research article dynamics of foreign exchange networks: a time...

Research ArticleDynamics of Foreign Exchange Networks A Time-VaryingCopula Approach

Gang-Jin Wang12 Chi Xie12 Peng Zhang1 Feng Han3 and Shou Chen12

1 College of Business Administration Hunan University Changsha 410082 China2 Center of Finance and Investment Management Hunan University Changsha 410082 China3 China Merchants Bank Shenzhen 518067 China

Correspondence should be addressed to Chi Xie xiechihnueducn

Received 11 March 2014 Accepted 16 April 2014 Published 6 May 2014

Academic Editor Fenghua Wen

Copyright copy 2014 Gang-Jin Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on a time-varying copula approach and the minimum spanning tree (MST) method we propose a time-varying correlationnetwork-based approach to investigate dynamics of foreign exchange (FX) networks In piratical terms we choose the daily FX ratesof 42 major currencies in the international FX market during the period of 2005ndash2012 as the empirical data The empirical resultsshow that (i) the distributions of cross-correlation coefficients (distances) in the international FX market (network) are fat-tailedand negatively skewed (ii) financial crises during the analyzed period have a great effect on the FX networkrsquos topology structure andlead to the US dollar becomingmore centered in theMST (iii) the topological measures of the FX network show a large fluctuationand display long-range correlations (iv) the FX network has a long-term memory effect and presents a scale-free behavior in themost of time and (v) a great majority of links between currencies in the international FXmarket survive from one time to the nextand multistep survive rates of FX networks drop sharply as the time increases

1 Introduction

Financial markets are accounted as complex dynamicalsystems with large quantities of interacting unties [1 2]Financial agents usually interact with each other and theirinterbehaviors change over time which means that theinterbehaviors are dynamics and widely found in economicsand finance [3 4] To capture the interactive behaviors orcross-correlations among heterogeneous entries in finan-cial markets many scholars generally resort to a powerfulanalytical tool namely correlation network-based methodswhich include the minimum spanning tree (MST) approachproposed byMantegna [5] the correlation thresholdmethodsdeveloped by Boginski et al [6] and Onnela et al [7] andthe approach of planar maximally filtered graph (PMFG)designed by Tumminello et al [8] The network analysisidea has been widely applied in financial markets such asstock markets [9ndash14] and foreign exchange (FX) market [15ndash20] Among correlation network-based approaches the MSTmethod is often preferred because of its robustness and

simplicity [10] However the MST method and its improve-ments ignore the volatiles and nonlinearities of financialtime series That is to say they cannot really detect thedynamic interbehaviors among different financial agents infinancial markets Therefore the purpose of this paper isto propose a dynamic correlation network-based approachby combining a time-varying copula method and the MSTapproach for studying the dynamic topology and marketnatures of financial networks In practical terms we focus ourstudy on networksrsquo dynamics of the international FX marketbecause it is the biggest and most liquid financial marketwhere foreign currencies are traded [21]

Themotivations that led us to combine the two aforemen-tioned methods to investigate dynamics of FX networks canbe summed up as follows On the one hand the MST andits improvements are usually used to identify the clusteringbehavior and dominant currencies in the FX network Toexamine the dynamic behavior of the network perviousworks often employ a rolling window analysis such as [1022] However some drawbacks are found in the dynamic

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 170921 11 pageshttpdxdoiorg1011552014170921

2 Discrete Dynamics in Nature and Society

MSTs using a rolling window analysis as follows (i) Thechoice of parameters of the rolling window is dependent onthe scholarsrsquo preference that is the window width and steplength are selected arbitrarily [23] For example Onnela et al[10] set the size of window as four calendar years (approxi-mately 1000 trading days) while Song et al [22] fix the lengthof window as 1250 trading days (ii) In the MST methodthe cross-correlation coefficient between two financial unitsis usually measured by the Pearsonrsquos correlation coefficient(PCC) that is a linear correlation method Nevertheless thePCC cannot quantify the nonlinear relationship between twoheterogeneous entries [21] and ignores financial variablesrsquovolatile and nonnormality features [24] It should be notedthat some works attempt to apply other dynamic approachesin the MST For example Lyocsa et al [23] employ thedynamic conditional correlations (DCC) multivariate gen-eralized autoregressive conditional heteroscedasticity (MV-GARCH) approach to construct the MST for the US stockmarket Trancoso [25] uses the Baba-Engle-Kraft-Kroner(BEKK) model to develop the conditional correlation matrixand apply it in the dynamic network analysis HoweverLyocsa et al [23] and Trancoso [25] assume that the financialvariables obey the normal distribution and neglect theirnonnormality characteristics

On the other hand copula methods proposed by Sklar[26 27] are awidely useful tool for deriving joint distributionsgiven the marginal distributions especially when the vari-ables follow nonnormal distributions [28] Besides copulascan be employed to investigate the dependence beyond thelinear cross-correlation by the PCC and allow for a time-varying nonlinear analysis Moreover copulas can be sup-plied as a powerful analytical instrument tomeasure dynamicdependence structures between financial variables In a wordcopulas have better properties and more advantages (see[29 30]) than the traditional linear correlation methodsand attract many scholars to use them in various financeapplications [31ndash37] For instance Patton [35] constructstime-varying copula models to examine the dependencebetween two FX rates that is Deutsche MarkUS Dollar(DMUSD) and Japanese YenUS Dollar (JPYUSD) He alsomakes a comparison between copulas and GARCH modelsand finds that the former can better describe the dependenceor cross-correlation of FX rates than the latter Diks et al [36]use several copula models to analyze the dependence amongfive currencies (ie Canadian Dollar (CAD) Swiss Franc(CHF) European Euro (EUR) British Pound (GBP) andJPY) against USD and show that the Studentrsquos 119905-copulamodelis evidently superior to the Gaussian Gumbel and Claytoncopula models Dias and Embrechts [37] investigate thedependence between EURUSD and JPYUSD fromOctober1 2000 to October 1 2008 by employing the time-varyingcopula-GARCH models Their empirical results suggest thatthe dependence between the two currencies is dynamicand a time-varying copula approach with given correlationspecifications has better outcomes than some conventionaldynamic methods (eg BEKK)

In consideration of the above-mentioned motives basedon a time-varying copula approach and the MST methodwe aim to construct time-varying FX networks and analyze

their topological dynamics andmarket propertiesWe choose42 major currenciesrsquo daily FX rate series in the internationalFX market during the years 2005ndash2012 as the empirical dataIn empirical process we first use a time-varying copula tocalculate the dynamic cross-correlation coefficients 120588

119905among

different currencies More specifically we adopt an AR(119901)-GARCH(11)-119905 model to characterize the marginal distribu-tion of returns for FX rates and then estimate the dependenceparameters of the time-varying Studentrsquos 119905-copula modeland obtain time-varying cross-correlation coefficients Nexton the basis of time-varying cross-correlation coefficientswe construct time-varying cross-correlation matrices (CMs)C119905for 42 major currencies in the international FX market

Then we transform the time-varying CMs into time-varyingFX networks by the MST approach Finally we examinetopological dynamics and statistic features for time-varyingFX networks

