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New Approach of Directional Dependence in Exchange Markets Using Generalized FGM Copula Function Yoon-Sung Jung a; Jong-Min Kim b; Jinhwa Kim c a Department of Statistics, Kansas State University, Manhattan, Kansas, USA b Division of Science and Mathematics, University of Minnesota-Morris, Morris, Minnesota, USA c School of Business, Sogang University, Mapo-Gu, Seoul, Korea
Online Publication Date: 01 April 2008 To cite this Article: Jung, Yoon-Sung, Kim, Jong-Min and Kim, Jinhwa (2008) 'New Approach of Directional Dependence in Exchange Markets Using Generalized FGM Copula Function', Communications in Statistics - Simulation and Computation, 37:4, 772 - 788 To link to this article: DOI: 10.1080/03610910701711091 URL: http://dx.doi.org/10.1080/03610910701711091
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Communications in Statistics—Simulation and Computation® , 37: 772–788, 2008 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0918 print/1532-4141 online DOI: 10.1080/03610910701711091
Multivariate Analysis
New Approach of Directional Dependence in Exchange Markets Using Generalized FGM Copula Function YOON-SUNG JUNG1 , JONG-MIN KIM2 , AND JINHWA KIM3 1
Department of Statistics, Kansas State University, Manhattan, Kansas, USA 2 Division of Science and Mathematics, University of Minnesota-Morris, Morris, Minnesota, USA 3 School of Business, Sogang University, Mapo-Gu, Seoul, Korea This article presents an application of copula methodology in exchange markets. In this article, we consider the concept of directional dependence given by Sungur (2005). We also consider and study directional dependence for generalized Farlie– Gumbel–Morgenstern (FGM) distributions, which are a member of the RodríguezLallena and Úbeda-Flores (2004) family, Cu v = uv + fugv. Examples of the generalized FGM distributions are provided with exchange market data of the Euro, Canadian dollar, Korean Won, Japanese Yen, and Hong Kong dollar against the U.S. dollar. Keywords Copulas; Directional dependence; Exchange markets; Generalized FGM family; MLE; Regression function. Mathematics Subject Classification 62H11; 62H20.
1. Introduction Copula gives us an alternative to the multivariate normal specification of the dependence between variables, and has gained much attention in several fields, such as asset pricing, portfolio management, and risk management applications. Furthermore, applications of copula in other fields are remarkably increasing. Breymann et al. (2003) is devoted to the modeling of the joint distribution to see how these models could be better than the joint normal specification. Longin Received June 10, 2007; Accepted September 26, 2007 Address correspondence to Jong-Min Kim, Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris, Morris, MN 56267, USA; E-mail:
[email protected]
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and Solnik (2001) studied the asymptotic behavior of the conditional correlation of equity returns the use of extreme value theory has shown these correlations deviate from what would be expected under multivariate normality. For other applications of extreme value dependence, one may look at Poon et al. (2003, 2004). Patton (2002) proposed a dynamic-copula approach in the foreign exchange market. Patton (2002) found a time variation to be significant in a copula model for asymmetric dependence between two exchange rates where the dependence parameter followed an ARMA-type process. In this article, we try to propose a new copula model which has a directional dependence using Generalized FGM Copula function. While Patton (2002) only considered the effect of asymmetric dependence on portfolio returns, our proposed copula model will include directional dependence as well as asymmetric dependence. This means our model will be simpler and more powerful than the Patton (2002) copula model. The definitions and properties of copulas are introduced in Sec. 2. Section 3 discusses the main results on directional dependence on Generalized FGM copulas. Its application to exchange market data is presented in Sec. 4. Section 5 concludes the article with discussion of advantages and limitations of the proposed model and future research plans.
