Research Article Measuring Contagion of Subprime Crisis

Research ArticleMeasuring Contagion of Subprime CrisisBased on MVMQ-CAViaR Method

Wuyi Ye Kebing Luo and Shaofu Du

School of Management University of Science and Technology of China Hefei 230026 China

Correspondence should be addressed to Shaofu Du dsoftustceducn

Received 13 March 2014 Accepted 8 May 2014 Published 9 June 2014

Academic Editor Fenghua Wen

Copyright copy 2014 Wuyi Ye et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The analysis of financial contagion is a topical issue in international finance and portfolio management In this paper we investigatewhether the global financial crisis originating from American subprime crisis spreads to China Japan UK France and GermanyFirstly multivariate conditional autoregressive value at risk (MV-CAViaR) models are applied to the whole sample to analyze thevariation of market risk among these countries By dividing the sampling period into three important subperiods (precrisis periodcrisis period and recovery period) we examine the changes of the dependence structure of risk during each period Comparingwith the situations in precrisis period if the estimated coefficients become significant or market risk increases during the crisis itimplies the existence of contagion from the angle of coefficient significance or riskThefindings show that the concerned coefficientsare significant or the market risks of the tested countries increase during the crisis except for China The results imply that thereis contagion from the US to all other countries except for China Furthermore the changes of the market risk are found to beconsistent with market events and media reports during that period

1 Introduction

An observation that stock markets behave very similarlyacross different continents and countries during the globalfinancial crisis of 2008-2009 casts serious doubts on thequestion whether financial contagion exists and if so to whatextent markets covary during the turmoil The financial con-tagion has vital impacts on the performance of internationalportfolios and risk management Investors are interested ininternational diversification of risks But if the financialmarkets are more closely linked during the crisis the oppor-tunities for international diversification are decreased Forregulators of financial markets it is particularly importantto comprehend such linkage increase among world financialmarkets As a result the topic issue has attracted a consider-able amount of interest among academic researchers

One may wonder about the existence of a contagionphenomenon across different financial markets worldwideFor this purpose it is necessary to define the concept of con-tagion despite the fact that several advanced results remainhard and complex to identify contagion In this paper wedraw lessons from the definition of contagion introduced

by Forbes and Rigobon [1] who defined it as a significantincrease in the market comovement after a shock happenedin a country It suggests that the financial contagion doesnot occur if two markets show a high degree of comovementduring both tranquil and turbulent phases In this paper wesay that there exists financial contagion only if one of thetwo following conditions is satisfied (i)The interested coeffi-cients are statistically significant during the crisis but remaininsignificant during the precrisis period (ii) The interestedcoefficients are significant during both precrisis period andthe crisis but compared with precrisis period themarket riskincreases during the crisis

The literature which focuses on changes of financial mar-ket dependence in a crisis period against a tranquil periodhas grown substantially in recent years and is generally relatedto the terminology ldquocontagionrdquo Baig and Goldfajn [2] andForbes and Rigobon [1] use a linear dependence measureBae et al [3] apply a multinomial logit model to estimatecoexceedances Baur [4] proposes an alternative quantileregression framework to decompose the dependence intothe degree and structure of dependence But the estimatedquantile is static and not dynamically varying with time at

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 386875 12 pageshttpdxdoiorg1011552014386875

2 Discrete Dynamics in Nature and Society

some given level To detect the dynamic characteristics offinancial contagion following from Engle and Manganelli[5] and White et al [6 7] we use multivariate conditionalautoregressive value at risk (MV-CAViaR) framework to esti-mate two dynamic conditional autoregressive quantile mod-els simultaneously We show that MV-CAViaR model pro-vides a flexible modeling and estimation method to identifythe financial contagion between two countries

We contribute to the existing literature in twomajor waysFirstly compared with single quantile regression modelwhich can only study the contagious relation from one coun-try to anotherMV-CAViaRmodel can study the interactionalcontagion between twomarkets in dynamic settings from theangle of risk with the help of VaR Moreover consideringthe information of two countries simultaneously we applyMV-CAViaR model to study financial contagion between theorigin of global financial crisis (the US market) and majormature equity markets namely Japan UK France Germanyand one of the biggest emergingmarkets (China) In this waywe can investigate not only the existence of contagion but alsothe changes of risk in two markets In order to capture thevariation of risk for the whole sampling period MV-CAViaRmodels are employed to the overall sample between the USand other countries We find that there is a high increase invalue at risk for all sampling countries during the crisisperiod

Secondly our research accounts for the financial con-tagion and changes of risk by using a comprehensive timeseries spanning three periods before during and after thecrisis In this way we increase the efficiency of our estimationand enable a comparison of different phases by dividingthe whole sample period into three periods (ie precrisiscrisis and recovery periods)Then we consider estimation ofthese models for three important subsamples just as Mollickand Assefa [8] did The first runs from January 2006 toDecember 2007 which is referred to as the precrisis periodThe second namely the crisis period starts at the beginningof 2008 right after NBER identified December of 2007 asthe start of the major recent financial crisis in the US (TheNational Bureau of Economic Research (NBER) is an Amer-ican private nonprofit research organization ldquocommitted toundertaking and disseminating unbiased economic researchamong public policymakers business professionals and theacademic communityrdquo The NBER is well known for provid-ing start and end dates for recessions in the United StatesNBER is the largest economics research organization in theUnited States Many of the American winners of the NobelMemorial Prize in Economic Sciences were NBER researchassociates Many of the chairmen of the Council of EconomicAdvisers have also been NBER research associates includingthe former NBER president and Harvard Professor MartinFeldstein) The US recession officially ended in June 2009although economic growth has since been sluggishThe thirdsubsample stretching from July 2009 to July 2013 refers tothe US economy entering a recovery phase and we call itrecovery period By comparing the coefficient significanceand the changes of risk during the three different phases wejudge whether there exists financial contagion from the USto the tested countries during the crisis and the economics

enter the phase of recovery The findings show that theAmerican subprime crisis spread to Japan UK France andGermany and increased their market risk during the crisisin contrast to precrisis period But we have not found statis-tically significant evidence that there is financial contagionfrom America to China During the recovery period thecomovement between the US and France exists but fromthe perspective of risk the market risks in two countriesbecome small in comparison with the risk in crisis periodFor UK there is no contagion phenomenon during therecovery period However there are risk spillovers from bothJapan andGermany to theUS due tomarket conditions of theday

The rest of the paper is organized as follows Section 2provides the literature review about financial contagion InSection 3 the multivariate multiquantiles CAViaR (MVMQ-CAViaR) framework and its estimation are detailed Thenin Section 4 we perform empirical analysis to examine thepossible financial contagion between the US and five testedcountries Finally Section 5 concludes

2 Literature Review

A rapid increase in global economic integration has acceler-ated financial integration spillovers and contagion in equitymarkets of different countries Current studies on financialcontagion offermanymethods tomeasure the propagation ofinternational shocks across countries Investigating the finan-cial transmission Kaminsky et al [9] categorize the theoriesinto three groups contagion caused by herding behavior (egBanerjee [10] Calvo and Mendoza [11]) contagion resultingfrom trade linkages (eg Lahiri and Vegh [12] Gerlach andSmets [13]) and contagion accelerated by financial links (egKodres and Pritsker [14] Fratzscher [15]) There are manyapproaches proposed to detect and identify the financialcontagion Next we review the main approaches aboutthe studies on financial contagion through financial linkages

Early work on detecting contagion is obtained mainlyby comparing the traditional constant Pearson correlationcoefficient between two countries in times of relative calmwith that in times of unusually high volatility (see forinstance Reinhart and Calvo [16] Bordo and Murshid [17]and Forbes and Rigobon [18]) However Corsetti et al[19] show how these results are flawed in the presence ofheteroscedasticity The findings show that an increase inthe variance of two variables will cause an increase in thecorrelation coefficient even if the propagation mechanismis unaltered Moreover Pearson correlation coefficient is astatic linear correlation measure and cannot measure thenonlinear relation Therefore an increase in the correlationcoefficient is not necessarily a proof of contagion Rigobon[20] introduces the DCC test which seems to provide a wayto test parameter stability in the models with simultaneousheteroscedasticity and endogeneity problems Recently thereare many literatures using DCC model to study financialcontagion (see eg Chiang et al [21] Naoui et al [22]Celik [23] and Dimitriou et al [24]) In addition thereare numerous traditional econometric models to investigatethe dependence or change of dependence These include

Discrete Dynamics in Nature and Society 3

the volatility spillovers methods based on ARCH andGARCH models [25ndash27] extreme value theory (EVT)method [28] probitlogit models [29 30] Markov switchingmodel [31ndash33] the factor model [34] the copula approach[35ndash39] andwavelet analysis [40 41] But none of these stud-ies analyze the financial contagion from the angle of risk withthe help of VaR directly