The remainder of the paper is organized as followsSection 2 represents the empirical data A time-varyingcorrelation network-based approach by combining a time-varying copula model with the MST method is describedin Section 3 In Section 4 the time-varying FX networks areconstructed and main empirical results are showed Someconclusions are drawn in Section 5

2 Data Set

As for the empirical data set we choose the daily FXrates of 42 major currencies in the international FX marketfrom January 4 2005 to December 31 2012 FollowingJang et al [18] and Wang et al [21] we select the specialdrawing right (SDR) as the numeraire The 42 currenciesare from 7 different continents or regions Their detailedinformation is showed as follows (1) Africa EgyptianPound (EGP) and South African Rand (ZAR) (2) AsiaChinese Renminbi (CNY) Indian Rupee (INR) IndonesianRupiah (IDR) Japanese Yen (JPY)MalaysianRinggit (MYR)Pakistani Rupee (PKR) Philippines Peso (PHP) SingaporeDollar (SGD) South Korean Won (KRW) Taiwanese Dollar(TWD) Thai Baht (THB) and Vietnamese Dong (VND)(3) Europe British Pound (GBP) Czech Koruna (CZK)European Euro (EUR) Hungarian Forint (HUF) IcelandicKrona (ISK) Norwegian Krone (NOK) Polish Zloty (PLN)Romanian New Leo (RON) Russian Rubles (RUB) SwedishKrona (SEK) Swiss Franc (CHF) and Turkish New Lira(TRY) (4) Latin America Argentine Peso (ARS) BrazilianReal (BRL) Chilean Peso (CLP) Colombian Peso (COP)Peruvian New Sole (PEN) and Mexican Peso (MXN) (5)Middle East Bahrain Dinar (BHD) Israeli New Shekel(ILS) Jordanian Dinar (JOD) Kuwaiti Dinar (KWD) SaudiArabian Riyal (SAR) and United Arab Emirates Dirham(AED) (6) North America Canadian Dollar (CAD) and USDollar (USD) (7) Pacific Ocean Australian Dollar (AUD)andNewZealandDollar (NZD) All the FX rates are obtainedfrom the website of the Pacific Exchange Rate Service(httpfxsauderubccadatahtml) We define the return ofcurrency 119894 on day 119905 as 119903

119894119905= 100(ln119875

119894119905minus ln119875

119894119905minus1) where 119875

119894119905is

the daily FX rate of currency 119894 on day 119905 During the analyzedperiod each currencyrsquos returns have 2003 observations

Discrete Dynamics in Nature and Society 3

3 Methodology

In this section we first introduce the time-varying copulamodel including the model for marginal distributions thedynamic Studentrsquos 119905-copula model and the estimation ofparameters After that we propose the time-varying corre-lation network-based approach by the MST method

31 Model for Marginal Distributions Following Patton [35]andDias and Embrechts [37] we use an AR(119901)-GARCH(11)-119905model which considers the influences of asymmetric infor-mation to characterize the returnsrsquo marginal distributions ofcurrency 119894 The proposed model is defined as follows

119903119894119905= 120583 +

119901

sum

119895=1

120601119895119903119894119905minus119895

+ 120576119894119905

120576119894119905= 120590119894119905119911119894119905 119911119894119905sim 119905 (])

1205902

119894119905= 120596119894+ 1205721198941205762

119894119905minus1+ 1205731198941205902

119894119905minus1

(1)

where 120601119895are autoregressive (AR) coefficients 119911

119894119905obeys a

Studentrsquos 119905-distribution ] is the degree of freedom and 1205902119894119905is

the conditional variance of 120576119894119905with the following parameter

restrictions 120596119894gt 0 120572

119894gt 0 120573

119894gt 0 and 120572

119894+ 120573119894lt 1 To be

simple and effective enough we use an AR(1) process

32 The Dynamic 119905-Copula Model According to Diks et al[36] and Dias and Embrechts [37] the dynamic 119905-copulamodel has a better capability for quantifying dynamic cor-relations in the FX rates data compared with GaussianGumbel and Clayton copula models Therefore in thispaper we employ the dynamic Studentrsquos 119905-copula model tocapture time-varying cross-correlations in the internationalFXmarket Let us briefly show the dynamic Studentrsquos 119905-copulamodel as follows

For all 119906119905 V119905isin [0 1] the density of dynamic Studentrsquos 119905-

copula is defined by

119888119905(119906119905 V119905| 120579119905 119899)

=1

radic1 minus 1205792

119905

Γ ((119899 + 2) 2) Γ (1198992)

[Γ ((119899 + 1) 2)]2

times [1 + (119879minus1

119899(119906119905)2

+ 119879minus1

119899(V119905)2

minus 2120579119905119879minus1

119899(119906119905) 119879minus1

119899(V119905))

times (119899 (1 minus 1205792

119905))minus1

]

minus(119899+2)2

times [(1 +119879minus1

119899(119906119905)2

119899)(1 +

119879minus1

119899(V119905)2

119899)]

(119899+1)2

(2)

where 119879minus1119899(sdot) represents the inverse of the cumulative distri-

bution function (CDF) of the Studentrsquos 119905-distribution with119899 degrees of freedom [39] 120579

119905isin (0 1) denotes the linear

correlation coefficient and Γ(sdot) is the Gamma function

As proposed in [31] the time-varying dependence coeffi-cients of the Studentrsquos 119905-copula is defined as

120588119905= Λ(120574

0+ 1205741120588119905minus1

+ 1205742

1

10

10

sum

119895=1

119879minus1

119899(119906119905minus119895) 119879minus1

119899(V119905minus119895)) (3)

where Λ(119909) = (1 minus 119890minus119909)(1 + 119890

119909) is the modified logistic func-

tion which can guarantee that cross-correlation coefficientsretain in the interval (minus1 1) at all times 120574

119896(119896 = 0 1 2) are

unknown parameters

33 Estimation of Copula Parameters Following Wang et al[34] and Lai et al [40] we adopt the inference-function-for-margins (IFM) method rather than the exact maxi-mum likelihood method to estimate the copula parametersbecause the former needs less computation than the latterThe IFM approach proposed by Joe and Xu [41] is a two-stepestimation which can be used to estimate the parameters ofmarginal distributions and the copula functions separatelyFor more detailed advantages see [34 41] The procedure ofIFM is showed as follows

Step 1 The marginal parameters are estimated by the maxi-mum likelihood (ML) as

120585119894= argmax

119879

sum

119905=1

ln119891119894119905(119911119894119905| Ω119905minus1 120585119894) (4)

where 119891119894119905(sdot | sdot) denotes the conditional marginal density of

currency 119894 at time 119905 120585119894is the marginal parameter of returns of

currency 119894 andΩ119905minus1

is the past information set

Step 2 Given 120585119894 suppose we have 120585

119906and 120585V the copula

parameters can be estimated as

120585119888= argmax

119879

sum

119905=1

ln 119888119905(119865119906119905(119911119906119905| Ω119905minus1120585119906)

119865V119905 (119911V119905 | Ω119905minus1120585V) 120585119888)

