uivi.

From (10),c(ui, vi)is obtained as

c(ui, vi) = 2C(ui, vi)

uivi

= 2

uivi

uv+ubvb(1 u)(1 v) (13)

= 1 + [1 ui(1 +)](1 ui)1 [1 vi(1 +)](1 vi)1.With (13), the empirical likelihood function become

L(; u, v) =n

i=1

1 + [1 ui(1 +)](1 ui)1 [1 vi(1 +)](1 vi)1

. (14)

Solving

log L(; u, v)

= 0and

log L(; u, v)

= 0 (15)

subject to, 1, one obtains estimates ofand denoted by and . Sincelog L(; u, v)is a linear function of

with and estimated from (15), there is not a closed form solution for MLE from the partial derivative functionwith respect to. As an alternative method, optimization techniques based on quasi-Newton (or Broyden-Fletcher-Goldfarb-Shanno (BFGS)), conjugate-gradient (CG), and Nelder-Mead algorithms could be considered for finding

a value ofto optimize the copula function. Method BFGS is a quasi-Newton method which uses function values

and gradients to build up a picture of the surface to be optimized. Method CG is a conjugate gradients methodby Fletcher and Reeves [6]. Nelder-Mead optimization [9] uses only function values of (14) and is robust even

though the computation is relatively slow. It has been selected for finding an estimate of because Nelder-Meadoptimization works reasonably well for non-differentiable functions of the parameter.

We can obtain the parameter estimates for, and by carrying out the following steps:

Step 1: Find the estimates, and , of andfrom (15).Step 2: Plug the estimates, and , into the empirical log-likelihood function,logL(; u, v).Step 3: Find the optimized value of by the Nelder-Mead optimization.

The Eqs (16)(20) for directional dependence and proportion of variation for directional dependence usethe estimated

parameter values.

For (10), we have f(u) =

u(1 u) and g (v) =

v(1v) . Jung et al. [8] introduced the directionaldependence fromU toVand fromV toUas

rU|V(v) =1

2 Beta(2, + 1)(1 v)1[1 (1 +)v] (16)

and

rV|U(u) = 1

2 Beta(2, + 1)(1 u)1[1 (1 +)u], (17)

whereBeta(, )is the beta function defined byBeta(a, b) = 10

ta1(1 t)b1dtfor a,b >0.By considering the proportion of variation for directional dependence, two types of measures can be derived. With

k= 2 on (9), Jung et al. [8] derived

(2)UV = 12

2[Beta(2, + 1)]2(Beta(1, 2 1)(18)

2(1 +)Beta(2, 2

1) + (1 +)2Beta(3, 2

1)),


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Table 8Forms off(u)andg(v)for each type

Type f(u) g(v)

I

ub(1 u)

vb(1 v)II

ub(1 u)

vb(1 v)

Table 9Forms ofrU|V(v)andrV|U(u)for each type

Type rU|V(v)

I 12

(b+1)(b+2)

bvb1(1 v) vb

II 12 Beta(b+ 1, + 1)v

b1(1 v)1 [b (b+)v]Type rV|U(u)

I 12

(b+1)(b+2) bub1(1 u) ub

II 12 Beta(b+ 1, + 1)ub1(1 u)1 [b (b+)u]Table 10

Equation ofc,(2)UV and

(2)VU

TYPE I TYPE II

c 12

(b+1)2(b+2)2 12beta(b+ 1, + 1)beta(b+ 1, + 1)

(2)UV

122b(b+1)2(b+2)2(2b1)(2b+1)

122 (beta(b+ 1, + 1))2 beta(2b, 2)

(2)VU

122b(b+1)2(b+2)2(2b1)(2b+1)

122(beta(b+ 1, + 1))2 beta(2b, 2)

and

(2)VU = 12

2[Beta(2, + 1)]2(Beta(1, 2 1)(19)

2(1 +)Beta(2, 2 1) + (1 +)2

Beta(3, 2 1)),where Spearmans correlation coefficient,c, is

c = 12

10

10

(C(u, v) uv) dudv = 12 Beta(2, + 1)Beta(2, + 1). (20)

Table 8 shows the forms off(u) andg (v) for each type of measure used in C(u, v) = uv+ f(u)g(v). Type Ihas a symmetric function form since both have the same functional form as a function ofu,f(u), and a functionofv,g(v), Type II has an asymmetric function form, and therefore Type II can show the directional dependence oftwo variables, u and v . It is more meaningful for a copula function to examine the asymmetry offinancial data.The directional dependenciesrU|V(v) andrV|U(u) are summarized for each type of measure in Table 9 and the

derivations ofc,(2)UV, and

(2)VUare given in Table 10.