More recently quantile regression methods have gainedpopularity in economics and finance fields due to theirappealing features in describing market risk directly Thequantile regression model was firstly proposed by Koenkerand Bassett in 1978 Baur and Schulze [42] apply the quan-tile regression framework to examine the occurrences anddegrees of coexceedances concentrating on Hong KongThailand and Malaysia as the origin countries of the Asiancrisis and test the contagion effects on theUS LatinAmericaEurope and other Asian countries Chuang et al [43] andLee and Li [44] apply quantile regression tomodel the depen-dence of financial variables for example trading volume andreturn volatility Engle and Manganelli [5] develop condi-tional autoregressive value at risk (CAViaR) model a classof models suitable for estimating conditional quantile indynamic settings Engle and Manganelli apply the approachto estimate the market risk White et al [6] extend CAViaRmodel to permit joint modeling of multiple quantiles thatis Multi-Quantile (MQ) CAViaR They apply MQ-CAViaRmodel to estimate measures of conditional skewness andkurtosis defined in terms of conditional quantiles Further-more White et al [7] extend MQ-CAViaR model to amultivariate version of MQ-CAViaR model called MVMQ-CAViaRmodelThe proposed framework can simultaneouslyaccommodate models with multiple random variables mul-tiple confidence levels and multiple lags of the associatedquantilesThey estimated a simple version of themodel usingdifferent market returns data and then constructed impulseresponse functions to study how financial institution specificand system wide shocks are absorbed by the system In thispaper MV-CAViaR model is employed to research whetherthere was financial contagion from the US to the testedcountries or not from the angle of the coefficients significanceand the variation of risk before and after the Americansubprime crisis in dynamic settings Moreover by comparingthe situation in the recovery period with that in the crisis weexplain how the tested countries affect American economicrecovery So far we find there is no literature investigatingthe financial contagion with the model from the perspectiveof coefficient significance or risk

3 Methodology

In this section we firstly introduce MVMQ-CAViaR modelfollowing fromWhite et al [7] The framework of the modelcan be conveniently thought of as a vector autoregressiveextension to quantile models Secondly we present theasymmetric Laplace distribution (see Yu and Zhang [45] fordetails) which is the only distribution associated with quan-tile regression estimation Then we outline the quasimax-imum likelihood method for estimating MVMQ-CAViaRmodel parameters

31 MVMQ-CAViaR Model It is assumed that (1198841015840119905 1198831015840

119905) 119905 =

0 plusmn1 plusmn2 is stationary and ergodic stochastic process onthe complete probability space (ΩF 119875) where119884

119905is a finitely

119899 times 1 dimensioned vector and119883119905is a countably dimensioned

vector whose first element is oneLet F

119905minus1be the 120590-algebra generated by

(1198841015840

119905minus1 1198831015840

119905minus1) (1198841015840

119905minus2 1198831015840

119905minus2) that is

F119905minus1

equiv 120590 ((1198841015840

119905minus1 1198831015840

119905minus1) (1198841015840

119905minus2 1198831015840

119905minus2) ) (1)

For 119894 = 1 2 119899 define 119865119894119905(119910) = 119875(119884

119894119905lt 119910 | F

119905minus1) as the

cumulative distribution function (CDF) of 119884119894119905conditional on

F119905minus1

Let 120579119894119895belong to (0 1) for 119894 = 1 2 119899 119895 = 1 2 119901

119894

The 120579119894119895th quantile of119884

119905conditional onF

119905minus1 denoted by 119902

119894119895119905

is

119902119894119895119905

= inf 119910 119865119894119905(119910) ge 120579

119894119895| F119905minus1

(2)

For simplicity we set 119902119905= (1199021015840

1119905 1199021015840

2119905 119902

1015840

119899119905)1015840 with 119902

119894119905=

(1199021198941119905

1199021198942119905

119902119894119901119894119905

)1015840 For the given finite integers 119898 and

119896 there exist a stationary ergodic sequence of random 119896 times

1 vectors Ψ119905 119905 = 1 2 119879 with Ψ

119905measurable F

119905minus1

and real vectors 120573 = 119888(1205731198941198951

1205731198941198952

120573119894119895119896

)1015840 and 120574

119894119895120591=

(1205741198941198951205911

1205741198941198951205912

120574119894119895120591119899

) where each 120574119894119895120591119896

is 119901119894times 1 vector

such that for 119894 = 1 2 119899 119895 = 1 2 119901119894and all 119905

119902119894119895119905

= Ψ1015840

119905120573119894119895+

119898

sum120591=1

1199021015840

119905minus120591120574119894119895120591

(3)

The structure of (3) is a multivariate multiquantiles con-ditional autoregressive value at risk (MVMQ-CAViaR)modelintroduced by White et al [7] itself a multivariate version ofthe MQ-CAViaR process of White et al [6] which is a mul-tiquantiles version of the CAViaR process proposed by Engleand Manganelli [5] Under suitable restrictions on the 120574

119894119895120591

we get the following three situations as special cases (1) uni-variate MQ-CAViaR process of each element of 119884

119905 (2) single

quantile CAViaR process of each element of 119884119905 and (3)mul-

tivariate CAViaR process in which a single quantile of eachelement of 119884

119905is dynamically related to single quantiles of the

other elements of 119884119905or lags of them

It is worthwhile to note that for MVMQ-CAViaR thenumber of relevant lags can differ across the elements of 119884

119905

and conditional quantiles In fact it is possible that for given119894 and 119895 elements of 120574

119894119895120591may be zero for the values of 120591 greater

than a certain integer For the simplicity of notation 119898 isnot denoted by 119898(119894 119895) In addition the finitely dimensionedrandom vectors Ψ

119905may include 119884

119905or the lagged values of

119884119905 as well as measurable functions of 119883

119905and lagged 119883

119905 In

special cases Ψ119905may contain Stinchcombe and White [46]

GCR transformations as discussed in White [47]For the analytical convenience in terms of a particular

quantile for example 120579119894119895 the coefficients to be estimated are

120573119894119895

and 120574119894119895

= (1205741015840

1198941198951 1205741015840

1198941198952 120574

1015840

119894119895119898) Let 120572

119894119895= (1205731015840

119894119895 1205741015840

119894119895) and

write 120572 = (1205721015840

11 1205721015840

12 120572

1015840

1119901119894 120572

1015840

1198991 1205721015840

1198992 120572

1015840

119899119901119899) an 119897 times 1

vector where 119897 = sum119899

119894=1119901119894(119896 + 119899119901

119894119898) 120572 is called the MVMQ-

CAViaR coefficient vector


To illustrate the usefulness of the models outlined abovewe give an example specifying the model as follows

1199021119905

= 1198881+ 11988611

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

+ 119887121199022119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988622

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887211199021119905minus1

+ 119887221199022119905minus1

(4)

Since 119875 (119884119894119905

le minusVaR119894119905

| F119905minus1

) = 120579 minusVaR119894119905is the 120579th

quantile of 119884119894119905conditional onF

119905minus1 Then we get

VaR1119905

= 119888lowast

1+ 119886lowast

11

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

12

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988711VaR1119905minus1

+ 11988712VaR2119905minus1

VaR2119905

= 119888lowast

2+ 119886lowast

21

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

22

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988721VaR1119905minus1

+ 11988722VaR2119905minus1

(5)

where 119888lowast119894= minus119888119894 120572lowast119894119895= minus120572119894119895 119894 = 1 2 119895 = 1 2

Hence the financial contagion can be detected from theperspective of the risk transmission The financial contagionbetween the two financial markets is measured by the off-diagonal coefficients 119887

12and 11988721 and the hypothesis of no

contagion can be tested by testing 1198670 11988712

= 11988721

= 0 Thedirection of the contagion can be captured by inspecting thesetwo coefficients For example if 119887

12= 0 and 119887

21= 0 then the

direction of financial contagion is from country 1 to country2 not the other way around At the same time the degreeof the contagion can be measured by the magnitude of 119887

12

and 11988721 The larger 119887

12or 11988721are the more serious contagion

between two markets is In a similar way the financial conta-gion can also be captured from the point of risk spilloversWe can investigate the existence of financial contagion bytesting the significance of coefficients 119867

0 119886lowast

12= 119886lowast

21=

0 The directions of contagion can be determined from thehypothesis test for example 119867

0 119886lowast

12= 0 and 119886

lowast

21= 0 which

means there is contagion from country 1 to country 2 and nocontagion from country 2 to county 1 Similarly the depen-dence structure can be analyzed from the sign andmagnitudeof the coefficients 119886

lowast

12and 119886

lowast

21 The larger coefficients 119886

lowast

12

and 119886lowast

21are the larger spillover risks are which implies that

the smaller coefficients 11988612

and 11988621

are the larger spilloverrisks are

32 The QMLE Estimate of MV-CAViaR Model A possibleparametric link between the minimization of the sum ofabsolute deviate for the estimation of quantile regression andthe maximum theory is given by the asymmetric Laplacedistribution The asymmetric Laplace distribution presentedhere is only a kind of the asymmetric Laplace distributions Itis useful for fitting quantile or quantile regression as well asfor data analysis in general see Yu and Moyeed [48] Yu andZhang [45] for details

We say that a random variable 119883 is distributed as ALDwith parameters 120583 120590 and 119901 denoted by119883 sim ALD(120583 120590 120591) ifit has the following probability density function (PDF)

119891 (119909 120583 120590 119901) =119901 (1 minus 119901)