(5)

34 The Time-Varying Correlation Network-Based ApproachAfter obtaining the time-varying cross-correlation coeffi-cients between any two currencies by a time-varying copulaapproach we can build119873times119873 time-varying cross-correlationmatrices (CMs) C

119905with elements 120588

119905(119894 119895) for currencies 119894 and

119895 where 1 le 119894 and 119895 le 119873 (in our case 119873 = 42) Accordingto the idea of MST proposed by Mantegna [5] we transformtime-varying CMs into the corresponding distance matricesD119905by a distancemeasure119889

119905(119894 119895) = radic2(1 minus 120588

119905(119894 119895)) that falls in

the interval [0 2] andmeets the three axioms of the Euclideandistance On the basis of time-varying distance matricesD119905 we can obtain time-varying networks for studying the

international FX market by using the Kruskalrsquos algorithm[42] that is time-varyingMSTs link119873 currencies with119873minus1

edges At each time 119905 the MST network extracts the mostimportant information (eg the strongest cross-correlationsamong currencies) in the international FX market Theproposed time-varying correlation network-based approach


can be used to examine dynamics of the international FXmarket over time

To investigate dynamics of FX networks we introducesome topological measures as follows We use a quantity ofaverage path length (APL) to quantify the MST networkrsquosdensity [43] which is defined by

APL119905=

2

119873 (119873 minus 1)

119873

sum

119894=1119895gt119894

119897119905

119894119895 (6)

where 119897119905

119894119895is the length of the shortest path between two

vertexes (currencies) 119894 and 119895 at time 119905 [21]The measure of mean occupation layer (MOL) proposed

by Onnela et al [9 10] which can be employed to analyzethe spread of nodes on the MST and characterize the densitychanges of the network is defined as

MOL119905(V119888) =

1

119873

119873

sum

119894=1

lev (V119905119894) (7)

where V119888is the central vertex of the MST at time 119905 and lev(V119905

119894)

defines the level of vertex V119894with reference to V

119888 whose level

is set as zeroWe introduce a concept of maximum degree 119896max which

is defined as the number of linkages of the central vertex inthe MST [21 43]

The scale-free behavior is widely found in differentnetworks [10 15 21 44 45] The scale-free network is suchthat the degree distribution of the network has a power-lawtail that is

119875 (119896) sim 119896minus120572 (8)

where 119875(119896) is the distribution function of vertex degrees 119896and 120572 is the exponent We adopt a powerful tool developedby Clauset et al [38] to estimate the power-law exponent andthe corresponding 119875 value This tool combines ML fittingmethods with goodness-of-fit tests using the Kolmogorov-Smirnov statistic and likelihood ratios

4 Empirical Results

41 Statistics of Cross-Correlation Coefficients and Distancesof MST Before studying dynamics of FX networks we firstanalyze statistical properties of cross-correlation coefficientsand distances of MST for 42 currencies in the internationalFX market The cross-correlation coefficient series contains119873(119873 minus 1)2 observations at each time while the distanceset of MST only contains the 119873 minus 1 most important linksIn Figures 1 and 2 we present the time evolution graphsfor four descriptive statistics (mean standard deviationskewness and kurtosis) of cross-correlation coefficients anddistances of MST respectively From each figure it can befound that the four descriptive statistics vary over time andhave a high volatile during the US subprime crisis and the2008 world financial crisis Especially in the period of June2007 to July 2009 the international FX market (network)has stronger cross-correlations or smaller distances among

2005 2006 2007 2008 2009 2010 2011 201201

015

02

Time (year)

Mea

n

035

04

045

Stan

dard

dev

iatio

n

minus06

minus04

minus02

0

Skew

ness

18

2

22

24

Kurt

osis

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

Time (year)

Time (year)

Figure 1 The mean standard deviation skewness and kurtosis ofcross-correlation coefficients of 42 currencies in the internationalFX market as functions of time

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

06

07

08

02

025

03

minus15

minus1

minus05

0

2

3

4

Mea

nSt

anda

rd d

evia

tion

Skew

ness

Kurt

osis

Figure 2 The mean standard deviation skewness and kurtosis ofdistances of MST of 42 currencies in the international FX market asfunctions of time


USD

EUR

CAD

GBP

JPY

CHF

ARS

AUD

BHD

BRL

CLP

CNY

COP CZK

EGP

HUF

ISK

INR

IDRILS

JODKWD

MYRMXN

NZD

NOK

PKR PEN

PHP

PLN

RON

RUB

SAR

SGDZAR

KRWSEK

TWDTHB

TRY

AED

VND

AfricaAsiaEuropeLatin America

Middle EastNorth AmericaPacific Ocean

Figure 3 MST of 42 currencies in the international FX market on January 5 2005 as a representative of the period of before financial crises

USD

EUR

CAD

GBP

JPYCHF

ARS

AUD

BHD

BRL

CLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN

NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND



Figure 4 MST of 42 currencies in the international FXmarket on January 2 2008 as a representative of the period of during financial crises


USD

EUR

CADGBP

JPY

CHF

ASR

AUD

BHD

BRLCLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND



Figure 5 MST of 42 currencies in the international FX market on January 3 2012 as a representative of the period of after financial crises

currencies than other periods This phenomenon confirmsthe proposal in [46] that the financial crisis often causesan increase of the marketrsquos cross-correlations As shown inFigures 1 and 2 it can be found that the values of skewnessof cross-correlation coefficients and distances at any time areless than 0 while values of kurtosis for most of time are notequal to 3This finding implies that the distributions of cross-correlation coefficients (distances) in the international FXmarket (network) are fat-tailed and negatively skewed

42 MST Results Considering that financial crises have astrong influence on the international FX market we choosethree days (ie January 5 2005 January 2 2008 and January3 2012) as representatives of three periods of before duringand after financial crises We present the three MSTs of 42currencies in the international FXmarket in Figures 3 4 and5 respectively In each MST figure currencies from the samecontinent (region) aremarked with the same color and shape

From Figure 3 which demonstrates the situation beforefinancial crises one can find that most of currencies aregathered together according to geographical distributionssuch as the European cluster Asian cluster Middle Easterncluster and Latin American cluster with EUR MYR AEDand MXN at their centers respectively In the FX networkthe most important cluster is the international cluster withUSD at its hub which is directly or indirectly connectedwith currencies from Asia Middle East Latin America andAfrica This outcome shows that USD is the predominantworld currency An interesting cluster is composed of GBPfrom Europe NZD and AUD from Pacific Ocean CAD fromNorth America and ZAR fromAfricaWe denote this clusteras the Commonwealth cluster because countries of the fivecurrencies are members of the Commonwealth of NationsIn the MST network we find that three major currencies inthe international FX market namely EUR CHF and JPY arelinked together


As illustrated in Figure 4 during the global financialcrisis it can be observed that a lot has changed in theFX network Notable changes are that USD becomes morecentered in the MST and the Latin American cluster andAsian cluster almost broke and their currencies directly orindirectly shift to USD That is to say during the financialcrisis most currencies from Asia Middle East and LatinAmerica are tightly linked to USD which indicates that thefinancial crisis can lead to a huge comovement effect amongcurrencies in the international FX market Although theEuropean cluster and the Commonwealth cluster still remainin the network their structure and currenciesrsquo positionchanged as a result of the influence by the financial crisis

