Asymmetric extreme interdependence in emerging equity markets

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRYAppl. Stochastic Models Bus. Ind., 2005; 21:483–498Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.602

Asymmetric extreme interdependence in emergingequity markets

Beatriz Vaz de Melo Mendes*,y,z

Federal University at Rio de Janeiro, Brazil

SUMMARY

We assess the extent of integration between stock markets during stressful periods using the concept ofcopulas. Our methodology consists of fitting copulas to simultaneous exceedances of high thresholds, andcomputing copula-based measures of interdependence and contagion. Using 21 pairs of emerging stockmarkets daily returns, we investigate if dependence increases with crisis, and analyse the chances of bothmarkets crashing together. Dependence at joint positive and negative extreme returns levels may differ.This type of asymmetry is captured by the upper and lower tail dependence coefficients. Propagation ofcrisis may be faster in one direction, and this feature is captured by asymmetric copulas. Copyright# 2005John Wiley & Sons, Ltd.

KEY WORDS: copulas; extreme events; tail dependence; asymmetry; non-exchangeability

1. INTRODUCTION

Recent financial crisis in emerging markets economies have raised questions concerning thebenefits of diversification, the robustness of domestic financial institutions, and the extent of thedomino effect with asymmetries in propagation of contaminations. All these points suggest thatthe measurement of cross-markets linkages and the assessment of changes in theirinterdependencies during crisis may be crucial for decision makers such as portfolio managers,central bankers, and regulatory authorities.

The problem of modelling cross-markets linkages may be approached by means of standardstatistical tools. For example, Lin et al. [1] investigated the transmission of returns and volatilityamong stock markets by formulating an aggregate shock model. Through the estimation of aGARCH model, they found that returns and volatility spillovers are typically symmetric.Longin and Solnik [2] found that a multivariate GARCH model with constant conditional

Received 17 January 2005Revised 24 April 2005

Copyright # 2005 John Wiley & Sons, Ltd. Accepted 6 May 2005

*Correspondence to: Beatriz Vaz de Melo Mendes, Rua Marquesa de Santos 22 apto. 1204, 22221080 Rio de Janeiro,RJ, Brazil.yE-mail: [email protected] Professor.

Contract/grant sponsor: CNPq-BrazilContract/grant sponsor: Convenio MCT/CNPq/FAPERJ

correlation is able to capture some of the evolution in the conditional covariance structure ofmonthly excess returns.

Many empirical studies ([3–5], among others) used the correlation or the conditionalcorrelation coefficients to measure dependence and change in dependence across markets.However, as shown in References [6, 7], the linear correlation coefficient r}the dependencemeasure most frequently used in practice}may be inappropriate and lead to erroneousconclusions in many situations. Boyer et al. [8] detect pitfalls in the conditional correlationcoefficient. They show that the conditional correlation, given a selected event or a (large)threshold value, possesses a systematic bias, and will differ from the (true) non-conditionalcorrelation coefficient even when the latter is constant.

Recently, appealing techniques derived from multivariate extreme value theory have gainedmore attention. Straetmans [9] computed probabilities of simultaneous stock market crashesand currency crises using a semi-parametric approach based on value theory (EVT). Longin andSolnik [10] empirically showed that international equity market correlation does not increasewith market volatility, but with market trend. In that work, they proposed a two-stepsprocedure to generate extreme (positive and negative) observations from a bivariate extremevalue model and tested the null hypothesis of zero correlation.

Malevergne and Sornette [11] used extreme value theory to obtain closed form expressions fortail dependence generated by factors models. Next, the stocks possessing minimal empirical taildependence coefficient with the market were selected to compose portfolios. The authorsdemonstrate that these portfolios possess superior behaviour (as measured by the Sharpe ratio)in stressed times.

More recently, copulas have been rediscovered as useful for modelling in finance. Schmidt [12]investigates the concept of tail dependence for elliptically contoured distributions. He showsthat the symmetric Pearson type VII distributions possess tail dependence, and may modelcredit risk in a more realistic way. Ane and Kharoubi [13] model parametrically the dependencestructure among stock index returns using copulas. They use non-parametric kernel densityestimates for the margins and fitted copulas by maximum likelihood. Breymann et al. [14] usecopulas to assess tail behaviour on high frequency data, discussing the problem of clustering ofextremes in bivariate data. See also References [15–17].

In this paper, we use the concept of copulas to empirically investigate the extent of emergingmarkets linkages during extreme events. We do not discuss economic or political issues causingfinancial crisis, nor apply financial theories concerning systematic risks to explain shockstransmission. We model the type and degree of interdependence, and study the phenomenon ofcontagion} during crisis, using copulas possessing two types of asymmetries: (i) the first oneallows for the strength of non-linear dependence existing in extreme joint positive and extremejoint negative events to be different, (ii) the second one models non-exchangeable variables andis used to assess asymmetries in propagation of crisis.