2. What is Directional Copula? Sungur (2005) proposed a directional dependence using copula function. In this paper, we briefly introduce the directional dependence using copula function. 2.1. Concepts of Copula A copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube 0 1n such that every marginal distribution is uniform on the interval 0 1. Definition 2.1 (Copula). A two-dimensional copula is the joint cumulative distribution function (cdf) of a pair of variables with marginal uniform distributions on [0, 1]. Thus a copula is satisfying the following properties: 1. ∀u ∈ 0 1, C1 1 u 1 1 = u, 2. ∀ui ∈ 0 1, Cu1 cn = 0 if at least one of the ui equals zero, 3. C is grounded and n-increasing. Sklar (1973) proposed that any multivariate distribution function, say F , can be represented as a function of its marginals, say G and H, by using a copula C, i.e., Fx y = CGx Hy. It can be defined as follows. Theorem 2.1 (Sklar’s Theorem). Given an n-dimensional distribution function F with continuous marginal (cumulative) distributions F1 Fn , there exists a unique n-copula C 0 1n → 0 1 such that Fx1 xn = CFx1 Fxn . 2.2. Concepts of Dependence Rodgers and Nicewander (1988) discussed 13 approaches of looking at the correlation coefficient. Rovine and Von Eye (1997) discussed a 14th way to look at a correlation coefficient as the proportion of matches. Many faces of the correlation
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coefficient have also been discussed by Falk and Well (1997). Recently, Dodge and Rousson (2000) studied more new faces of the correlation coefficient and also derived some formulae to denote the relationship between two random variables in linear regression setting. Their results have implications for selecting the response variable in a linear regression setting. Definition 2.2 (Pearson’s Correlation). Under assumption that is symmetric, the Pearson’s correlation is as follows: 3XY =
Y X = 0
X
(1)
where X and Y are the coefficients of skewness of the variables X and Y as
X =
EX − EX3 EY − EY3 and
= Y X3 Y3
where X2 and Y2 are the variances of the corresponding variables. The above equations given by Dodge and Rousson (2000) are used to decide the directional dependence of a regression line by generalizing the type of covariance between two variables to that of three variables. Muddapur (2003) provided an alternative simple method to prove Eq. (1) of the correlation coefficient. The alternative equation is defined as 2XY 3 =
2Y
2X
(2)
with following the same definitions of X Y X2 , and Y2 . Since the left-hand side of (2) is always less than or equal to 1, 2Y ≤ 2X . Thus, Y is dependent on X with being symmetric. Similarly, Dodge and Rousson (2000) and Muddapur (2003) changed the location of variables X and Y to show the other directional dependence which is X depending on Y . This contradicts a real directional dependence. For more detailed explanation, one may look at Dodge and Rousson (2000) and Muddapur (2003). Definition 2.3 (Rodríguez-Lallena and Úbeda-Flores, 2004). The bivariate copula form proposed by the Rodríguez-Lallena and Úbeda-Flores (2004) is as follows: Cu v = uv + fugv for all u v
(3)
Bairamov et al. (2001) studied distributional properties of concomitants for the generalized FGM distribution, and presented recurrence relations between moments of concomitants. Bairamov and Kotz (2002) presented dependence structure and symmetry of Huang-Kotz FGM distributions. Bairamov et al. (2003) proposed some properties of the local dependence function. 2.3. Concepts of Directional Dependence Sungur (2005) proposed two types of directional dependence with the RodríguezLallena and Úbeda-Flores (2004) family of copula, Cu v = uv + fugv, in a
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regression setting. The difference between direction of dependence and directional dependence was emphasized in Sungur (2005). The direction of dependence means a property of marginal distributions, and the directional dependence means a property of joint distribution. Sungur (2005) also considered the general measurement of the directional dependence because directional dependence can happen from a marginal or a joint behavior or both. Using Eq. (3), Sungur (2005) defined 1 −1 1 EV U = u = rV U u = − C f u 12 fudu 2 0 1 −1 1 EU V = v = rU V v = − C g v 12 gvdu 2 0 1 1 where C = 12 0 0 Cu vdu dv − 3 is the Pearson’s correlation. The directional dependence in regression setting as the above is: when there exists two different functions f and g, the copula will not be symmetric and the forms of the regression functions for V and U will differ. Therefore, one can consider two different kinds of directional dependence. The first one is the directional dependence from U to V , and the second is the directional dependence from V to U . These are new attractive concepts in applied statistics fields. In this paper, we want to introduce these interesting concepts to readers and apply the concepts to a real data. So we use two types of directional dependence structure as Sungur (2005) defined: one is directional dependence in joint behavior and the other is directional dependence in marginals. Definition 2.4. We can define that any pair U V is directionally dependent in joint behavior: 1. if the form of the regression functions for V U = u and U V = v differ, i.e., rV U w = rU V w; 2. by the conditional copula, if CV U w z = CU V w z, where CV U v u =
Cuv CU V u v = Cuv .