120590exp minus

119909 minus 120583

120590[119901 minus 119868

(119909le120583)] (6)

where minusinfin lt 120583 lt +infin is the location parameter 0 lt 119901 lt 1

is the skew parameter 120590 gt 0 is the scale parameter and 119868(sdot)

is the indication function which takes the value one if theargument is true and zero otherwise

Following from Koenker and Machado [49] we assumethat the error terms in quantile regression are distributed asasymmetric Laplace distribution in this paper that is 119884

119894119905minus

119902119894119895119905

sim ALD(0 120590 120579119894119895) Then we can estimate the unknown

parameters by the method of quasimaximum likelihoodThelikelihood function for 119879 independent observations is

119871 (120572 1198841 1198842) prop 120590

minus1 exp

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

120590

(7)

where 120588120579

= 119890Ψ120579(119890) is the standard check function (see

Koenker and Basset [50]) defined by the usual quantilestep function Ψ

120579(119890) = 120579 minus 119868

(119890le0) Because 119884

119894119905minus 119902119894119895119905

arenot necessarily distributed as this distribution we call itquasimaximum likelihood estimator rather than maximumlikelihood estimator as described in White et al [7] If weconsider120590 as a nuisance parameter the quasimaximum likeli-hood estimator is equivalent to the solution to the followingoptimization problem

= argmin120572isinR

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

(8)

Thus ALD proves to be useful as unifying bridge betweenthe likelihood and inference for QR estimation For exampleKoenker and Machado [49] introduced a goodness-of-fitprocess for QR and related inference process and they alsoconsidered the likelihood ratio statistic under the parametricassumption of a Laplacean distribution for the error term ForBayesian approach see Yu and Moyeed [48] among othersThe consistency and asymptotic normality for the estimatedcoefficients are derived and detailed in White et al [7]Following this we can get the significance test of coefficients

4 Data and Empirical Analysis

41 Data Our empirical study of 2008-2009 financial con-tagion focuses on equity market indices of six countries(SampP 500 (US) CSI300 (China) Nikkei 225 (Japan) FTSE-100 (UK) CAC-40 (France) and DAX (Germany)) fromJanuary 1 2006 to July 25 2013 which include the episode ofthe financial crisis of 2008-2009 We call China Japan UKFrance and Germany tested countries because their marketrisk is subject to the impact of the SampP 500 index return inUSThedaily datawere downloaded fromdata stream Equityindex risk during the subprime crisis is analyzed based on themethodology outlined above Daily returns are calculated as

119903119905= 100 times (ln (119901

119905) minus ln (119901

119905minus1)) (9)

In addition because the US market opens later than theother countries in order to analyze the impact of SampP 500index returns on the tested markets we use the returns onday 119905 minus 1 for the SampP 500 index and the returns on day 119905 forthe tested markets when calculating Kendall tau and Pearsoncorrelation coefficients Meanwhile it should be noted that


Table 1 Summary statistics of log daily index returns This table provides summary statistics of log daily index returns for six marketsincluding the US (SampP 500 index) China (CSI 300 index) Japan (Nikkei 225 index) UK (FTSE-100 index) France (CAC-40 index) andGermany (DAX index) from June 1 2006 to July 25 2013 The Kendall tau correlation coefficient (120588

120591) and Pearson correlation coefficient (120588)

between the US and tested countries are also reported

US China Japan UK France GermanyMean 001 00345 001 00129 minus9119890 minus 04 00288Median 00808 01176 0048 00469 00297 01026Min minus94695 minus96952 minus12111 minus92656 minus94715 minus7335Max 109572 89309 132346 93843 105946 107975Stdev 13736 18993 16547 12979 15349 14629Skewness minus01751 minus03594 minus0581 minus0098 01327 01375Kurtosis 95992 27969 83085 76485 61511 63971KS (119875 value) 0 0 0 0 0 0120588120591

00873 03317 01206 01086 01125120588 01476 05223 02542 02354 02013Note the overall sample period is from 12006 to 72013 Standard errors are in parentheses

if one of the markets has a holiday that is when it is closedfor any trading day other than the weekend we delete theobservations 119905 minus 1 for the other markets Hence each of thetwomarkets between Americanmarket andmarkets of testedcountries has exactly the same trading days

Summary statistics of the index returns are reportedin Table 1 We find that all returns except for France andGermany are negatively skewed and the excess kurtosis ofall returns is larger than zero In terms of the Kolmogorov-Smirnov test all the distribution of the series of six returnsdeviates from normal distribution Returns from all sixindices exhibit the characteristics of high kurtosis and fat tailTable 1 also reports theKendall correlation coefficient120588

120591 and

Pearson correlation coefficient 120588 between return of the SampP500 index and lagged returns from the tested stock marketsWe observed a relatively loose Kendall tau correlation coef-ficient between the US and China stock markets (00873)but tighter correlation between the US and Japan (03317)UK (01206) France (01086) and Germany (01125) Pearsoncorrelation coefficient presents the same pattern as Kendalltau correlation coefficient Perhaps because stock market inChina is weak from efficiency and highly regulated changesin market value of listed companies are difficult to react tothose in their intrinsic value quickly and accurately

Figure 1 illustrates the evolution of stock market returnsover timeThe figure indicates that all markets have trembledsince 2008 Stockmarkets show volatility clustering revealingthe presence of heteroscedasticity This characteristic sup-ports the usefulness of dynamic conditional autoregressivemodel to analyze dynamic character in stock returns Mean-while due to the effect of European debt crisis there arehigh volatilities around 2011 However there is no obviousevidence that high volatilities appear in Chinese and Japanesestock returns in Figure 1 around the period It may be dueto the little impact of the European debt crisis on China andJapan

42 Model Specification To provide evidence of the volatilityclustering phenomenon in financial time series and financial

contagion from the American subprime crisis to marketindices of the tested countries the MV-CAViaR models areestimated as a function of one lagged quantiles and one laggedreturns of other countries Next we estimate the followingsystem of equations for the conditional quantiles of the USand one of the tested countries (ie China JapanUK Franceand Germany) Consider

1199021119905

= 1198881+ 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119887221199022119905minus1

(10)

where 1198841119905minus1

1198842119905minus1

are separately denoted as the daily returnon one of the indices of the tested countries and that on theindex of America at time 119905minus1 119902

1119905 1199022119905represent 5 quantiles

of one of the returns on the indices of tested countriesand American stock index return at time 119905 respectively Inthe framework of the specified model (10) 119887

11 11988722

can beused to test whether there are the characteristics of volatilityclustering in the responding stock markets If 119887

11 11988722are sig-

nificantly different from zero at some given significance levelit indicates that there is volatility clustering phenomenonin the responding stock markets Similar to the discussionabove the coefficients 119886

12 11988621can be used to capture financial

contagion If 11988612is significantly different from zero it shows

that there is financial contagion from America to the testedcountries Similarly if 119886

21is significantly different from zero

it indicates that there is risk spillover from equity markets ofthe tested countries to American equity marketThus we call11988612 11988621contagious coefficients Meanwhile contagious coef-

ficients can be used to analyze the variation of marketrisk From the model (10) we can see that the smaller thecontagious coefficients are the larger the risks in stockmarketof the infected countries are Furthermore the negative con-tagious coefficients imply that themarket risks in the infectedcountries increase

In addition estimating the systemof (10) is not trivialWeperform the computations in a step-wise fashion as follows


2006

-01-

03

2007

-01-

04

2008

-01-

04

2009

-01-

02

2009

-12-

30

2010

-12-

26

2011

-12-

22

2012

-12-

21

SampP500

minus10

minus5

0

5

10

(a)

0

5

2006

-01-

04

2007

-01-

17

2008

-01-

25

2009

-02-

10

2010

-02-

10

2011

-03-

01

2012

-03-

08

2013

-03-

20

CSI300

minus10

minus5

(b)

2006

-01-

04

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-02-

01

2011

-02-

08

2012

-02-

15

2013

-02-

20

Nikkei225

0

5

minus10

10

minus5

(c)

2006

-01-

03

2006

-12-

28

2007

-12-

21

2008

-12-

17

2009

-12-

14

2010

-12-

09

2011

-12-

07

2012

-12-

04

FTSE-100

0

5

minus10

10

minus5

(d)

2006

-01-

02

2006

-12-

21

2007

-12-

14

2008

-12-

08

2009

-12-

01

2010

-11-

22

2011

-11-

09

2012

-10-

30

CAC-40

0

5

minus10

10

minus5

(e)

2006

-01-

02

2006

-12-

21

2007

-12-

17

2008

-12-

11

2009

-12-

08

2010

-12-

01

2011

-11-

22

2012

-11-

12

DAX

0

5

10

minus5

(f)

Figure 1 Daily returns of SampP500 (US) CSI300 (China) Nikkei225 (Japan) FTSE-100 (UK) CAC-40 (France) and DAX (Germany) Notesdaily data from January 2006 to July 2013


Table 2 Estimates of MV-CAViaR model for the overall sample

Parameters 1198881

11988612

11988711

1198882

11988621

11988722

USChina minus00350 minus00157 09852 00034 minus00116 09942(00107)lowastlowastlowast (00047)lowastlowastlowast (00041)lowastlowastlowast (00015)lowastlowast (00016) (00007)lowastlowastlowast