The copula of a multivariate distribution is the distribution function of their marginstransformed into Uniformð0; 1Þ and, as such, it contains all the information about itsdependence structure. Copula-based measures of dependence may be defined to reveal eachspecific aspect of the dependence. For example, the copula-based non-parametric Kendall’s rankcorrelation coefficient provides some advantages over the use of the canonical measure of linear

}For a literature review on the concepts of dependence and contagion in financial markets see Reference [18] andreferences therein.

Copyright # 2005 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind., 2005; 21:483–498

B. V. DE MELO MENDES484

dependence in the elliptical world (see Nelsen, 1999 [19]). On the other hand, the strength ofnon-linear dependence at joint extreme levels may be informed by the tail dependencecoefficients. Unlike the linear correlation coefficient r; this measure defined on the upper rightquadrant, the upper tail dependence coefficient lU; may differ from the lower tail dependencecoefficient lL; defined on the lower left quadrant. In addition, non-exchangeability between thevariables may be modelled by asymmetric copulas, but may not be detected by using neither r;nor the conditional correlation coefficient rC:

In particular, extreme value copulas may be used to model interdependencies during extremeevents. Costinot et al. [18] use extreme value copulas, following a methodology given inReference [20]. They model the margins using asymptotic results from EVT concerning the limitdistribution of block maxima, build up a set of parametric extreme value copulas, and estimatethe marginal distributions and copula parameters by maximum likelihood. To measurecontagion they compute the tail dependence function lUðaÞ:

Related works focusing on the estimation of tail dependence using EVT are References [9, 10].Straetmans [9] applies asymptotic results concerning the limit distribution of block maxima anduses the Hill estimator to estimate the tail index and the stable tail dependence function. Longinand Solnik [10] used EVT results concerning the limit distribution of scaled excesses over highthresholds. They model the margins using the generalized Pareto distribution (GPD) andsuggest using the symmetric logistic dependence function [21] to model the pairs of exceedances.All parameters are estimated by maximum likelihood. Costinot et al. [18] show the relationbetween these last cited articles and their approach based on copulas.

Our approach is similar to Costinot et al. [18] in the sense that we model the marginal tailsusing results from EVT, and model the dependence structure using copulas. Our approach,however, differs in the following. First, we select thresholds in each margin and collectsimultaneous exceedances. Our joint tail data are observed pairs, which may not be the casewhen using block maxima. Second, we consider a large set of copulas with some desirablecharacteristics. Finally, we compute, besides lUðaÞ (and lLðaÞ), alternative measures ofinterdependence and contagion. In this paper, our applications involve daily returns, whosedistribution is typically hard to determine using simple models (due to the existence of extremeobservations, heavy tails, asymmetries, etc.). Our approach has the advantage of leaving thedistribution of daily data unspecified. We focus on the tails, for which suitable univariate modelsfrom the extreme value theory are available. We are not modelling volatility periods, eventhough extreme simultaneous (positive or negative) returns may occur during these periods.

Our work differs from Straetmans [9] because empirical estimation of the Hill estimator and adependence function are used there. Even though our objectives are similar, the modellingstrategies and estimation methods are different, and the applications consider different equitymarkets. We understand papers may be seen as complementing each other.

In summary, we assess the type and strength of linkages across emerging markets, and thestrength and direction of propagation of (asymmetric) transmission of shocks by modelling co-exceedances (beyond fixed high thresholds) of pairs of stock index returns using (a)symmetricextreme value copulas, and computing copula-based probabilities of single and joint crashes, taildependence, and alternative definitions of contagion. Formal statistical goodness of fit tests areused to select the best copula and univariate fits.

This article contributes to the existing literature in the following ways: (i) showing that themodified generalized Pareto distribution (MGPD) adjusts properly (as verified by a goodness offit test) to the marginal excess data; (ii) taking a closer look at the asymmetric copula associated


ASYMMETRIC EXTREME INTERDEPENDENCE 485

to the asymmetric logistic model of Reference [21] and obtaining the expression of its coefficientof tail dependence; (iii) illustrating the methods using emerging markets data, and comparingestimates of the copula-based lU; of the Pearson’s linear correlation coefficient r and itsconditional version rC; and of alternative definitions of (asymmetric) contagion; (iv) Finally, bycarrying on a small simulation study to assess properties of the copulas parameters maximumlikelihood estimates at small samples.

The remaining part of the paper is structured as follows. In Section 2, we start by giving abrief introduction to copulas and provide the expressions of five copulas possessing differenttypes of asymmetry. Then we define contagion, asymptotic contagion, and propose copula-based alternative dependence measures. In Section 3 we perform the empirical analysis ofextreme co-exceedances from 21 pairs of emerging markets index returns. In the last section, wesummarize the results and conclude.