u
v Definition 2.5. Under the following conditions: 1. G and H are the distribution functions of X and Y ; 2. U = GX and V = HY has copula C; 3. rV U w = rU V w = rc w where rY X x = Ec Y X = x = FY−1 rC FX x −1 rX Y y = Ec Y X = x = FX rC FY y, then the random pair X Y is directionally dependent in marginals if three conditions and rY X z = rX Y z are satisfied. In Sungur (2005), we can illustrate the following comments: • The directional dependence of Dodge and Rousson (2000) is only working at the skewed X and Y . • It is not possible to see whether there is any direction of regression with the same marginals using the directional dependence of Dodge and Rousson (2000).
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Since directional dependence can arise from a marginal or joint behavior or both, the following general measurement of the directional dependence can be considered: k
X→Y = k X→Y
=
ErY X X − EYk EEY X − EYk = k Y = 0 EY − EYk k Y ErX Y Y − EXk k X
(4)
k X = 0
The properties and examples of the general measurement of the directional dependence can be found in Sungur (2005). Here are some remarks on directional dependence: • Copula approach to directional dependence removes the influence of marginals. • When we want to construct a directional dependence model in joint behavior, one needs to use asymmetric copula because it can lead to different forms of copula regression function depending the direction.
3. Directional Dependence on Generalized FGM Distribution In this section, we consider several different types of the Farlie–Gumbel– Morgenstern (FGM) distribution which has a specific form of Rodríguez-Lallena and Úbeda-Flores (2004) copula family. Each type of FGM distribution is defined as: • Type I: Cu v = uv + ub vb 1 − u1 − v where 0 ≤ u v ≤ 1 • Type II: Cu v = uv + ub vb 1 − u 1 − v where ≥ 1; ≥ 1; 0 ≤ u v ≤ 1 • Type III: Cu v = uv + ub vb 1 − up q 1 − vp q where p ≥ 1; q ≥ 1; 0 ≤ u v ≤ 1, where b, p, and q are any given values and are parameters. Here, we need to estimate parameters, , and . Let FX x and FY y be cdfs with density function fX x and fY y. We consider Fx y = CFX x FY y, shown by Sklar (1973). It yields the joint density function as the likelihood fx y = fX xfY ycFX x FY y
(5)
where cu v =
2 Cu v
u v
We define Ui = FX Xi and Vi = FY Yi for the continuous empirical marginal distribution function FX and FY . Note that Ui and Vi have uniform distribution U0 1. Hence we can reduce our empirical likelihood function to L U V =
n i=1
cUi Vi
(6)
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The estimation of the parameter is determined by maximizing the likelihood (6) for the data set. For maximizing the likelihood from the data set, we consider the logarithmic form as ˆ = argmax∈R
n
log L Ui Vi
(7)
i=1
where R is the set of all possible ’s. Therefore, we use the logarithmic form to estimate parameters, and , in Type II. Instead of using the above procedure of the maximum likelihood estimation, we propose the following form to estimate parameters, and : n n b1 − v log1 − vi − vi i=1 b1 − ui log1 − ui − ui ˆ = ˆ i=1 n i n i=1 ui log1 − ui i=1 vi log1 − vi In this article, to estimate the parameter, , we consider Eq. (6) because of a simple form of likelihood function more than the logarithmic form. But we find there is a difficulty to estimate by the method of the maximum likelihood estimation.