Compared with the MSTs in Figures 3 and 4 as drawn inFigure 5 the FX network recovered to the precrisis state butits structure and currenciesrsquo position have a lot of changesFor instance the Asian cluster and Latin American clusterare formed again At this point the Commonwealth clusterhas reappeared in the network with the same structure andposition of their currencies as they appeared in Figure 3One can see a remarkable change that JPY deviates from theEuropean cluster and connects to the international clusterwith USD at its centre It is interesting to note that CNY linkswith USD TWD and SAR One possible interpretation ofthe linkages is that US Taiwan and Saudi Arabia are Chinarsquosimportant and top trading partners

From Figures 3 4 and 5 we can obtain some conclusionsas follows (i) USD is the predominant world currency andhas a powerful influence in the monetary system (ii) TheEuropean cluster has a relatively stable structure and thismay be ascribed to the influence of EUR (iii) Currenciesfrom the Middle East except for ILS always form a clusterand link to USD Possible explanations are that Saudi Arabiathe United Arab Emirates Kuwait Jordanian and Bahrainare oil-producing countries (the former three countries aremembers of the Organization of the Petroleum ExportingCountries) and have a mass of USD holdings and mostof their currencies peg to USD (iv) The Commonwealthcluster is formed in the FX network suggesting that theCommonwealth nations maybe have the same currencymechanism

43 Dynamics of Topological Features In this subsection weaim to investigate the dynamical evolution of time-varyingFX networksrsquo topological features To begin with it we showthe calculation results of the average path length (APL)mean occupation layer (MOL) and maximum degree 119896maxin Figure 6 As for the density measures of APL and MOLboth of their patterns do not show any tendency but witha fluctuation above and below The values of maximumdegree 119896max also have a large volatility especially during theperiod of financial crises Then we estimate the power-lawexponent and the corresponding 119875 value for each MST andpresent their outcomes in Figure 7 The estimated power-lawexponent also changes over time and varies from 209 to 35Although a handful of (about 309) 119875 values are less than 01the power-law hypothesis can be accepted for most MSTsThis finding suggests the FX network is a scale-free network

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km

axFigure 6 The average path length (APL) mean occupation layer(MOL) and maximum degree kmax of MST of 42 currencies in theinternational FX market as functions of time In each panel the redsolid line stands for the corresponding statistical average value overthe time investigated

2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

Figure 7The estimated power-law exponent 120572 and the correspond-ing 119875 value of degree distribution of MST of 42 currencies in theinternational FX market as functions of time In each panel thered solid line stands for the corresponding statistical average valueover the time investigated In the bottom panel the yellow solidline represents the value of 01 As proposed in [38] if the 119875 valueis greater than 01 the power-law hypothesis is accepted for theinvestigated data otherwise it is rejected

in most of the time That is a small number of vertexes(currencies such as USD) always have the vast majority ofconnections while most of vertexes have a very few links

Similar to Qiu et al [47] we further examine dynamicsof topological features of FX networks by analyzing the timecorrelations In practical terms we employ the detrended


s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)

Figure 8The DFA functions of the average path length (APL) mean occupation layer (MOL) and maximum degree 119896max and the estimatedpower-law exponent 120572 on log-log plots In each panel the red solid line stands for the corresponding linear fitting curve and the estimatedHurst exponent119867 and its corresponding coefficient of determination 1198772 are presented The Hurst exponent 05 lt 119867 lt 10 implies that thetime series is long-range correlated or has a long-term memory

fluctuation analysis (DFA) method proposed by Peng et al[48] which can be used to quantify long-range correlationsof a nonstationary time series The DFA approach providesa relationship between the DFA function 119865(119904) and the timescale 119904 characterized by a power-law 119865(119904) sim 119904

minus119867 where119867 is the well-known Hurst exponent The Hurst exponent119867 = 05 0 lt 119867 lt 05 and 05 lt 119867 lt 10means uncorrelatedlong-term correlated and anticorrelated time series respec-tivelyTheDFA functions of theAPLMOLmaximumdegree119896max and the estimated power-law exponent are drawn inFigure 8 We calculate the Hurst exponents for APL MOL119896max and the power-law exponent as 076 plusmn 002 080 plusmn

002 086 plusmn 002 and 073 plusmn 001 respectively which are alllarger than 05 These results mean that the four topological

measures are long-range correlated and thus suggest that theFX network has a long-term memory effect

44 Single- andMultistep Survival Rates In order to study therobustness of links over time and the long-term evolution ofFX networks respectively we use two measures that is thesingle-step survival rate (SSR) and themultistep survival ratio(MSR) proposed by Onnela et al [9 10] The measure of SSRis defined as the fraction of links found in two consecutiveMST at times 119905 and 119905 + 1 that is

SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

Figure 9 The single-step survival ratio (SSR) of MST of 42currencies in the international FX market as a function of timeThered solid line stands for the corresponding statistical average valueover the time investigated

where 119864(119905) represents the set of edges of the MST at time 119905cap is the intersection operator and | sdot sdot sdot | gives the number ofelements in the set [10] The MSR measure is defined by

MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)

where 1199050stands for the initial time and 120575 is the step length

In Figure 9 we plot the time-varying SSRs for the MSTThe mean value of SSR is close to 092 which shows that agreatmajority of links between currencies in the internationalFX market survive from one time to the next Moreover wefind that about 80 SSRs are equal to 1 indicating that the twoconsecutive networks at times 119905 and 119905+1 are identicalWe alsoinvestigate the time correlations of the SSR series by the DFAmethod and present the results in Figure 10 One can find thatthe Hurst exponent for the SSR series is 068 plusmn 001 onceagain suggesting that the long-range memory effect exists inthe FX network In Figure 11 we show the MSR of MST of42 currencies in the international FX market as a function oftime for different initial time 119905

0 In Figure 11 the initial time

1199050is the first trading date of the year and 8 curves of MSR are

presented For each curve ofMSR it drops rapidly as the timeincreases which implies that the long-term stability of the FXnetwork is falling as the time is increasing However we alsofind that each MSR is usually unchanged and moves towarda constant in the last or middle period of time meaning thatsome structures or clusters (eg the Middle Eastern cluster)of the FX network are always preserved and stabilized

5 Conclusions

In this paper we investigate the daily FX rates of 42 majorcurrencies in the international FX market during the periodof 2005ndash2012 and construct time-varying FX networks bya time-varying copula approach and the MST method Indetail we first use the AR(119901)-GARCH(11)-119905 model to char-acterize the returnsrsquo marginal distributions of FX rates Then

SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)

Figure 10 The DFA function of the single-step survival ratio (SSR)on a log-log plot The red solid line stands for the associatedlinear fitting curve and the estimated Hurst exponent 119867 and itscorresponding coefficient of determination 1198772 are presented