USJapan minus07877 minus10917 03103 minus00039 00089 10045(01674)lowastlowastlowast (00853)lowastlowastlowast (00679)lowastlowastlowast (00013)lowastlowastlowast (00025)lowastlowastlowast (00009)lowastlowastlowast

USAustralia minus04195 minus06613 04356 minus07129 minus12766 02390(00740)lowastlowastlowast (00769)lowastlowastlowast (00575)lowastlowastlowast (01552)lowastlowast (00862)lowastlowastlowast (00758)lowastlowastlowast

USUK minus16514 minus03998 00293 00357 minus03188 08806(03451)lowastlowastlowast (01074)lowastlowastlowast (01702)lowastlowastlowast (00173)lowastlowast (00431)lowastlowastlowast (00140)lowastlowastlowast

USFrance minus02077 minus03289 079422 minus16113 minus05503 00132(00466)lowastlowastlowast (00626)lowastlowastlowast (00368)lowastlowastlowast (02641)lowastlowastlowast (01143)lowastlowastlowast (01151)

USGermany minus01907 minus02937 08031 minus00022 minus02847 08669(00476)lowastlowastlowast (00608)lowastlowastlowast (00384)lowastlowastlowast (00223) (00574)lowastlowastlowast (00224)lowastlowastlowast

Note the overall sample period is from 12006 to 72013 Standard errors are in parentheseslowastlowastSignificant at 5 levellowastlowastlowastSignificant at 1 level

Step 1 We use the QMLE method to estimate single CAViaRmodels 119902

1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1

respectively with the initial value of 119888

119894(119894 = 1 2) equal to

the responding 120579th sample quantiles of 119884119894(119894 = 1 2) We

initialize the remaining parameters at zero

Step 2 Weuse the parameter estimates of step 1 as the startingvalues for the optimization of the CAViaR models 119902

1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1

settingto zero the remaining parameters

Step 3 We use the estimates from step 2 as the starting valuesfor the full MV-CAViaR model (10) optimization whichcontains two financial institutions of interest simultaneously

43 Whole Sample Analysis To capture the changes of mar-ket risk throughout the sampling period we estimate the sys-temof (10) for the entire sampling period Table 2 displays theresults of the model specified in (10) for tested countries andAmerica

Table 2 shows that the coefficients 11988711and 11988722 are positive

in each case and almost significant It indicates that risk ofstock markets is positive autocorrelation In other wordsthere is volatility clustering phenomenon in stock marketsThis is consistent with the conclusions of the existing researchliteratures for example Teyssiere and Kirman [51] Niu andWang [52] and so on Meanwhile the contagious coefficientssignificantly deviate from zero in each case and are negativeexcept for Japan It implies that there may be financial conta-gion between the US and the tested countries and one coun-tryrsquos risk increases due to another country except for JapanIt is interesting to find that the magnitude of the contagiouscoefficients is the smallest for China among all the examinedcountries The parameter estimates of 119886

12and 119886

21are only

minus00157 and minus00116 respectively This means that the degreeof risk spillovers for China is smallest and America has alighter effect on China than the other investigated countriesIt is consistent with the results in Table 1 that there is the

smallest Kendall tau correlation coefficient between Chinaand America

Figure 2 illustrates the evolution of the estimated VaR atthe level of 5 under the structure of the system of (10) fromJanuary 2006 to July 2013The figure exhibits that all marketsare subjected to high risks since 2008 Stockmarkets show thecharacteristics of high risk during the period of the financialcrisis of 2008-2009

However it should be noted that estimated samplingperiod includes the crisis period and noncrisis period Thuswe cannot distinguish whether the contagion is caused by thefinancial crisis originating fromAmerican subprime crisis Inorder to overcome the defect we consider the estimations forthree important subsamples in the structure of model (10)The first runs from January 2006 to December 2007 whichwe refer to as the precrisis period The second starts at thebeginning of 2008 right after NBER identified December of2007 as the start of the major recent financial crisis in the USThe US recession officially ended in June 2009 although eco-nomic growth has since been sluggish The third subsamplefrom July 2009 to July 2013 refers to the US economy enteringa recovery phase

44 Financial Contagion Analysis As discussed in theprevious section there exists financial contagion from theUS stock market to the foreign stock markets if one ofthe two following conditions is satisfied (i) The contagiouscoefficient 119886

12 is insignificant during the precrisis period

but it is significant during the crisis (ii) If the contagiouscoefficient 119886

12 is significant during both the precrisis and

crisis period but compared to the contagious coefficient 11988612

during the precrisis period the estimated coefficient 11988612

during the crisis becomes smaller which implies that there isrisk increase due to America during the crisis From the twoaspects we analyze whether there is financial contagion fromone country to another The detailed results are shown inTables 3ndash7


5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)

Figure 2 Estimated VaR at the level of 5 by the systems of (3) Notes the overall sample period is from 12006 to 72013


Table 3 Financial contagion analysis between the US and China

Parameters 1198881

11988612

11988711

1198882

11988621

11988722

Precrisis minus00298 00580 10016 00166 minus00015 10118(00078)lowastlowastlowast (00193)lowastlowastlowast (00019)lowastlowastlowast (00046)lowastlowastlowast (00032) (00053)

Crisis minus05988 00294 08932 01006 minus00537 09954(14327) (00719) (02574)lowastlowastlowast (00425)lowastlowastlowast (00151)lowastlowastlowast (00032)lowastlowastlowast

Recovery minus48570 00554 minus08753 minus30359 minus00569 minus07033(01735)lowastlowastlowast (00671) (00360)lowastlowastlowast (07243)lowastlowastlowast (00600) (03949)lowast

Notes standard errors are in parentheses The overall sample period is portioned into three subsamples a precrisis period spanning from 12006 to 122007 acrisis period spanning from 12008 to 62009 and a recovery period spanning from 72009 to 72013lowastSignificant at 10 levellowastlowastlowastSignificant at 1 level

Table 4 Financial contagion analysis between the US and Japan

Parameters 1198881

11988612

11988711

1198882

11988621

11988722

Precrisis minus35184 minus00866 minus07231 minus01383 minus00239 08897(12479)lowastlowastlowast (01806) (06273) (04116) (00557) (03054)lowastlowastlowast

Crisis minus09344 minus10917 03553 minus00333 00011 09949(02990) (00956)lowastlowastlowast (00872)lowastlowastlowast (00059)lowastlowastlowast (00094) (00042)lowastlowastlowast

Recovery minus27990 03103 minus03753 00161 minus00146 10001(13049)lowastlowast (01331) (06186) (00047)lowastlowastlowast (00049)lowastlowastlowast (00010)lowastlowastlowast

Notes standard errors are in parentheses The overall sample period is portioned into three subsamples a precrisis period spanning from 12006 to 122007 acrisis period spanning from 12008 to 62009 and a recovery period spanning from 72009 to 72013lowastlowastSignificant at 5 levellowastlowastlowastSignificant at 1 level

Table 5 Financial contagion analysis between the US and UK

Parameters 1198881

11988612

11988711

1198882

11988621

11988722

Precrisis 00270 00292 10255 00171 00110 10208(00208) (00366) (00227)lowastlowastlowast (00070)lowastlowast (00169) (00107)lowastlowastlowast

Crisis 00148 00516 10291 00254 minus00516 09888(00089)lowast (00128)lowastlowastlowast (00038)lowastlowastlowast (00109)lowastlowast (00171)lowastlowastlowast (00046)lowastlowastlowast

Recovery minus34809 minus00129 minus09977 minus34290 00071 minus08669(00969)lowastlowastlowast (00150) (00035)lowastlowastlowast (07464)lowastlowastlowast (00550) (04042)lowastlowastlowast

Notes standard errors are in parentheses The overall sample period is portioned into three subsamples a precrisis period spanning from 12006 to 122007 acrisis period spanning from 12008 to 62009 and a recovery period spanning from 72009 to 72013lowastSignificant at 10 levellowastlowastSignificant at 5 levellowastlowastlowastSignificant at 1 level

Table 6 Financial contagion analysis between US and France

Parameters 1198881

11988612

11988711

1198882

11988621

11988722

Precrisis minus30343 00071 minus09769 00766 00000 10678(02408)lowastlowastlowast (00399) (00859)lowastlowastlowast (00000)lowastlowastlowast (00000) (00000)lowastlowastlowast

Crisis minus02327 minus04096 07474 00254 minus00664 09751(00782)lowastlowastlowast (00954)lowastlowastlowast (00500)lowastlowastlowast (00122) (00196)lowastlowastlowast (00065)lowastlowastlowast

Recovery minus01490 minus02760 08462 minus34290 02311 minus05932(00693)lowastlowast (01155)lowastlowast (00583)lowastlowastlowast (03309)lowastlowastlowast (00990)lowastlowast (01592)lowastlowastlowast



Table 7 Financial contagion analysis between the US and Germany

Parameters 1198881

11988612

11988711

1198882

11988621

11988722

Precrisis minus32792 00204 minus09912 00126 00019 10125(01669)lowastlowastlowast (00152 (00111)lowastlowastlowast (00070)lowast (00077) (00026)lowastlowastlowast