5. Application of directional dependence

To examine the asymmetry of currency exchange data in detail, we have applied a copula directional dependence

approach to the daily exchange rates for the Korean won (KRW-USD) and the Japanese yen against the U.S. dollar

(JPY-USD) from January 2, 2001 to December 31, 2010. Tables 11 and 12 showV ar(ru|v ), V ar(rv|u), (2)uv ,

(2)vu, and2c for Type I at four different periods from January 2001 through December 2005, January 2006 throughDecember 2010, January 2006 through August 2008, and September 2008 through December 2010. The V ar(ru|v ),V ar(rv|u), and the directional dependencies in joint behavior, ru|v and rv|uat b=1, 1.5, 2,3 whereuis KRW-USD

andv is JPY-USD, are given. The(2)uv and

(2)vu are the general measures of directional dependence fromu =

KRW-USD to v= JPY-USD and from v= JPY-USD to u= KRW-USD, and 2cis Spearmans correlation coefficientof KRW-USD and JPY-USD. Type I is a symmetric copula function so that we cant detect the directional dependence


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Table 11V ar(ru|v),V ar(rv|u),

(2)uv ,

(2)vu, and

2c for Type I Model

Jan. 2001 through Dec. 2005

b= 1.0 b= 1.5 b= 2.0 b= 3.0

Var(ru|v) 8.36297e-03 2.45085e-03 9.26591e-04 2.14418e-04Var(rv|u) 8.36298e-03 2.45090e-03 9.26631e-04 2.14437e-04

(2)uv 1.00278e-01 2.93878e-02 1.11111e-02 2.57143e-03

(2)vu 1.00278e-01 2.93878e-02 1.11111e-02 2.57143e-03

2c 3.16667e-01 1.56735e-01 8.33333e-02 3.00000e-02

Jan. 2006 through Dec. 2010

b= 1.0 b= 1.5 b= 2.0 b= 3.0

Var(ru|v) 3.70655e-04 6.27413e-05 4.54023e-06 5.36028e-05Var(rv|u) 3.70657e-04 6.27420e-05 4.54035e-06 5.36076e-05

(2)uv 4.44444e-03 7.52327e-04 5.44444e-05 6.42857e-04

(2)vu 4.44444e-03 7.52327e-04 5.44444e-05 6.42857e-04

2c 6.66667e-02 2.50776e-02 5.83333e-03 1.50000e-02


(2)uv ,

(2)vu, and

2c for Type I Model

Jan. 2006 through Aug. 2008

b= 1.0 b= 1.5 b= 2.0 b= 3.0

V ar(ru|v) 2.31813e-05 3.53156e-05 1.00968e-04 1.89561e-04V ar(rv|u) 2.31807e-05 3.53117e-05 1.00950e-04 1.89503e-04

(2)uv 2.77778e-04 4.23184e-04 1.21000e-03 2.27211e-03

(2)vu 2.77778e-04 4.23184e-04 1.21000e-03 2.27211e-03

2c 1.66667e-02 1.88082e-02 2.75000e-02 2.82000e-02

Sep. 2008 through Dec. 2010

b=1.0

b=1.5

b=2.0

b=3.0

V ar(ru|v) 1.55901e-03 5.65102e-04 1.00968e-04 2.06121e-05V ar(rv|u) 1.55901e-03 5.65176e-04 1.00986e-04 2.06195e-05

(2)uv 1.86778e-02 6.77094e-03 1.21000e-03 2.47114e-04

(2)vu 1.86778e-02 6.77094e-03 1.21000e-03 2.47114e-04

2c 1.36667e-01 7.52327e-02 2.75000e-02 9.30000e-03

of KRW-USD and JPY-USD. But from Type I we observe a meaningful result, which is the change of the values of

2c depending on the different time groups; that is2c =0.31667 in Jan. 2001 through Dec. 2005,

2c =0.06667 in

Jan. 2006 through Dec. 2010, 2c =0.016667 in Jan. 2006 through Aug. 2008, and 2c =0.13667 in Sep. 2008

through Dec. 2010 atb= 1.0. We observe a trend that values ofV ar(ru|v )andV ar(rv|u)decrease as the value ofbincreases from 1 to 3 before January 2006. After 2005, values ofV ar(ru|v )and V ar(rv|u)decrease as the valueofb increases from 1 to 2. However, whenb is 3, the values ofV ar(ru|v )and V ar(rv|u)increase little, but these

are still smaller than whenb = 1.5. For the values of(2)uv ,(2)vu, and2c , the trends are similar withV ar(ru|v)andV ar(rv|u)for the corresponding time periods. For the time period January 2006 to August 2008, we observe a

trend in which values ofV ar(ru|v ),V ar(rv|u),(2)uv ,

(2)vu, and 2cincrease as the value ofbincreases from 1 to 3.