101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012

Figure 11 The multistep survival ratio (MSR) of MST of 42currencies in the international FX market as a function of time fordifferent initial time 119905

0 For each curve the initial time 119905

0is the first

trading date of the year for example 2005 stands for January 5 2005

we employ the time-varying Studentrsquos 119905-copula to calculatethe dynamic cross-correlation coefficients between each pairof rates Finally we adopt the MST to build time-varying FXnetworks and analyze the networks properties including thedynamics and time correlations of topological features andsurvival rates of the MST

Some basic finding for examining FX networks in thisresearch can be summarized as follows (i) By analyzing thedescriptive statistics of cross-correlation coefficients and dis-tances of MST we find that distributions of cross-correlationcoefficients (distances) in the international FX market (net-work) are fat-tailed and negatively skewed (ii) On basis


of MSTs for three different periods we observe that somecurrencies gather together and form into several clusterssuch as the international cluster with USD at its centerthe Middle Eastern cluster and the European cluster Thefinancial crises have a great influence on the FX networkrsquostopology structure and lead to USD becomingmore centeredin the MST because lots of currencies from Asia LatinAmerica Middle East and Africa are directly or indirectlylinked to USD (iii) The topological measures of the FXnetwork present a large fluctuation and have a long-termmemory effect By estimating the degree distribution ofMSTwe find that the FX network is a scale-free network in mostof the time (iv) A great majority of links between currenciesin the international FX market survive from one time to thenext and multistep survive rates descend sharply as the timeincreases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank C Yu who works in the Guosen SecuritiesCo Ltd for helpful discussionsThis work was supported bythe Fundamental Research Funds for the Central Universitiesof HunanUniversity the Hunan Provincial Innovation Foun-dation for Postgraduate (Grant no CX2013A006) the Schol-arship Award for Excellent Doctoral Student granted by theMinistry of Education of China the National Natural ScienceFoundation of China (Grant no 71373072) the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion (Grant no 20130161110031) the China Postdoctoral Sci-ence Foundation (Grant no 2013M530376) and the Founda-tion for Innovative Research Groups of the National NaturalScience Foundation of China (Grant no 71221001)

References

[1] R N Mantegna and H E Stanley Introduction to EconophysicsCorrelations and Complexity in Finance Cambridge UniversityPress Cambridge UK 1999

[2] J Kwapien and S Drozdz ldquoPhysical approach to complexsystemsrdquo Physics Reports vol 515 no 3-4 pp 115ndash226 2012

[3] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[4] C Huang H Kuang X Chen and F Wen ldquoAn LMI approachfor dynamics of switched cellular neural networks with mixeddelaysrdquo Abstract and Applied Analysis vol 2013 Article ID870486 8 pages 2013

[5] R N Mantegna ldquoHierarchical structure in financial marketsrdquoEuropean Physical Journal B vol 11 no 1 pp 193ndash197 1999

[6] V Boginski S Butenko and P M Pardalos ldquoStatistical analysisof financial networksrdquo Computational Statistics and Data Anal-ysis vol 48 no 2 pp 431ndash443 2005

[7] J-P Onnela K Kaski and J Kertesz ldquoClustering and informa-tion in correlation based financial networksrdquo European PhysicalJournal B vol 38 no 2 pp 353ndash362 2004

[8] M Tumminello T Aste T Di Matteo and R N Mantegna ldquoAtool for filtering information in complex systemsrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 102 no 30 pp 10421ndash10426 2005

[9] J-P Onnela A Chakraborti K Kaski and J Kertesz ldquoDynamicasset trees and portfolio analysisrdquo European Physical Journal Bvol 30 no 3 pp 285ndash288 2002

[10] J-P Onnela A Chakraborti K Kaski J Kertesz and A KantoldquoDynamics of market correlations taxonomy and portfolioanalysisrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 68 no 5 Article ID 056110 12 pages 2003

[11] J G Brida and W A Risso ldquoDynamics and structure of the 30largest North American companiesrdquoComputational Economicsvol 35 no 1 pp 85ndash99 2010

[12] W-Q Huang X-T Zhuang and S Yao ldquoA network analysis ofthe Chinese stock marketrdquo Physica A Statistical Mechanics andIts Applications vol 388 no 14 pp 2956ndash2964 2009

[13] C K Tse J Liu and F C M Lau ldquoA network perspective of thestock marketrdquo Journal of Empirical Finance vol 17 no 4 pp659ndash667 2010

[14] D Y Kenett M Tumminello A Madi G Gur-GershgorenR N Mantegna and E Ben-Jacob ldquoDominating clasp of thefinancial sector revealed by partial correlation analysis of thestock marketrdquo PLoS ONE vol 5 no 12 Article ID e15032 2010

[15] A Z Gorski S Drozdz and J Kwapien ldquoScale free effectsin world currency exchange networkrdquo The European PhysicalJournal B vol 66 no 1 pp 91ndash96 2008

[16] J Kwapien S Gworek S Drozdz and A Gorski ldquoAnalysis ofa network structure of the foreign currency exchange marketrdquoJournal of Economic Interaction and Coordination vol 4 no 1pp 55ndash72 2009

[17] J Kwapien A Gorski and S Drozdz ldquoStructure and evolutionof the foreign exchange networksrdquo Acta Physica Polonica B vol40 no 1 pp 175ndash194 2009

[18] W Jang J Lee and W Chang ldquoCurrency crises and theevolution of foreign exchangemarket evidence fromminimumspanning treerdquo Physica A Statistical Mechanics and Its Applica-tions vol 390 no 4 pp 707ndash718 2011

[19] G-J Wang C Xie F Han and B Sun ldquoSimilarity measure andtopology evolution of foreign exchange markets using dynamictime warping method evidence from minimal spanning treerdquoPhysica A StatisticalMechanics and Its Applications vol 391 no16 pp 4136ndash4146 2012

[20] D Matesanz and G J Ortega ldquoNetwork analysis of exchangedata Interdependence drives crisis contagionrdquoQualityampQuan-tity 2013

[21] G-J Wang C Xie Y-J Chen and S Chen ldquoStatisticalproperties of the foreign exchange network at different timescales evidence from detrended cross-correlation coefficientand minimum spanning treerdquo Entropy vol 15 no 5 pp 1643ndash1662 2013

[22] D-M Song M Tumminello W-X Zhou and R N MantegnaldquoEvolution of worldwide stock markets correlation structureand correlation-based graphsrdquo Physical Review EmdashStatisticalNonlinear and Soft Matter Physics vol 84 no 2 Article ID026108 9 pages 2011

[23] S Lyocsa T Vyrost and E Baumohl ldquoStock market networksthe dynamic conditional correlation approachrdquo Physica A


Statistical Mechanics and Its Applications vol 391 no 16 pp4147ndash4158 2012

[24] C Huang X Gong X Chen and F Wen ldquoMeasuring andforecasting volatility in Chinese stock market using HAR-CJ-M modelrdquo Abstract and Applied Analysis vol 2013 Article ID143194 13 pages 2013