using
log L UV The problem is L UV or is not a function of . Therefore, we cannot
estimate a parameter by likelihood function. As an alternative method, we used a numerical method to find a value of with maximizing the copula function in this article. The directional dependence for the direction U to V and for the direction V to U is defined as 1 Cu v
1 EV U = u = rV U u = 1 − dv = 1 − Cu vdv (8)
u
u 0 0 and EU V = v = rU V v = 1 −
1 0
Cu v
1 Cu vdu du = 1 −
v
v 0
(9)
Sungur (2005) has derived two kinds of measurement of the directional dependence: 1. From (4), the proportion of variation for the directional dependence is defined as: 2
U →V =
VarrV U U VarV
= 12 VarrV U U
Table 1 Forms of fu and gv for each type Type I II III
fu √ b √ ub 1 − u √ ub 1 − u u 1 − up q
gv √ b √ vb 1 − v √ vb 1 − v v 1 − vp q
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Jung et al. Table 2 Forms of rU V v and rV U u for each type Type
rU V v
I III
1 2 1 2 1 2
I
rV U u 1 − b+1b+2 bub−1 1 − u − ub 2
II
II III
1 2 1 2
−
bvb−1 1 b+1b+2
− v − vb
− Betab + 1 + 1vb−1 1 − v−1 b − b + v
− Beta b+1 q + 1 vb−1 1 − vp q−1 b − b + pqvp p
− Betab + 1 + 1ub−1 1 − u−1 b − b + u − Beta b+1 q + 1ub−1 1 − up q−1 b − b + pqup p
= 12
ErV2 U u
1 = 12ErV2 U u − 3 − 4
2
V →U = 12ErV2 U u − 3
(10)
2. The proportion of variation for the direction U to V and for the direction V to U is also defined as 1 f u2 du 2 2 U →V = c 0 1 12 0 fudu2 1 g v2 du 2 2 V →U = c 0 1 (11) 12 0 gvdv2 where c = 12
1 0
1 0
Cu vdu dv − 3 k
Since, from Eq. (4), the positive version of the X→Y can be interpreted as the proportion of kth central moment of Y that can be explained by the regression 2 of Y on X, X→Y can be seen as the proportion of variation of the paired variable. From Sungur (2005), we can get the following fact, if both of the copula regression functions are linear, the random variable pair U V cannot be directionally dependent. Table 1 shows the forms of fu and gv per each type used in Cu v = uv + fugv. Using the proposed three 1 types of 1fu and gv, we can derive the additional results of f u, g v, 0 f udu, 0 g vdv, 1 1 f u2 udu, 0 g v2 vdv. These derivation results helped us calculate the 0 directional dependence, rU V v and rV U u for each type. So we summarized the directional dependencies, rU V v and rV U u for each type in Table 2. We should explain that rU V v depends on the value of V and rV U u depends on the value of U .
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Figure 1. Monthly exchange rates of rit from January 1993–April 2006.
4. Data Analysis This section demonstrates how our proposed directional copula method can apply to real data. Data used in this article is the monthly exchange market data from January 1993–April 2006, from the Pacific Exchange Rate Service in Sauder School of Business, University of British Columbia. Five traded currencies quoted against the U.S. Dollar (USD) are used. They are the Korean Won (KRW/USD), the EURO (EUR/USD), the Canadian Dollar (CAD/USD), the Japanese Yen (JPY/USD), and the Hong Kong Dollar (HKD/USD). These currencies are grouped into four pairs with KRW/USD as the base; any pair consists of KRW/USD and another quote against USD. That is, four currency pairs (KRW, EUR), (KRW,CAD), (KRW,JPY), and (KRW,HKD) against USD will be analyzed to check whether there exists any directional dependence in at least one of four pairs by using our proposed directional copula function in this paper. The following figure is for five exchange ratio plots based on USD. The ratios were expressed by using x log-ratio, rit = 100 × log x it . it−1 Figure 1 shows the exchange ratio plot for KRW against USD has an unusual peak hike in 1997 because the Republic of Korea faced a temporary financial crisis