Crisis minus00550 minus01400 09218 minus00044 00863 10386(00336)lowast (00477)lowastlowastlowast (00266)lowastlowastlowast (00162) (00256)lowastlowastlowast (00077)lowastlowastlowast

Recovery minus43511 00133 minus08860 minus14618 minus04146 00097(13772)lowastlowastlowast (00846) (06013) (04062)lowastlowastlowast (01496)lowastlowastlowast (02052)


In Table 3 we report the results between the US andChina for the three subsample periods (precrisis periodcrisis period and recovery period) The result indicates thatcompared to the precrisis period there is no evidence thatthere is contagion from America to China This is consistentwith the fact that the impact of the subprime crisis onChina isrelatively small In reality the Chinese government hadmain-tained a relatively calm attitude and did not show panic thatappeared in other developed countries throughout the pro-cess which to a large extent relieved the negative factorsbrought by the stagnant international financial market Butto some extent Chinese market has a relationship withAmerican market due to the economic globalization Thereasonwhy there is only one significant contagious coefficient11988621

during the crisis is that the linkage between the twomarkets increased during the crisis Moreover it should benoted that 119886

21is negative This is consistent with the fact that

the market risk in America increased during the crisis As isknown to all the risk increase in America is caused by its ownmarket conditions of the day During the recovery period thecontagious coefficients are insignificant which suggests thatthere is weaker link and no significant effect during theperiod

Table 4 summarizes the financial contagion analysisbetween American and Japanese stock markets We find thatduring the precrisis period the contagious coefficients 119886

12

and 11988621

are not statistically significant but during the crisis11988612

is significant and 11988621

is insignificant This indicates thatthere is financial contagion from American equity marketto Japanese equity market but no contagion from Japan toAmerica It should be noted that during the recovery period11988621is the only significant contagious coefficientThis suggests

that there may be contagion effect from Japanese equity mar-ket to American equity market during the recovery periodIn fact America is the second largest country for Japan andJapanese economy is affected by American economy Afterthe crisis there is a strong economic resurgence in the USHowever in recent years because the demand inside Japanis low and confused economy is long-term backwater As aresult American economy recovery is cumbered by Japanesesluggish economyThe significance of 119886

21cannot be explained

from the perspective of financial contagion It should beexplained from the standpoint that Japan has a negative effect

on American economic recovery and increases the recoveryrisk of American economy

The results in Table 5 reveal the contagion effects betweenthe US and UK The contagion coefficients are statisticallysignificant at the level of 1 only during the crisis Itmeans that there is strong financial contagion from the USto UK during the financial crisis It is obvious that thecomovements between the US and UK become stronger incontrast with noncrisis period It is possible that there arestrong historical ties between the two countries Britishcorporations are one of the largest foreign investors in theUSThe tighter relations between the two countries may lead tothe significant contagion coefficient 119886

21during the period of

crisis At last the insignificance of the contagion coefficientssuggests that the relations between the countries return to thestate of the precrisis

Table 6 shows the contagion test results between theUS and France There is strong evidence of spillover risksbetween the US and French markets The contagious coeffi-cients are statistically significant at the level of 1 during thecrisis period Moreover they are negative and the contagiouscoefficient 119886

12from America to France is minus04096 and

contagious coefficient 11988621fromFrance toAmerica isminus00664

Because the financial crisis of 2008-2009 is caused by Ameri-can subprime crisis we only say that there is financial conta-gion from America to France In other words the existenceof the financial contagion from the US to France leads toa stronger linkage between the two countries In additionfrom the magnitude of contagion coefficients America hasa stronger effect on France which is consistent with the factthat the financial crisis is originating fromAmerica At last incomparison with the crisis period the contagion coefficientsare significant at level of 5 but larger It means that thecomovements between the two countries become weaker andrisk spillovers become smaller during the recovery periodThe global economy steps in a recovery phase

In Table 7 we present estimatedmodel parameters for theUS andGermany Estimates of the contagious coefficients 119886

12

and 11988621 show that there is evidence of financial contagion

between the US and Germany Furthermore in terms of thesign of the contagious coefficients we see clearly thatAmericaincreases stock market risk in Germany and Germanydecreases the risk in American stock market Similar to


the discussion above we say that the direction of contagion isfromAmerica to Germany and there exists a stronger linkagebetween the US and Germany During the recovery periodonly one contagion coefficient 119886

21 is significant Further-

more the sign of it is negative which implies that Germanyincreases the market risk in America during the period Thatis to say Germany becomes a burden on the recovery ofAmerican economy It is possible that European debt crisishas more serious effect on German economy than on Amer-ican economy and the recovery speed of American economyis faster than that of German economy

5 Conclusions

In this paper we examine whether the subprime crisis origi-nating from theUS led to risk spillovers in financial indices inmajormaturemarkets (JapanUK FranceGermany) and oneof the largest emerging markets (China) We apply the MV-CAViaR model to the whole sample period from 1 January2006 to 25 July 2013 The findings show that there is a signif-icant increase in market risk during the crisis To investigatethe changes of dependence structure during the crisis andnoncrisis period we divide the sampling period into threeimportant subperiods precrisis period crisis period andrecovery period according to the announcements by NBERand government

The piecewise estimated results show that Americansubprime crisis spilled over to Japan UK France andGermany and increased their market risks during the crisisin comparison with the turmoil period But we have notfound statistically significant evidence showing that thereis financial contagion from America to China During therecovery period the strong linkages between the US andFrance still exist but there is a simultaneous decrease in themarket risks of both countries As for UK there is no conta-gion during the recovery period But it should be noted thatthere are spillover risks from Japan and Germany to the USdue to market conditions of the day

The findings of our paper contribute to the ongoingdebate on the detection modeling and especially risk anal-ysis of financial contagion It is of great significance forinvestors and portfolio managers to determine the existenceof financial contagion as well as risk variation before and afterfinancial crisis In near future it would be interesting researchareas to study how the methodology could be applied to assetand risk management despite the existing contagion in someregions

Conflict of Interests

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

Acknowledgments

This research is supported by National Natural Science Foun-dation of China (nos 71371007 and 71271199) Program for

New Century Excellent Talents in University (no NCET-13-0538) and the Fundamental Research Funds for the CentralUniversities of China (no WK2040160008)

References

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[2] T Baig and I Goldfajn ldquoFinancial market contagion in theAsian crisisrdquo IMF Staff Papers vol 46 no 2 pp 167ndash195 1999

[3] K-H Bae G A Karolyi and R M Stulz ldquoA new approach tomeasuring financial contagionrdquoReview of Financial Studies vol16 no 3 pp 717ndash763 2003

[4] D G Baur ldquoThe structure and degree of dependence a quantileregression approachrdquo Journal of Banking and Finance vol 37no 3 pp 786ndash798 2013

[5] R F Engle and SManganelli ldquoCAViaR conditional autoregres-sive value at risk by regression quantilesrdquo Journal of Business ampEconomic Statistics vol 22 no 4 pp 367ndash381 2004

[6] H White T H Kim and S Manganelli ldquoModeling autore-gressive conditional skewness and kurtosis with multi-quantileCAViaRrdquo European Central Bank Working Paper Series 09572008

[7] H White T H Kim and S Manganelli ldquoVAR for VaR mea-suring systemic risk using multivariate regression quantilesrdquoMPRA Paper 35372 2010

[8] A VMollick and T A Assefa ldquoUS stock returns and oil pricesthe tale from daily data and the 2008-2009 financial crisisrdquoEnergy Economics vol 36 pp 1ndash18 2013

[9] G L Kaminsky C M Reinhart and C A Vegh ldquoThe unholytrinity of financial contagionrdquo Journal of Economic Perspectivesvol 17 no 4 pp 51ndash74 2003

[10] A V Banerjee ldquoA simplemodel of herd behaviorrdquoTheQuarterlyJournal of Economics vol 107 no 3 pp 797ndash817 1992

[11] G A Calvo and E G Mendoza ldquoMacroeconomics of emerg-ing markets capital-markets crises and economic collapse inemerging markets an informational-frictions approachrdquo TheAmerican Economic Review vol 90 no 2 pp 59ndash64 2000

[12] A Lahiri and C A Vegh ldquoDelaying the inevitable interestrate defense and balance of payments crisesrdquo Journal of PoliticalEconomy vol 111 no 2 pp 404ndash424 2003

[13] S Gerlach and S Frank ldquoContagious speculative attacksrdquo CEPRDiscussion Paper 1055 1996

[14] L E Kodres and M Pritsker ldquoA rational expectations model offinancial contagionrdquo Journal of Finance vol 57 no 2 pp 769ndash799 2002

[15] M Fratzscher ldquoOn currency crises and contagionrdquo Interna-tional Journal of Finance and Economics vol 8 no 2 pp 109ndash129 2003

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testrdquo Yale University Economic Growth Center working paper822 2005

[20] R Rigobon ldquoOn the measurement of the international prop-agation of shocks is the transmission stablerdquo Journal ofInternational Economics vol 61 no 2 pp 261ndash283 2003

[21] T C Chiang B N Jeon and H Li ldquoDynamic correlation anal-ysis of financial contagion evidence from Asian marketsrdquo Jour-nal of International Money and Finance vol 26 no 7 pp 1206ndash1228 2007