[25] T Trancoso ldquoEmerging markets in the global economic net-work real(ly) decouplingrdquo Physica A Statistical Mechanics andIts Applications vol 395 pp 499ndash510 2014

[26] M Sklar ldquoFonctions de repartition a 119899 dimensions et leursmargesrdquo Publications de lrsquoInstitut de Statistique de lrsquoUniversite deParis vol 8 pp 229ndash231 1959

[27] A Sklar ldquoRandom variables joint distribution functions andcopulasrdquo Kybernetika vol 9 pp 449ndash460 1973

[28] A C Cameron T Li P K Trivedi and D M ZimmerldquoModelling the differences in counted outcomes using bivariatecopula models with application to mismeasured countsrdquo TheEconometrics Journal vol 7 no 2 pp 566ndash584 2004

[29] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journalof Information Technology amp Decision Making vol 8 no 4 pp787ndash801 2009

[30] J Hu ldquoDependence structures in Chinese and US financialmarkets a time-varying conditional copula approachrdquo AppliedFinancial Economics vol 20 no 7 pp 561ndash583 2010

[31] A J Patton ldquoEstimation of multivariate models for time seriesof possibly different lengthsrdquo Journal of Applied Econometricsvol 21 no 2 pp 147ndash173 2006

[32] R Aloui M S Ben Aıssa and D K Nguyen ldquoConditionaldependence structure between oil prices and exchange rates acopula-GARCH approachrdquo Journal of International Money andFinance vol 32 pp 719ndash738 2013

[33] R Aloui S Hammoudeh and D K Nguyen ldquoA time-varyingcopula approach to oil and stock market dependence the caseof transition economiesrdquoEnergy Economics vol 39 pp 208ndash2212013

[34] K Wang Y-H Chen and S-W Huang ldquoThe dynamic depen-dence between the Chinese market and other internationalstock markets a time-varying copula approachrdquo InternationalReview of Economics and Finance vol 20 no 4 pp 654ndash6642011

[35] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006

[36] C Diks V Panchenko and D van Dijk ldquoOut-of-samplecomparison of copula specifications in multivariate densityforecastsrdquo Journal of Economic Dynamics and Control vol 34no 9 pp 1596ndash1609 2010

[37] A Dias and P Embrechts ldquoModeling exchange rate dependencedynamics at different time horizonsrdquo Journal of InternationalMoney and Finance vol 29 no 8 pp 1687ndash1705 2010

[38] A Clauset C R Shalizi and M E J Newman ldquoPower-lawdistributions in empirical datardquo SIAM Review vol 51 no 4 pp661ndash703 2009

[39] YWei YWang andD Huang ldquoA copula-multifractal volatilityhedging model for CSI 300 index futuresrdquo Physica A StatisticalMechanics and Its Applications vol 390 no 23-24 pp 4260ndash4272 2011

[40] Y Lai C W S Chen and R Gerlach ldquoOptimal dynamichedging via copula-threshold-GARCH modelsrdquo Mathematicsand Computers in Simulation vol 79 no 8 pp 2609ndash26242009

[41] H Joe and J J Xu ldquoThe estimation method of inferencefunctions for margins for multivariate modelsrdquo Tech Rep166 Department of Statistics University of British ColumbiaVancouver Canada 1996

[42] J B Kruskal Jr ldquoOn the shortest spanning subtree of agraph and the traveling salesman problemrdquo Proceedings of theAmerican Mathematical Society vol 7 pp 48ndash50 1956

[43] C Yang Y Shen and B Xia ldquoEvolution of Shanghai stockmarket based on maximal spanning treesrdquo Modern PhysicsLetters B vol 27 no 3 Article ID 135002 19 pages 2013

[44] N Vandewalle F Brisbois and X Tordoir ldquoNon-randomtopology of stock marketsrdquo Quantitative Finance vol 1 no 3pp 372ndash374 2001

[45] R Albert and A-L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[46] T Aste W Shaw and T Di Matteo ldquoCorrelation structure anddynamics in volatilemarketsrdquoNew Journal of Physics vol 12 no8 Article ID 085009 21 pages 2010

[47] T Qiu B Zheng and G Chen ldquoFinancial networks with staticand dynamic thresholdsrdquo New Journal of Physics vol 12 no 4Article ID 043057 16 pages 2010

[48] C-K Peng S V Buldyrev S Havlin M Simons H EStanley and A L Goldberger ldquoMosaic organization of DNAnucleotidesrdquo Physical Review E vol 49 no 2 pp 1685ndash16891994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


MSTs using a rolling window analysis as follows (i) Thechoice of parameters of the rolling window is dependent onthe scholarsrsquo preference that is the window width and steplength are selected arbitrarily [23] For example Onnela et al[10] set the size of window as four calendar years (approxi-mately 1000 trading days) while Song et al [22] fix the lengthof window as 1250 trading days (ii) In the MST methodthe cross-correlation coefficient between two financial unitsis usually measured by the Pearsonrsquos correlation coefficient(PCC) that is a linear correlation method Nevertheless thePCC cannot quantify the nonlinear relationship between twoheterogeneous entries [21] and ignores financial variablesrsquovolatile and nonnormality features [24] It should be notedthat some works attempt to apply other dynamic approachesin the MST For example Lyocsa et al [23] employ thedynamic conditional correlations (DCC) multivariate gen-eralized autoregressive conditional heteroscedasticity (MV-GARCH) approach to construct the MST for the US stockmarket Trancoso [25] uses the Baba-Engle-Kraft-Kroner(BEKK) model to develop the conditional correlation matrixand apply it in the dynamic network analysis HoweverLyocsa et al [23] and Trancoso [25] assume that the financialvariables obey the normal distribution and neglect theirnonnormality characteristics

On the other hand copula methods proposed by Sklar[26 27] are awidely useful tool for deriving joint distributionsgiven the marginal distributions especially when the vari-ables follow nonnormal distributions [28] Besides copulascan be employed to investigate the dependence beyond thelinear cross-correlation by the PCC and allow for a time-varying nonlinear analysis Moreover copulas can be sup-plied as a powerful analytical instrument tomeasure dynamicdependence structures between financial variables In a wordcopulas have better properties and more advantages (see[29 30]) than the traditional linear correlation methodsand attract many scholars to use them in various financeapplications [31ndash37] For instance Patton [35] constructstime-varying copula models to examine the dependencebetween two FX rates that is Deutsche MarkUS Dollar(DMUSD) and Japanese YenUS Dollar (JPYUSD) He alsomakes a comparison between copulas and GARCH modelsand finds that the former can better describe the dependenceor cross-correlation of FX rates than the latter Diks et al [36]use several copula models to analyze the dependence amongfive currencies (ie Canadian Dollar (CAD) Swiss Franc(CHF) European Euro (EUR) British Pound (GBP) andJPY) against USD and show that the Studentrsquos 119905-copulamodelis evidently superior to the Gaussian Gumbel and Claytoncopula models Dias and Embrechts [37] investigate thedependence between EURUSD and JPYUSD fromOctober1 2000 to October 1 2008 by employing the time-varyingcopula-GARCH models Their empirical results suggest thatthe dependence between the two currencies is dynamicand a time-varying copula approach with given correlationspecifications has better outcomes than some conventionaldynamic methods (eg BEKK)