Table 3 Basic statistics of log-ratio, rit
EUR CAD JPY KRW HKD
Mean
Median
Minimum
Maximum
St.D
Skewness
Kurtosis
−0007447417 −0069527667 −0041618518 0114017021 0001688580
0107313157 0000000000 0162898134 −0072611098 0001282109
−62571857 −52069447 −104245730 −86111204 −05461609
43080834 31561719 82060438 374537308 02773176
2.25594663 1.38303302 2.73887559 3.83776508 0.08034702
−2.816353e-47 −5.564461e-46 −4.657973e-47 6.557292e-48 −6.426970e-48
4.651820e-62 2.484682e-60 9.098489e-62 6.663045e-63 6.487066e-63
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Jung et al. Table 4 Parameter estimates KRW vs EUR
KRW vs JPY
ˆ ˆ KRW
b 1 1.5 2 3
0.9932854 1.1588377 1.3243899 1.6554944
ˆ EUR
ˆ
Type I Type II Type III
1.0072005 4545 4584 1.1751018 3151 18181 1.3430030 18750 18670 1.6788055 2334 31500
36905 33690 73820 320600
ˆ JPY
1.0100544 4745 1.1784735 19129 1.3468926 4644 1.6837309 8865
KRW vs CAD
ˆ KRW
1 1.5 2 3
0.9932854 1.1588377 1.3243899 1.6554944
ˆ CAD
Type I Type II Type III
1.0092792 4471 45155 0642 1.1775502 18506 91650 3060 1.3458212 18782 399880 7699 1.6823631 7971 500000 28490
4795 4275 13640 34300
45585 28400 49300 176700
KRW vs HKD
ˆ b
Type I Type II Type III
ˆ ˆ HKD
Type I Type II Type III
1.0084998 97540 95940 185290 1.1766215 288400 44890 70080 1.3447431 17400 69950 146210 1.6809865 20340 204400 146040
in that year. Table 3 provides basic descriptive statistics of log-ratio, rit . From Table 3, we can see that the skewness and the kurtosis of each country are almost zero. We also notice that the maximum of KRW is much higher than the maximums of the other ones because of the effect of the temporary financial crisis in that year in the Republic of Korea. The simulation work of estimating parameters, , , and , are a main part in this article. To do the simulation, we needed to fix the power values of b such as b = 1 15 2 3 for Types I, II, and III in Table 1. The next step of estimating
Figure 2. Plots of rU V v (left) and rV U u (right) for KRW vs EUR with b = 15, 2.
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Table 5 2 2 Equation of c , U →V , and V →U c 2 U →V 2 V →U
TYPE I
TYPE II
12 b+12 b+22 122 b b+12 b+22 2b−12b+1 122 b b+12 b+22 2b−12b+1
12betab + 1 + 1betab + 1 + 1 2 122 betab + 1 + 1 beta2b 2 122 betab + 1 + 12 beta2b 2
TYPE III
2 12 beta b+1 q + 1 p
b+1 2 2
12 beta p q + 1 2q + 1 b beta 2b−1 p
2b−1 + 2 2q − 1 −2bpqbeta p + 1 2q + p2 q 2 beta 2b−1 p
2 2
12 beta b+1 q + 1 2q + 1 b beta 2b−1 p p
2b−1 + 2 2q − 1 −2bpqbeta p + 1 2q + p2 q 2 beta 2b−1 p
c 2
U →V 2
V →U
Table 6 Variance of directional dependence in joint behavior KRW vs EUR VarrU V v b
VarrV U u
Type I
Type II
Type III
Type I
Type II
Type III
1 1.5 2 3
0.19235610 0.02447416 0.32750766 0.00117366
0.19557348 0.41314786 0.07123836 0.00677517
0.26278545 0.03213116 0.03216871 0.04666579
0.19237543 0.02446888 0.32743808 0.00117442
0.19559687 0.41218663 0.07122000 0.00680115
0.26293743 0.03213487 0.03216065 0.04666600
1 1.5 2 3
0.20965537 0.90084346 0.02005046 0.01687131
0.21303662 0.02270966 0.03775389 0.00794974
0.21300605 0.02265123 0.03773234 0.00798224
0.40116766 0.02283553 0.01434401 0.01417577
1 1.5 2 3
0.18615045 0.84351977 0.32832355 0.01365968
0.18916157 0.10453937 0.32527922 0.01695482
0.18914076 0.10428140 0.32496252 0.01701002
0.00795705 0.02651046 0.03498205 0.03685177
1 1.5 2 3
0.88676863 2.04870822 0.00281650 0.00088811
0.85576209 0.02513786 0.00998460 0.00284563
0.85493849 0.02505923 0.00996423 0.00285077