However, for the time period September 2008 to December 2010 in Table 12, we observe the reverse trend in which

values ofV ar(ru|v ),V ar(rv|u),(2)uv, and

(2)vu decrease as the value ofb increases from 1 to 3. One difference

for September 2008 to December 2010, is for2cin that the trend of2c still increases as the value ofbincreases from

1 to 3.

To examine the directional dependence of the Korean won (KRW-USD) and the Japanese yen against the U.S.

dollar (JPY-USD) from January 2006 to December 2010, we compared two time groups, January 2006 to August

2008 and September 2008 to December 2010, in Table 13 by using a Type II model.

The main part on the simulation work in the Type II model is to estimate parameters for this model. The model

parameters are estimated under fixed values of power,b, such thatb = 1, 1.5, 2, and 3. When the estimated values


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(2)uv ,

(2)vu, and

2c for Type II Model

Jan 2006 through Aug 2008

b= 1.0 b= 1.5 b= 2.0 b= 3.0

1.01630 1.54047 1.91962 2.52630Var(ru|v) 9.49750e-03 2.91991e-03 7.40657e-04 4.27447e-05Var(rv|u) 9.49722e-03 2.92012e-03 7.40739e-04 4.27510e-05

(2)uv 5.70384e-02 1.86964e-02 4.76466e-03 2.72249e-04

(2)vu 5.70516e-02 1.87002e-02 4.76555e-03 2.72294e-04

2c 3.37347e-01 1.75877e-01 8.05911e-02 1.61808e-02

Sep 2008 through Dec 2010

b= 1.0 b= 1.5 b= 2.0 b= 3.0

1.01695 1.54081 1.91995 2.52662Var(ru|v) 9.50294e-03 2.91905e-03 7.40201e-04 4.26908e-05Var(rv|u) 9.50300e-03 2.91836e-03 7.39954e-04 4.26792e-05

(2)uv 5.70976e-02 1.86939e-02 4.76210e-03 2.71919e-04(2)vu 5.70663e-02 1.86849e-02 4.76001e-03 2.71815e-04

2c 3.37410e-01 1.75814e-01 8.05481e-02 1.61672e-02

are put into the equations given in Tables 9 and 10, we get the values ofV ar(ru|v),V ar(rv|u),(2)uv ,

(2)vu, and

2c . Table 13 gives results for ,V ar(ru|v ),V ar(rv|u),(2)uv ,

(2)vu, and 2c . For the period January 2006 to August

2008, the value of(2)uv is always smaller than (2)vu as the value ofb increases from 1 to 3. This means that

the Korean Won against the U.S. dollar is more dependent on the Japanese yen against the U.S. dollar before the

financial crisis. However, for the period September 2008 to December 2010, the trend is reversed since the value of

(2)uv is always larger than

(2)vu as the value ofb increases from 1 to 3. This means that the Japanese yen against

the U.S. dollar is more dependent on the Korean Won against the U.S. dollar after the financial crisis. Therefore, the

Type II model gives us a meaningful result that there exists a different directional dependence between KRW-USD

and JPY-USD, before and after the financial crisis.

6. Conclusion

Tables 6 and 7 showed the tail dependence of KRW-USD and JPY-USD before and after the financial crisis,

but we failed to find meaningful information of the asymmetry of KRW-USD and JPY-USD. Since the directional

dependence only when existing an asymmetry of data can be found, we have used the copula directional dependence

approach proposed by Jung et al. [8] in this paper. We found the meaning fact that there exists a directional

dependence of the currency exchange rates of the Republic of Korea and Japan against the US dollar before and after

the 2008 financial crisis. It means this directional dependence by copula explained the existence of the asymmetry

of data clearly. In future study, we will use Aas et al.s [1] pair-copula method for the directional dependence of

multivariate data with several countries currency exchange rate data or stock index data.

Acknowledgments

The authors are thankful to the Guest Editor Professor Arkady Shemyakin and an anonymous learned referee for

the valuable comments on the original version of this manuscript which led to substantial improvement.

References

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large asymmetry and directional dependence by using copula modeling to currency exchange rates

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