In consideration of the above-mentioned motives basedon a time-varying copula approach and the MST methodwe aim to construct time-varying FX networks and analyze

their topological dynamics andmarket propertiesWe choose42 major currenciesrsquo daily FX rate series in the internationalFX market during the years 2005ndash2012 as the empirical dataIn empirical process we first use a time-varying copula tocalculate the dynamic cross-correlation coefficients 120588

119905among

different currencies More specifically we adopt an AR(119901)-GARCH(11)-119905 model to characterize the marginal distribu-tion of returns for FX rates and then estimate the dependenceparameters of the time-varying Studentrsquos 119905-copula modeland obtain time-varying cross-correlation coefficients Nexton the basis of time-varying cross-correlation coefficientswe construct time-varying cross-correlation matrices (CMs)C119905for 42 major currencies in the international FX market

Then we transform the time-varying CMs into time-varyingFX networks by the MST approach Finally we examinetopological dynamics and statistic features for time-varyingFX networks

The remainder of the paper is organized as followsSection 2 represents the empirical data A time-varyingcorrelation network-based approach by combining a time-varying copula model with the MST method is describedin Section 3 In Section 4 the time-varying FX networks areconstructed and main empirical results are showed Someconclusions are drawn in Section 5

2 Data Set

As for the empirical data set we choose the daily FXrates of 42 major currencies in the international FX marketfrom January 4 2005 to December 31 2012 FollowingJang et al [18] and Wang et al [21] we select the specialdrawing right (SDR) as the numeraire The 42 currenciesare from 7 different continents or regions Their detailedinformation is showed as follows (1) Africa EgyptianPound (EGP) and South African Rand (ZAR) (2) AsiaChinese Renminbi (CNY) Indian Rupee (INR) IndonesianRupiah (IDR) Japanese Yen (JPY)MalaysianRinggit (MYR)Pakistani Rupee (PKR) Philippines Peso (PHP) SingaporeDollar (SGD) South Korean Won (KRW) Taiwanese Dollar(TWD) Thai Baht (THB) and Vietnamese Dong (VND)(3) Europe British Pound (GBP) Czech Koruna (CZK)European Euro (EUR) Hungarian Forint (HUF) IcelandicKrona (ISK) Norwegian Krone (NOK) Polish Zloty (PLN)Romanian New Leo (RON) Russian Rubles (RUB) SwedishKrona (SEK) Swiss Franc (CHF) and Turkish New Lira(TRY) (4) Latin America Argentine Peso (ARS) BrazilianReal (BRL) Chilean Peso (CLP) Colombian Peso (COP)Peruvian New Sole (PEN) and Mexican Peso (MXN) (5)Middle East Bahrain Dinar (BHD) Israeli New Shekel(ILS) Jordanian Dinar (JOD) Kuwaiti Dinar (KWD) SaudiArabian Riyal (SAR) and United Arab Emirates Dirham(AED) (6) North America Canadian Dollar (CAD) and USDollar (USD) (7) Pacific Ocean Australian Dollar (AUD)andNewZealandDollar (NZD) All the FX rates are obtainedfrom the website of the Pacific Exchange Rate Service(httpfxsauderubccadatahtml) We define the return ofcurrency 119894 on day 119905 as 119903

119894119905= 100(ln119875

119894119905minus ln119875

119894119905minus1) where 119875

119894119905is

the daily FX rate of currency 119894 on day 119905 During the analyzedperiod each currencyrsquos returns have 2003 observations


3 Methodology



119903119894119905= 120583 +

119901

sum

119895=1

120601119895119903119894119905minus119895

+ 120576119894119905

120576119894119905= 120590119894119905119911119894119905 119911119894119905sim 119905 (])

1205902

119894119905= 120596119894+ 1205721198941205762

119894119905minus1+ 1205731198941205902

119894119905minus1

(1)


119894119905obeys a



restrictions 120596119894gt 0 120572

119894gt 0 120573

119894gt 0 and 120572

119894+ 120573119894lt 1 To be





119888119905(119906119905 V119905| 120579119905 119899)

=1


119905

Γ ((119899 + 2) 2) Γ (1198992)

[Γ ((119899 + 1) 2)]2


119899(119906119905)2

+ 119879minus1

119899(V119905)2

minus 2120579119905119879minus1

119899(119906119905) 119879minus1

119899(V119905))

times (119899 (1 minus 1205792

119905))minus1

]

minus(119899+2)2


119899(119906119905)2

119899)(1 +

119879minus1

119899(V119905)2

119899)]

(119899+1)2

(2)






120588119905= Λ(120574

0+ 1205741120588119905minus1

+ 1205742

1

10

10

sum

119895=1

119879minus1

119899(119906119905minus119895) 119879minus1

119899(V119905minus119895)) (3)




119896(119896 = 0 1 2) are

unknown parameters



120585119894= argmax

119879

sum

119905=1

ln119891119894119905(119911119894119905| Ω119905minus1 120585119894) (4)








120585119888= argmax

119879

sum

119905=1

ln 119888119905(119865119906119905(119911119906119905| Ω119905minus1120585119906)

119865V119905 (119911V119905 | Ω119905minus1120585V) 120585119888)

(5)





119905(119894 119895) = radic2(1 minus 120588

119905(119894 119895)) that falls in







APL119905=

2

119873 (119873 minus 1)

119873

sum

119894=1119895gt119894

119897119905

119894119895 (6)

where 119897119905




MOL119905(V119888) =

1

119873

119873

sum

119894=1

lev (V119905119894) (7)


119894)


119888 whose level




119875 (119896) sim 119896minus120572 (8)


4 Empirical Results


2005 2006 2007 2008 2009 2010 2011 201201

015

02

Time (year)

Mea

n

035

04

045

Stan

dard

dev

iatio

n

minus06

minus04

minus02

0

Skew

ness

18

2

22

24

Kurt

osis

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

Time (year)

Time (year)


2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

06

07

08

02

025

03

minus15

minus1

minus05

0

2

3

4

Mea

nSt

anda

rd d

evia

tion

Skew

ness

Kurt

osis



USD

EUR

CAD

GBP

JPY

CHF

ARS

AUD

BHD

BRL

CLP

CNY

COP CZK

EGP

HUF

ISK

INR

IDRILS

JODKWD

MYRMXN

NZD

NOK

PKR PEN

PHP

PLN

RON

RUB

SAR

SGDZAR

KRWSEK

TWDTHB

TRY

AED

VND




USD

EUR

CAD

GBP

JPYCHF

ARS

AUD

BHD

BRL

CLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN

NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND





USD

EUR

CADGBP

JPY

CHF

ASR

AUD

BHD

BRLCLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND












2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km


2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)





s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3 Methodology



119903119894119905= 120583 +

119901

sum

119895=1

120601119895119903119894119905minus119895

+ 120576119894119905

120576119894119905= 120590119894119905119911119894119905 119911119894119905sim 119905 (])