[22] K Naoui S Khemiri and N Liouane ldquoCrises and financialcontagion the subprime crisisrdquo Journal of Business StudiesQuaterly vol 2 no 1 pp 15ndash28 2010

[23] S Celik ldquoThe more contagion effect on emerging markets theevidence of DCC-GARCHmodelrdquo EconomicModelling vol 29no 5 pp 1946ndash1959 2012

[24] D Dimitriou D Kenourgios and T Simos ldquoGlobal financialcrisis and emerging stock market contagion a multivariateFIAPARCH-DCC approachrdquo International Review of FinancialAnal-ysis vol 30 pp 46ndash56 2013

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[26] M Billio andM Caporin ldquoMarket linkages variance spilloversand correlation stability empirical evidence of financial conta-gionrdquo Computational Statistics amp Data Analysis vol 54 no 11pp 2443ndash2458 2010

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[28] J A Chan-Lau D J Mathieson and J Y Yao ldquoExtremecontagion in equity marketsrdquo IMF Staff Papers vol 51 no 2 pp386ndash408 2004

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[32] G M Gallo and E Otranto ldquoVolatility spillovers interde-pendence and comovements a Markov switching approachrdquoComputational StatisticsampDataAnalysis vol 52 no 6 pp 3011ndash3026 2008

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[34] G Corsetti M Pericoli and M Sbracia ldquoSome contagionsome interdependence more pitfalls in tests of financial con-tagionrdquo Journal of International Money and Finance vol 24 no8 pp 1177ndash1199 2005

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[36] L Hu ldquoDependence patterns across financial markets a mixedcopula approachrdquo Applied Financial Economics vol 16 no 10pp 717ndash729 2006

[37] J C Rodriguez ldquoMeasuring financial contagion a Copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

[38] S Jayech and N B Zina ldquoA copula-based approach to finan-cial contagion in the foreign exchange marketsrdquo InternationalJournal of Mathematics in Operational Research vol 3 no 6 pp636ndash657 2011

[39] W Ye X Liu and B Miao ldquoMeasuring the subprime crisis con-tagion evidence of change point analysis of copula functionsrdquoEuropean Journal of Operational Research vol 222 no 1 pp 96ndash103 2012

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[41] S DajcmanM Festic and A Kavkler ldquoComovement dynamicsbetween central and eastern European and developed Europeanstock markets during European integration and amid financialcrisesmdasha wavelet analysisrdquo Engineering Economics vol 23 no1 pp 22ndash32 2012

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[43] C-C Chuang C-M Kuan and H-Y Lin ldquoCausality in quan-tiles and dynamic stock return-volume relationsrdquo Journal ofBanking and Finance vol 33 no 7 pp 1351ndash1360 2009

[44] B S Lee andM-Y L Li ldquoDiversification and risk-adjusted per-formance a quantile regression approachrdquo Journal of Bankingand Finance vol 36 no 7 pp 2157ndash2173 2012

[45] K Yu and J Zhang ldquoA three-parameter asymmetric Laplace dis-tribution and its extensionrdquo Communications in Statistics The-ory and Methods vol 34 no 9-10 pp 1867ndash1879 2005

[46] M B Stinchcombe and H White ldquoConsistent specificationtesting with nuisance parameters present only under the alter-nativerdquo Econometric Theory vol 14 no 3 pp 295ndash325 1998

[47] H White ldquoApproximate nonlinear forecasting methodsrdquo inHandbook of Economic Forecastin G Elliott C W J Grangerand A Timmermann Eds pp 460ndash512 Elsevier New YorkNY USA 2006

[48] K Yu and R A Moyeed ldquoBayesian quantile regressionrdquoStatistics amp Probability Letters vol 54 no 4 pp 437ndash447 2001

[49] R Koenker and J A F Machado ldquoGoodness of fit and relatedinference processes for quantile regressionrdquo Journal of theAmerican Statistical Association vol 94 no 448 pp 1296ndash13101999

[50] R Koenker and G Bassett Jr ldquoRegression quantilesrdquo Econo-metrica vol 46 no 1 pp 33ndash50 1978

[51] G Teyssiere and A P Kirman Long Memory in EconomicsSpringer Berlin Germany 2007

[52] H Niu and J Wang ldquoVolatility clustering and long memory offinancial time series and financial price modelrdquo Digital SignalProcessing vol 23 no 2 pp 489ndash498 2013

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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


some given level To detect the dynamic characteristics offinancial contagion following from Engle and Manganelli[5] and White et al [6 7] we use multivariate conditionalautoregressive value at risk (MV-CAViaR) framework to esti-mate two dynamic conditional autoregressive quantile mod-els simultaneously We show that MV-CAViaR model pro-vides a flexible modeling and estimation method to identifythe financial contagion between two countries

We contribute to the existing literature in twomajor waysFirstly compared with single quantile regression modelwhich can only study the contagious relation from one coun-try to anotherMV-CAViaRmodel can study the interactionalcontagion between twomarkets in dynamic settings from theangle of risk with the help of VaR Moreover consideringthe information of two countries simultaneously we applyMV-CAViaR model to study financial contagion between theorigin of global financial crisis (the US market) and majormature equity markets namely Japan UK France Germanyand one of the biggest emergingmarkets (China) In this waywe can investigate not only the existence of contagion but alsothe changes of risk in two markets In order to capture thevariation of risk for the whole sampling period MV-CAViaRmodels are employed to the overall sample between the USand other countries We find that there is a high increase invalue at risk for all sampling countries during the crisisperiod

Secondly our research accounts for the financial con-tagion and changes of risk by using a comprehensive timeseries spanning three periods before during and after thecrisis In this way we increase the efficiency of our estimationand enable a comparison of different phases by dividingthe whole sample period into three periods (ie precrisiscrisis and recovery periods)Then we consider estimation ofthese models for three important subsamples just as Mollickand Assefa [8] did The first runs from January 2006 toDecember 2007 which is referred to as the precrisis periodThe second namely the crisis period starts at the beginningof 2008 right after NBER identified December of 2007 asthe start of the major recent financial crisis in the US (TheNational Bureau of Economic Research (NBER) is an Amer-ican private nonprofit research organization ldquocommitted toundertaking and disseminating unbiased economic researchamong public policymakers business professionals and theacademic communityrdquo The NBER is well known for provid-ing start and end dates for recessions in the United StatesNBER is the largest economics research organization in theUnited States Many of the American winners of the NobelMemorial Prize in Economic Sciences were NBER researchassociates Many of the chairmen of the Council of EconomicAdvisers have also been NBER research associates includingthe former NBER president and Harvard Professor MartinFeldstein) The US recession officially ended in June 2009although economic growth has since been sluggishThe thirdsubsample stretching from July 2009 to July 2013 refers tothe US economy entering a recovery phase and we call itrecovery period By comparing the coefficient significanceand the changes of risk during the three different phases wejudge whether there exists financial contagion from the USto the tested countries during the crisis and the economics

enter the phase of recovery The findings show that theAmerican subprime crisis spread to Japan UK France andGermany and increased their market risk during the crisisin contrast to precrisis period But we have not found statis-tically significant evidence that there is financial contagionfrom America to China During the recovery period thecomovement between the US and France exists but fromthe perspective of risk the market risks in two countriesbecome small in comparison with the risk in crisis periodFor UK there is no contagion phenomenon during therecovery period However there are risk spillovers from bothJapan andGermany to theUS due tomarket conditions of theday

The rest of the paper is organized as follows Section 2provides the literature review about financial contagion InSection 3 the multivariate multiquantiles CAViaR (MVMQ-CAViaR) framework and its estimation are detailed Thenin Section 4 we perform empirical analysis to examine thepossible financial contagion between the US and five testedcountries Finally Section 5 concludes

2 Literature Review

A rapid increase in global economic integration has acceler-ated financial integration spillovers and contagion in equitymarkets of different countries Current studies on financialcontagion offermanymethods tomeasure the propagation ofinternational shocks across countries Investigating the finan-cial transmission Kaminsky et al [9] categorize the theoriesinto three groups contagion caused by herding behavior (egBanerjee [10] Calvo and Mendoza [11]) contagion resultingfrom trade linkages (eg Lahiri and Vegh [12] Gerlach andSmets [13]) and contagion accelerated by financial links (egKodres and Pritsker [14] Fratzscher [15]) There are manyapproaches proposed to detect and identify the financialcontagion Next we review the main approaches aboutthe studies on financial contagion through financial linkages

Early work on detecting contagion is obtained mainlyby comparing the traditional constant Pearson correlationcoefficient between two countries in times of relative calmwith that in times of unusually high volatility (see forinstance Reinhart and Calvo [16] Bordo and Murshid [17]and Forbes and Rigobon [18]) However Corsetti et al[19] show how these results are flawed in the presence ofheteroscedasticity The findings show that an increase inthe variance of two variables will cause an increase in thecorrelation coefficient even if the propagation mechanismis unaltered Moreover Pearson correlation coefficient is astatic linear correlation measure and cannot measure thenonlinear relation Therefore an increase in the correlationcoefficient is not necessarily a proof of contagion Rigobon[20] introduces the DCC test which seems to provide a wayto test parameter stability in the models with simultaneousheteroscedasticity and endogeneity problems Recently thereare many literatures using DCC model to study financialcontagion (see eg Chiang et al [21] Naoui et al [22]Celik [23] and Dimitriou et al [24]) In addition thereare numerous traditional econometric models to investigatethe dependence or change of dependence These include