6.62805273 0.13904734 0.12616266 0.00968316
KRW vs JPY 0.40124017 0.20967868 0.02283822 0.90178250 0.01435075 0.02008681 0.01418025 0.01694257 KRW vs CAD 0.00795020 0.18616206 0.02648388 0.84399988 0.03499065 0.32855669 0.03688536 0.01369769 KRW vs HKD 6.62695738 0.88602777 0.13889199 2.04978251 0.12613955 0.00281984 0.00969056 0.00089192
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parameters, and , is to use the form we proposed: n n i=1 b1 − ui log1 − ui − ui i=1 b1 − vi log1 − vi − vi ˆ ˆ = n n i=1 ui log1 − ui i=1 vi log1 − vi The last step is to estimate the parameter, , by a numerical method. The results of estimating parameters, , and is Table 4. Table 4 shows the estimated values of the parameters, , and based on the data of four currency pairs, (KRW, EUR), (KRW,CAD), (KRW,JPY), and (KRW,HKD) against USD. We set values, p = 2 and q = 2 for estimating for Type III. In Table 4, we first estimated the parameter, , then estimated the parameter, . So the estimate of is fixed for a function of KRW and the estimate of is used for functions of CAD, JPY, and HKD, respectively. In terms of the estimated values of , EURO has the smallest value, Hong Kong is second, next Canada, and Japan has largest estimated value at each of b = 1, 1.5, 2, 3. We may make an inference from this fact that the higher
Table 7 2 2 2c , U →V and V →U KRW vs EUR Type I
b
1
1.5
2
3
1
1.5
2
3
2c 2 U →V 2 V →U 2c 2 U →V 2 V →U 2c 2 U →V 2 V →U 2c 2 U →V 2 V →U 2c 2 U →V 2 V →U 2c 2 U →V 2 V →U 2c 2 U →V 2 V →U 2c 2 U →V 2 V →U
Type II
22952250 2.33297544 22952250 1.15867421 22952250 1.17490891 02439086 4.35034304 02917852 2.60860109 02917852 2.64050339 24414063 0.62155570 39062500 0.45269985 39062500 0.45771312 00049028 0.04134784 00140080 0.04272653 00140080 0.04313877 KRW vs CAD 22210934 2.2559704 22210934 1.1204296 22210934 1.1384871 84130873 1.1006176 100644843 0.6599646 100644843 0.6692575 24497467 2.8360333 39195947 2.0655782 39195947 2.0919350 00571832 0.1034130 01633804 0.1068612 01633804 0.1080504
KRW vs JPY Type III
Type I
Type II
Type III
7.6611320 6.2697296 6.2697296 0.9300584 0.7658470 0.7658470 0.8357344 0.7670639 0.7670639 0.9250592 1.1133110 1.1133110
25016694 25016694 25016694 89890711 107535274 107535274 01497690 02396304 02396304 00707294 02020840 02020840
11.6887063 9.5658224 9.5658224 0.6609139 0.5442227 0.5442227 0.3727469 0.3421191 0.3421191 0.2810060 0.3381914 0.3381914
0.2318423 0.1897354 0.1897354 0.7672750 0.6318047 0.6318047 0.9090522 0.8343574 0.8343574 0.7305121 0.8791730 0.8791730
105711684 105711684 105711684 204324663 244431360 244431360 00210250 00336400 00336400 00037234 00106384 00106384
254062284 126180247 12831300 023906803 014335263 01454713 032930006 023984027 02430545 004852832 005014644 00507331 KRW vs HKD 1019725148 506447351 514209616 026448281 015859213 016071383 008696050 006333625 006410356 001733139 001790929 001809833
193.11966 158.04558 158.04558 4.0243560 3.3138145 3.3138145 3.2784936 3.0091069 3.0091069 0.1919491 0.2310112 0.2310112
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estimated value of is, the more influence the KRW receives from other country currency because the Republic of Korea has more exchange currency trading with Japan compared to other countries. Figure 2 is plots of rU V v and rV U u for KRW vs. EUR at b = 15. From Fig. 2, we notice that the directional dependencies, rU V v and rV U u, are clearly different for Type II compared to Types I and III because fu and gv in Type II have asymmetric functions. Table 5 is for the
Figure 3. Plots of (a) MLE, (b) rU V v and rV U u, (c) 3D, (d) contour for KRW vs EUR with b = 15.