1205902

119894119905= 120596119894+ 1205721198941205762

119894119905minus1+ 1205731198941205902

119894119905minus1

(1)


119894119905obeys a



restrictions 120596119894gt 0 120572

119894gt 0 120573

119894gt 0 and 120572

119894+ 120573119894lt 1 To be





119888119905(119906119905 V119905| 120579119905 119899)

=1


119905

Γ ((119899 + 2) 2) Γ (1198992)

[Γ ((119899 + 1) 2)]2


119899(119906119905)2

+ 119879minus1

119899(V119905)2

minus 2120579119905119879minus1

119899(119906119905) 119879minus1

119899(V119905))

times (119899 (1 minus 1205792

119905))minus1

]

minus(119899+2)2


119899(119906119905)2

119899)(1 +

119879minus1

119899(V119905)2

119899)]

(119899+1)2

(2)






120588119905= Λ(120574

0+ 1205741120588119905minus1

+ 1205742

1

10

10

sum

119895=1

119879minus1

119899(119906119905minus119895) 119879minus1

119899(V119905minus119895)) (3)




119896(119896 = 0 1 2) are

unknown parameters



120585119894= argmax

119879

sum

119905=1

ln119891119894119905(119911119894119905| Ω119905minus1 120585119894) (4)








120585119888= argmax

119879

sum

119905=1

ln 119888119905(119865119906119905(119911119906119905| Ω119905minus1120585119906)

119865V119905 (119911V119905 | Ω119905minus1120585V) 120585119888)

(5)





119905(119894 119895) = radic2(1 minus 120588

119905(119894 119895)) that falls in







APL119905=

2

119873 (119873 minus 1)

119873

sum

119894=1119895gt119894

119897119905

119894119895 (6)

where 119897119905




MOL119905(V119888) =

1

119873

119873

sum

119894=1

lev (V119905119894) (7)


119894)


119888 whose level




119875 (119896) sim 119896minus120572 (8)


4 Empirical Results


2005 2006 2007 2008 2009 2010 2011 201201

015

02

Time (year)

Mea

n

035

04

045

Stan

dard

dev

iatio

n

minus06

minus04

minus02

0

Skew

ness

18

2

22

24

Kurt

osis

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

Time (year)

Time (year)


2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

06

07

08

02

025

03

minus15

minus1

minus05

0

2

3

4

Mea

nSt

anda

rd d

evia

tion

Skew

ness

Kurt

osis



USD

EUR

CAD

GBP

JPY

CHF

ARS

AUD

BHD

BRL

CLP

CNY

COP CZK

EGP

HUF

ISK

INR

IDRILS

JODKWD

MYRMXN

NZD

NOK

PKR PEN

PHP

PLN

RON

RUB

SAR

SGDZAR

KRWSEK

TWDTHB

TRY

AED

VND




USD

EUR

CAD

GBP

JPYCHF

ARS

AUD

BHD

BRL

CLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN

NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND





USD

EUR

CADGBP

JPY

CHF

ASR

AUD

BHD

BRLCLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND












2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km


2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)





s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












APL119905=

2

119873 (119873 minus 1)

119873

sum

119894=1119895gt119894

119897119905

119894119895 (6)

where 119897119905




MOL119905(V119888) =

1

119873

119873

sum

119894=1

lev (V119905119894) (7)


119894)


119888 whose level




119875 (119896) sim 119896minus120572 (8)


4 Empirical Results


2005 2006 2007 2008 2009 2010 2011 201201

015

02

Time (year)

Mea

n

035

04

045

Stan

dard

dev

iatio

n

minus06

minus04

minus02

0

Skew

ness

18

2

22

24

Kurt

osis

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

Time (year)

Time (year)


2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

06

07

08

02

025

03

minus15

minus1

minus05

0

2

3

4

Mea

nSt

anda

rd d

evia

tion

Skew

ness

Kurt

osis



USD

EUR

CAD

GBP

JPY

CHF

ARS

AUD

BHD

BRL

CLP

CNY

COP CZK

EGP

HUF

ISK

INR

IDRILS

JODKWD

MYRMXN

NZD

NOK

PKR PEN

PHP

PLN

RON

RUB

SAR

SGDZAR

KRWSEK

TWDTHB

TRY

AED

VND




USD

EUR

CAD

GBP

JPYCHF

ARS

AUD

BHD

BRL

CLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN

NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND





USD

EUR

CADGBP

JPY

CHF

ASR

AUD

BHD

BRLCLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND












2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km


2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)





s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










USD

EUR

CAD

GBP

JPY

CHF

ARS

AUD

BHD

BRL

CLP

CNY

COP CZK

EGP

HUF

ISK

INR

IDRILS

JODKWD

MYRMXN

NZD

NOK

PKR PEN

PHP

PLN

RON

RUB

SAR

SGDZAR

KRWSEK

TWDTHB

TRY

AED

VND




USD

EUR

CAD

GBP

JPYCHF

ARS

AUD

BHD

BRL

CLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN

NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND





USD

EUR

CADGBP

JPY

CHF

ASR

AUD

BHD

BRLCLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND












2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km


2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)





s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










USD

EUR

CADGBP

JPY

CHF

ASR

AUD

BHD

BRLCLP

CNY

COP

CZK

EGP

HUF

ISK

INR

IDR

ILS

JOD

KWD

MYR

MXN NZD

NOK

PKR

PEN

PHP

PLN

RON

RUB

SAR

SGD

ZAR

KRW

SEK

TWD

THB

TRY

AED

VND












2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km


2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)





s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

4

5

6

7

8

APL

3

4

5

6

7

MO

L

5

10

15

km


2

25

3

35

120572

0

02

04

06

08

1

P-v

alue

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)





s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










s

F(s)

H = 076 plusmn 002

R2= 097

101

102

10minus1

100

101

102

APL

(a)

H = 080 plusmn 002

R2= 097

s

101

102

10minus1

100

101

102

F(s)

MOL

(b)

H = 086 plusmn 002

R2= 098

s

101

102

100

101

102

F(s)

kmax

(c)

H = 073 plusmn 001

R2= 099

s

101

102

100

F(s)120572

(d)







SSR (119905) = 1

119873 minus 1|119864 (119905) cap 119864 (119905 + 1)| (9)


065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










065

07

075

08

085

09

095

1

Sing

le-s

tep

surv

ival

ratio

(SSR

)

2005 2006 2007 2008 2009 2010 2011 2012

Time (year)



MSR (1199050 120575) =

1

119873 minus 1

1003816100381610038161003816119864 (1199050) cap 119864 (119905

0+ 1) sdot sdot sdot 119864 (119905

0+ 120575 minus 1)

cap 119864 (1199050+ 120575)

1003816100381610038161003816

(10)






5 Conclusions


SSR

H = 068 plusmn 001

R2= 099

s

101

102

10minus1

10minus2

100

F(s)


101

100

102

103

10minus1

100

120575 (days)

Mul

tiste

p su

rviv

al ra

tio (M

SR)

2005

2006

2007

2008

2009

2010

2011

2012



0is the first








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article dynamics of foreign exchange networks: a time...

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