3 Methodology



119905) 119905 =


119905is a finitely




(1198841015840

119905minus1 1198831015840

119905minus1) (1198841015840

119905minus2 1198831015840


F119905minus1

equiv 120590 ((1198841015840

119905minus1 1198831015840

119905minus1) (1198841015840

119905minus2 1198831015840

119905minus2) ) (1)

For 119894 = 1 2 119899 define 119865119894119905(119910) = 119875(119884

119894119905lt 119910 | F



F119905minus1

Let 120579119894119895belong to (0 1) for 119894 = 1 2 119899 119895 = 1 2 119901

119894




119894119895119905

is

119902119894119895119905

= inf 119910 119865119894119905(119910) ge 120579

119894119895| F119905minus1

(2)


1119905 1199021015840

2119905 119902

1015840

119899119905)1015840 with 119902

119894119905=

(1199021198941119905

1199021198942119905

119902119894119901119894119905



1 vectors Ψ119905 119905 = 1 2 119879 with Ψ

119905measurable F

119905minus1

and real vectors 120573 = 119888(1205731198941198951

1205731198941198952

120573119894119895119896

)1015840 and 120574

119894119895120591=

(1205741198941198951205911

1205741198941198951205912

120574119894119895120591119899

) where each 120574119894119895120591119896



119902119894119895119905

= Ψ1015840

119905120573119894119895+

119898

sum120591=1

1199021015840

119905minus120591120574119894119895120591

(3)


119894119895120591


119905 (2) single






119905







119905and lagged 119883

119905 In




120573119894119895

and 120574119894119895

= (1205741015840

1198941198951 1205741015840

1198941198952 120574

1015840

119894119895119898) Let 120572

119894119895= (1205731015840

119894119895 1205741015840

119894119895) and

write 120572 = (1205721015840

11 1205721015840

12 120572

1015840

1119901119894 120572

1015840

1198991 1205721015840

1198992 120572

1015840

119899119901119899) an 119897 times 1


119894=1119901119894(119896 + 119899119901





1199021119905

= 1198881+ 11988611

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

+ 119887121199022119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988622

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887211199021119905minus1

+ 119887221199022119905minus1

(4)

Since 119875 (119884119894119905


| F119905minus1




VaR1119905

= 119888lowast

1+ 119886lowast

11

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

12

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988711VaR1119905minus1

+ 11988712VaR2119905minus1

VaR2119905

= 119888lowast

2+ 119886lowast

21

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

22

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988721VaR1119905minus1

+ 11988722VaR2119905minus1

(5)





= 11988721


12= 0 and 119887

21= 0 then the


12




0 119886lowast

12= 119886lowast

21=


0 119886lowast

12= 0 and 119886

lowast

21= 0 which


lowast

12and 119886

lowast


lowast

12

and 119886lowast



and 11988621




119891 (119909 120583 120590 119901) =119901 (1 minus 119901)

120590exp minus

119909 minus 120583

120590[119901 minus 119868

(119909le120583)] (6)





119894119905minus

119902119894119895119905



119871 (120572 1198841 1198842) prop 120590

minus1 exp

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

120590

(7)

where 120588120579



120579(119890) = 120579 minus 119868

(119890le0) Because 119884

119894119905minus 119902119894119895119905


= argmin120572isinR

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

(8)




119903119905= 100 times (ln (119901

119905) minus ln (119901

119905minus1)) (9)










120591 and





1199021119905

= 1198881+ 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119887221199022119905minus1

(10)


1198842119905minus1




11 11988722


11 11988722are sig-










2006

-01-

03

2007

-01-

04

2008

-01-

04

2009

-01-

02

2009

-12-

30

2010

-12-

26

2011

-12-

22

2012

-12-

21

SampP500

minus10

minus5

0

5

10

(a)

0

5

2006

-01-

04

2007

-01-

17

2008

-01-

25

2009

-02-

10

2010

-02-

10

2011

-03-

01

2012

-03-

08

2013

-03-

20

CSI300

minus10

minus5

(b)

2006

-01-

04

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-02-

01

2011

-02-

08

2012

-02-

15

2013

-02-

20

Nikkei225

0

5

minus10

10

minus5

(c)

2006

-01-

03

2006

-12-

28

2007

-12-

21

2008

-12-

17

2009

-12-

14

2010

-12-

09

2011

-12-

07

2012

-12-

04

FTSE-100

0

5

minus10

10

minus5

(d)

2006

-01-

02

2006

-12-

21

2007

-12-

14

2008

-12-

08

2009

-12-

01

2010

-11-

22

2011

-11-

09

2012

-10-

30

CAC-40

0

5

minus10

10

minus5

(e)

2006

-01-

02

2006

-12-

21

2007

-12-

17

2008

-12-

11

2009

-12-

08

2010

-12-

01

2011

-11-

22

2012

-11-

12

DAX

0

5

10

minus5

(f)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722









1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1


119894(119894 = 1 2) equal to




1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1






12and 119886

21are only













5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3 Methodology



119905) 119905 =


119905is a finitely




(1198841015840

119905minus1 1198831015840

119905minus1) (1198841015840

119905minus2 1198831015840


F119905minus1

equiv 120590 ((1198841015840

119905minus1 1198831015840

119905minus1) (1198841015840

119905minus2 1198831015840

119905minus2) ) (1)

For 119894 = 1 2 119899 define 119865119894119905(119910) = 119875(119884

119894119905lt 119910 | F



F119905minus1

Let 120579119894119895belong to (0 1) for 119894 = 1 2 119899 119895 = 1 2 119901

119894




119894119895119905

is

119902119894119895119905

= inf 119910 119865119894119905(119910) ge 120579

119894119895| F119905minus1

(2)


1119905 1199021015840

2119905 119902

1015840

119899119905)1015840 with 119902

119894119905=

(1199021198941119905

1199021198942119905

119902119894119901119894119905



1 vectors Ψ119905 119905 = 1 2 119879 with Ψ

119905measurable F

119905minus1

and real vectors 120573 = 119888(1205731198941198951

1205731198941198952

120573119894119895119896

)1015840 and 120574

119894119895120591=

(1205741198941198951205911

1205741198941198951205912

120574119894119895120591119899

) where each 120574119894119895120591119896



119902119894119895119905

= Ψ1015840

119905120573119894119895+

119898

sum120591=1

1199021015840

119905minus120591120574119894119895120591

(3)


119894119895120591


119905 (2) single






119905







119905and lagged 119883

119905 In




120573119894119895

and 120574119894119895

= (1205741015840

1198941198951 1205741015840

1198941198952 120574

1015840

119894119895119898) Let 120572

119894119895= (1205731015840

119894119895 1205741015840

119894119895) and

write 120572 = (1205721015840

11 1205721015840

12 120572

1015840

1119901119894 120572

1015840

1198991 1205721015840

1198992 120572

1015840

119899119901119899) an 119897 times 1


119894=1119901119894(119896 + 119899119901





1199021119905

= 1198881+ 11988611

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

+ 119887121199022119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988622

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887211199021119905minus1

+ 119887221199022119905minus1

(4)

Since 119875 (119884119894119905


| F119905minus1




VaR1119905

= 119888lowast

1+ 119886lowast

11

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

12

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988711VaR1119905minus1

+ 11988712VaR2119905minus1

VaR2119905

= 119888lowast

2+ 119886lowast

21

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

22

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988721VaR1119905minus1

+ 11988722VaR2119905minus1

(5)





= 11988721


12= 0 and 119887

21= 0 then the


12




0 119886lowast

12= 119886lowast

21=


0 119886lowast

12= 0 and 119886

lowast

21= 0 which


lowast

12and 119886

lowast


lowast

12

and 119886lowast



and 11988621




119891 (119909 120583 120590 119901) =119901 (1 minus 119901)

120590exp minus

119909 minus 120583

120590[119901 minus 119868

(119909le120583)] (6)





119894119905minus

119902119894119895119905



119871 (120572 1198841 1198842) prop 120590

minus1 exp

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

120590

(7)

where 120588120579



120579(119890) = 120579 minus 119868

(119890le0) Because 119884

119894119905minus 119902119894119895119905


= argmin120572isinR

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

(8)




119903119905= 100 times (ln (119901

119905) minus ln (119901

119905minus1)) (9)










120591 and





1199021119905

= 1198881+ 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119887221199022119905minus1

(10)


1198842119905minus1




11 11988722


11 11988722are sig-










2006

-01-

03

2007

-01-

04

2008

-01-

04

2009

-01-

02

2009

-12-

30

2010

-12-

26

2011

-12-

22

2012

-12-

21

SampP500

minus10

minus5

0

5

10

(a)

0

5

2006

-01-

04

2007

-01-

17

2008

-01-

25

2009

-02-

10

2010

-02-

10

2011

-03-

01

2012

-03-

08

2013

-03-

20

CSI300

minus10

minus5

(b)