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Figure 4. Plots of (a) MLE, (b) rU V v and rV U u, (c) 3D (d) contour for KRW vs CAD with b = 15.
variance table of directional dependence in joint behavior, rU V v and rV U u. The 2 2 following table is a summary table of equations derived for c , U →V and V →U . 2 2 Table 5 explains the relationship among c , U →V , and V →U in Table 7. Table 6 provides variances of directional dependence in joint behavior, rU V v and rV U u, which are for the Euro, Japanese Yen, Canadian dollar, and the Hong Kong dollar against the Korean Won. In case of b = 1, variances of Types I, II, and III for KRW vs. HKD are larger than the other. One of the reasons may be that Korea and Hong
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Figure 5. Plots of (a) MLE, (b) rU V v and rV U u, (c) 3D, (d) contour for KRW vs JPY with b = 15.
Kong can have an inconsistent currency trading in both countries compared to the other currencies. Next is for KRW vs. JPY. The Japanese Yen gives the Korean Won consistent effect than Hong Kong dollar. When the value of b is larger than 1, we can find a trend that values of rU V v and rV U u are decreased as the value of b is increased at Types I, II, and III, respectively. But, in the Type III cases of
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Figure 6. Plots of (a) MLE, (b) rU V v and rV U u, (c) 3D, (d) contour for KRW vs HKD with b = 15.
KRW vs EUR and KRW vs. CAD, values of rU V v, and rV U u are increased as 2 2 the value of b is increased. We can compare values of 2c , U →V , and V →U from Table 6. Also, there are some trends from Table 7 that we have to note. First, it is 2 2 that Type I has a same value of 2c , U →V , and V →U at b = 1. Next, we have to look 2 2 at Types I and III for U →V and V →U with b = 1, 1.5, 2, 3. We can find the reason
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from Table 5. It is the reason that U →V and V →U have the same equation form. We carefully looked at Type I. When b is larger than 1, the proportions of variation for the directional dependence from V to U and from U to V are larger than 2c . 2 Finally, Types II and III give us that the value of 2c is larger than those of U →V and 2 V →U when b = 1 15 2. At b = 3, the proportions of variation for the directional 2 2 dependence from U to V and V to U , U →V and V →U , are larger than 2c . The plots from Figs. 3–6 compared four currency pairs, (KRW, EUR), (KRW, CAD), (KRW, JPY), and (KRW, HKD) against USD for the case of b = 15. In each figure, we presented a MLE of for each type, plots for variance of rU V v and rV U u, and 3D plots and contour plots of Types I, II, and III.
5. Conclusion The copula approach to directional dependence has an advantage that eliminates the effects of marginals. With three types of generalized FGM distribution functions, we try to get the trend of the directional dependence and the form of the proportion of variation for the direction U to V and for the direction V to U against 2c . Furthermore, the visualization of each type of the generalized FGM distribution, from Figs. 3–6, classifies the directional dependence for the direction U to V and for the direction V to U by 3D plot and contour plot. An important fact presented in this article. We can get the directional dependence between two countries compared to Pearson correlation, c . The directional dependence can give us important information in deciding the relationship between two countries in this article.
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