2. MODELLING INTERDEPENDENCE

We start with a brief review of copulas. For a comprehensive discussion of their mathematicalproperties see References [15, 19].

2.1. Copulas

Let X ¼ ðX1;X2Þ be a random variable (r.v.) in R2 with joint distribution function (d.f.) F andmargins Fi; i ¼ 1; 2: Suppose that the margins X1 and X2 are, respectively, transformed intoUniformð0; 1Þ r.v.s U and V : If F1 and F2 are continuous, this transformation may be obtainedthrough the probability integral transformation ðx1; x2Þ/ðF1ðx1Þ;F2ðx2ÞÞ:

The joint distribution function CF ð�Þ of ðF1ðX1Þ;F2ðX2ÞÞ is the copula of the random vectorðX1;X2Þ; or equivalently, the copula pertaining to F : It follows that

Fðx1; x2Þ ¼ CF ðF1ðx1Þ;F2ðx2ÞÞ ð1Þ

If F1 and F2 are continuous, then CF is unique. Conversely, if CF is a copula and F1 and F2 ared.f.s, the function F defined in (1) is a joint d.f with margins Fi; i ¼ 1; 2, [22]. If we assume thateach Fi and CF are differentiable, then the joint density f ðx1; x2Þ of X can be written as

f ðx1; x2Þ ¼ cðF1ðx1Þ;F2ðx2ÞÞ �Y2i¼1

fiðxiÞ ð2Þ

where cðu; vÞ ¼ @C2ðu; vÞ=@u@v is the copula density, and fi is the density function of Xi; i ¼ 1; 2:In (2) we note the decomposition of the joint density in two parts. One describes the dependencestructure, and the other describes the marginal behaviour of each component.

In this work we are interested in joint bivariate extreme events. According to Tawn [21], ifthere exists a positive association between extreme events of X1 and X2; then the conditionalprobability PrfX1 > F�11 ð1� aÞjX2 > F�12 ð1� aÞg is greater than zero and decreases as a # 0:Asymptotic dependence at joint (negative and positive) quantiles may be summarized by theconcept of tail dependence.

The upper tail dependence coefficient lU between X1 and X2 is defined as

lima!0þ

lUðaÞ ¼ lima!0þ

PrfX1 > F�11 ð1� aÞjX2 > F�12 ð1� aÞg ð3Þ



if this limit exists. The two variables X1 and X2 are said to be asymptotically dependent in theupper tail if lU 2 ð0; 1�; and asymptotically independent if lU ¼ 0: The same concept is appliedto define the lower tail dependence coefficient lL: Both the upper and the lower tail dependencecoefficients may be expressed using the pertaining copula

lU ¼ limu"1

%CF ðu; uÞ1� u

and lL ¼ limu#0

CF ðu; uÞu

ð4Þ

if these limits exist, and where %CF ðu; vÞ ¼ PrfU > u;V > vg: The measure lU quantifies theamount of extremal dependence within the class of asymptotically dependent distributions. Itprovides a natural way for ordering copulas. If copula C2 is more concordant than copula C1;that is, C1ðu; vÞ4C2ðu; vÞ; 8u; v; 2 I2 ¼ ½0; 1�2; than lU of C2 is greater than lU of C1 [15].

A copula C is symmetric if it satisfies Cðu; vÞ ¼ Cðv; uÞ; 8ðu; vÞ in I2; which makes their quantilelines symmetric with respect to the main diagonal of I2: An asymmetric (differentiable) copulapossesses asymmetric density vertical section, cðu; 1� uÞ; along the secondary diagonal.Moreover, a copula C may be classified into three types according to the shape of its densityvertical section on the main diagonal, cðu; uÞ: The ‘J’ copula possess only lU > 0: The ‘L’ typepossess only lL > 0; and the ‘U’ copula possess both lL > 0 and lU > 0: The asymmetry impliedby lL=lU is the other type of asymmetry we are interested in.