2006

-01-

04

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-02-

01

2011

-02-

08

2012

-02-

15

2013

-02-

20

Nikkei225

0

5

minus10

10

minus5

(c)

2006

-01-

03

2006

-12-

28

2007

-12-

21

2008

-12-

17

2009

-12-

14

2010

-12-

09

2011

-12-

07

2012

-12-

04

FTSE-100

0

5

minus10

10

minus5

(d)

2006

-01-

02

2006

-12-

21

2007

-12-

14

2008

-12-

08

2009

-12-

01

2010

-11-

22

2011

-11-

09

2012

-10-

30

CAC-40

0

5

minus10

10

minus5

(e)

2006

-01-

02

2006

-12-

21

2007

-12-

17

2008

-12-

11

2009

-12-

08

2010

-12-

01

2011

-11-

22

2012

-11-

12

DAX

0

5

10

minus5

(f)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722









1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1


119894(119894 = 1 2) equal to




1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1






12and 119886

21are only













5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199021119905

= 1198881+ 11988611

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

+ 119887121199022119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 11988622

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887211199021119905minus1

+ 119887221199022119905minus1

(4)

Since 119875 (119884119894119905


| F119905minus1




VaR1119905

= 119888lowast

1+ 119886lowast

11

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

12

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988711VaR1119905minus1

+ 11988712VaR2119905minus1

VaR2119905

= 119888lowast

2+ 119886lowast

21

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119886lowast

22

10038161003816100381610038161198842119905minus11003816100381610038161003816

+ 11988721VaR1119905minus1

+ 11988722VaR2119905minus1

(5)





= 11988721


12= 0 and 119887

21= 0 then the


12




0 119886lowast

12= 119886lowast

21=


0 119886lowast

12= 0 and 119886

lowast

21= 0 which


lowast

12and 119886

lowast


lowast

12

and 119886lowast



and 11988621




119891 (119909 120583 120590 119901) =119901 (1 minus 119901)

120590exp minus

119909 minus 120583

120590[119901 minus 119868

(119909le120583)] (6)





119894119905minus

119902119894119895119905



119871 (120572 1198841 1198842) prop 120590

minus1 exp

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

120590

(7)

where 120588120579



120579(119890) = 120579 minus 119868

(119890le0) Because 119884

119894119905minus 119902119894119895119905


= argmin120572isinR

119879

sum119905=1

119899

sum119894=1

119901119894

sum119895=1

120588120579119894119895(119884119894119905minus119902119894119895119905)

(8)




119903119905= 100 times (ln (119901

119905) minus ln (119901

119905minus1)) (9)










120591 and





1199021119905

= 1198881+ 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119887221199022119905minus1

(10)


1198842119905minus1




11 11988722


11 11988722are sig-










2006

-01-

03

2007

-01-

04

2008

-01-

04

2009

-01-

02

2009

-12-

30

2010

-12-

26

2011

-12-

22

2012

-12-

21

SampP500

minus10

minus5

0

5

10

(a)

0

5

2006

-01-

04

2007

-01-

17

2008

-01-

25

2009

-02-

10

2010

-02-

10

2011

-03-

01

2012

-03-

08

2013

-03-

20

CSI300

minus10

minus5

(b)

2006

-01-

04

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-02-

01

2011

-02-

08

2012

-02-

15

2013

-02-

20

Nikkei225

0

5

minus10

10

minus5

(c)

2006

-01-

03

2006

-12-

28

2007

-12-

21

2008

-12-

17

2009

-12-

14

2010

-12-

09

2011

-12-

07

2012

-12-

04

FTSE-100

0

5

minus10

10

minus5

(d)

2006

-01-

02

2006

-12-

21

2007

-12-

14

2008

-12-

08

2009

-12-

01

2010

-11-

22

2011

-11-

09

2012

-10-

30

CAC-40

0

5

minus10

10

minus5

(e)

2006

-01-

02

2006

-12-

21

2007

-12-

17

2008

-12-

11

2009

-12-

08

2010

-12-

01

2011

-11-

22

2012

-11-

12

DAX

0

5

10

minus5

(f)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722









1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1


119894(119894 = 1 2) equal to




1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1






12and 119886

21are only













5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















120591 and





1199021119905

= 1198881+ 11988612

10038161003816100381610038161198842119905minus11003816100381610038161003816 + 119887111199021119905minus1

1199022119905

= 1198882+ 11988621

10038161003816100381610038161198841119905minus11003816100381610038161003816 + 119887221199022119905minus1

(10)


1198842119905minus1




11 11988722


11 11988722are sig-










2006

-01-

03

2007

-01-

04

2008

-01-

04

2009

-01-

02

2009

-12-

30

2010

-12-

26

2011

-12-

22

2012

-12-

21

SampP500

minus10

minus5

0

5

10

(a)

0

5

2006

-01-

04

2007

-01-

17

2008

-01-

25

2009

-02-

10

2010

-02-

10

2011

-03-

01

2012

-03-

08

2013

-03-

20

CSI300

minus10

minus5

(b)

2006

-01-

04

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-02-

01

2011

-02-

08

2012

-02-

15

2013

-02-

20

Nikkei225

0

5

minus10

10

minus5

(c)

2006

-01-

03

2006

-12-

28

2007

-12-

21

2008

-12-

17

2009

-12-

14

2010

-12-

09

2011

-12-

07

2012

-12-

04

FTSE-100

0

5

minus10

10

minus5

(d)

2006

-01-

02

2006

-12-

21

2007

-12-

14

2008

-12-

08

2009

-12-

01

2010

-11-

22

2011

-11-

09

2012

-10-

30

CAC-40

0

5

minus10

10

minus5

(e)

2006

-01-

02

2006

-12-

21

2007

-12-

17

2008

-12-

11

2009

-12-

08

2010

-12-

01

2011

-11-

22

2012

-11-

12

DAX

0

5

10

minus5

(f)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722









1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1


119894(119894 = 1 2) equal to




1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1






12and 119886

21are only













5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2006

-01-

03

2007

-01-

04

2008

-01-

04

2009

-01-

02

2009

-12-

30

2010

-12-

26

2011

-12-

22

2012

-12-

21

SampP500

minus10

minus5

0

5

10

(a)

0

5

2006

-01-

04

2007

-01-

17

2008

-01-

25

2009

-02-

10

2010

-02-

10

2011

-03-

01

2012

-03-

08

2013

-03-

20

CSI300

minus10

minus5

(b)

2006

-01-

04

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-02-

01

2011

-02-

08

2012

-02-

15

2013

-02-

20

Nikkei225

0

5

minus10

10

minus5

(c)

2006

-01-

03

2006

-12-

28

2007

-12-

21

2008

-12-

17

2009

-12-

14

2010

-12-

09

2011

-12-

07

2012

-12-

04

FTSE-100

0

5

minus10

10

minus5

(d)

2006

-01-

02

2006

-12-

21

2007

-12-

14

2008

-12-

08

2009

-12-

01

2010

-11-

22

2011

-11-

09

2012

-10-

30

CAC-40

0

5

minus10

10

minus5

(e)

2006

-01-

02

2006

-12-

21

2007

-12-

17

2008

-12-

11

2009

-12-

08

2010

-12-

01

2011

-11-

22

2012

-11-

12

DAX

0

5

10

minus5

(f)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722









1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1


119894(119894 = 1 2) equal to




1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1






12and 119886

21are only













5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameters 1198881

11988612

11988711

1198882

11988621

11988722









1119905= 1198881+ 119887111199021119905minus1

and 1199022119905

= 1198882+ 119887221199022119905minus1


119894(119894 = 1 2) equal to




1119905= 1198881+

11988612|1198842119905minus1

|+119887111199021119905minus1

and 1199022119905

= 1198882+11988621|1198841119905minus1

|+119887221199022119905minus1






12and 119886

21are only













5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

2

3

4

2006

-01-

04

2007

-01-

31

2008

-02-

29

2009

-03-

23

2010

-04-

12

2011

-05-

06

2012

-05-

30

2013

-06-

28

China versus US

ChinaUS

(a)

Japan versus US

8

2

4

6

2006

-01-

04

2007

-01-

23

2008

-02-

14

2009

-03-

05

2010

-03-

26

2011

-04-

14

2012

-05-

07

2013

-05-

28

JapanUS

(b)

UK versus US

8

2

4

6

2006

-01-

03

2007

-01-

10

2008

-01-

16

2009

-01-

20

2010

-01-

22

2011

-01-

27

2012

-02-

01

2013

-02-

08

12

10

0

UKUS

(c)

France versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

14

2009

-01-

15

2010

-01-

15

2011

-01-

13

2012

-01-

12

2013

-01-

16

FranceUS

(d)

Germany versus US

8

2

4

6

2006

-01-

03

2007

-01-

09

2008

-01-

16

2009

-01-

22

2010

-01-

20

2011

-01-

23

2012

-01-

24

2013

-01-

30

10

GermanyUS

(e)




Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722






Parameters 1198881

11988612

11988711

1198882

11988621

11988722







Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameters 1198881

11988612

11988711

1198882

11988621

11988722










12

and 11988621
















12







5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 Conclusions






Acknowledgments



References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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