For any given data set, the problem of identifying the right copula is complicated and hasbeen addressed by several authors, including Frees and Valdez [23]. Here we overcome thisdifficulty by restricting our attention to ‘J’ or ‘U’ type copulas allowing for positive dependence.Choice of the most suitable copula is based on formal statistical tests. In the applications ofSection 3 we consider the following parametric copula families (in the expressions below, *u ¼�lnðuÞ and *v ¼ �lnðvÞ):

(i) Gumbel copula: CdGuðu; v; dÞ ¼ expf�ð*ud þ *vdÞ1=dg; where d51: It is a ‘J’ copula related to

the symmetric logistic model (see Reference [24]), with lU ¼ 2� 21=d:(ii) Galambos copula: Cd

Galðu; v; dÞ ¼ uv expfð*u�d þ *v�dÞ�1=dg; where 04d51: It is asymmetric ‘J’ copula with lU ¼ 2� 21=d; for d > 1:

(iii) Associated to Kimeldorf and Sampson copula: CdAKSðu; v; dÞ ¼ uþ v� 1þ ½ð1� uÞ�1=d þ

ð1� vÞ�1=d � 1��d; where d50: This is a symmetric ‘J’ copula, with lU ¼ 2�1=d;associated to the Kimeldorf and Sampson copula Cðu; vÞ ¼ ðu�d þ v�d � 1Þ�1=d (FamilyB4 in Reference [15]). Frees and Valdez [23] obtain the CAKSðu; v; dÞ as the copulapertaining to joint Pareto-dependent variables.

(iv) Asymmetric Logistic Model copula: Cd;p1;p2ALM ðu; v; d; p1; p2Þ ¼ exp½�ðpd1 *u

d þ pd2*vdÞ1=d � ð1�

p1Þ*u� ð1� p2Þ*v�; for ðp1; p2Þ 2 ½0; 1�2 and d51: This is a mixture of the Gumbel and theindependence copulas, obtained using the asymmetrization technique given in Reference[24]. It is a flexible asymmetric copula. In finance, for example, the non-exchangeabilityimplies that the conditional probability that a crash occurs in a market given that acatastrophic event has occurred in some other one is different of this probabilitycomputed the other way around. Figure 1 illustrates the asymmetries on both diagonals.From the asymmetrization technique we obtain the expression for the dependencefunction Að�Þ of the asymmetric logistic model given in Reference [21]. It follows thatlU ¼ p1 þ p2 � ðpd1 þ pd2Þ

1=d; for ðp1; p2Þ 2 ½0; 1�2 and d51: If p1 or p2 is equal to zero, thenlU ¼ 0; the independence case. When p1 ¼ p2 ¼ 1; by letting d " 1; lU approaches 1, theperfect dependence case.



(v) Joe–Clayton copula: Cd;yJC ðu; v; d; yÞ ¼ 1� ½1� ð½1� ð1� uÞy��d þ ½1� ð1� vÞy��d � 1Þ�1=d

�1=y; where y51; d50: This is a symmetric ‘U’ shape extreme value copula possessinglU ¼ 2� 21=y independent of d; and lL ¼ 2�1=d; independent of y:

2.2. Modelling interdependence and contagion

When studying co-movements of equity markets, an important issue is whether or not thestrength of cross-market linkages during periods of stability would remain the same after ashock to one country. Any upward change characterizes contagion} and it does not have to beexplained by economic theories. Contagion may show up among markets with weak trade linksbut with similar economic, geographical, or political positions. In this section, we developalternative ways to assess interdependencies and contagion by modelling co-exceedances of highthresholdsk using asymmetric copulas.

Let Xi represent market i stock index daily log return, i ¼ 1; 2: Let A and B represent,respectively, the excess random variables, i.e. A ¼ ðX1 � q1ÞIðX1>q1Þ; and B ¼ ðX2 � q2ÞIðX2>q2Þ;where qi; is a selected threshold in margin i; i ¼ 1; 2; and IðEÞ is the indicator function of event E:

ALM

den

sity

on

mai

n di

agon

al

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0

2

4

6

ALM

den

sity

on

seco

ndar

y di

agon

al

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Figure 1. At left, the ‘J’ shape of the ALM copula density on main diagonal. At right, theasymmetric density shape on secondary diagonal (ð1; 0Þ to ð0; 1Þ). In grey, d ¼ 1:73; p1 ¼ 0:37;

p2 ¼ 0:99; In black, d ¼ 2:06; p1 ¼ 0:25; p2 ¼ 0:99:

}The definition of contagion varies greatly in the literature. The reader may consult References [1, 7] and referencestherein.kWe stress that the methodology developed here is explained within the context of financial returns, whose distribution istypically centred at zero. All methods derived for joint positive quantiles apply to joint negative returns as well. Inpractice, we multiply the data by ð�1Þ:



Regardless of the type of association existing between markets 1 and 2; whenever there existspositive dependence, the expression

PrfA > ajB > bg > PrfA > ag ð5Þ

holds for large a and b: In fact, the random vector ðA;BÞ is said to be positive quadrantdependent if (5) holds 8a; b 2 R [15].

Let Hi; i ¼ 1; 2; be a Pareto-type distribution** of the excesses A and B; and let a and b besuch that H1ðaÞ ¼ H2ðbÞ ¼ u; u 2 ð0; 1Þ: Rewriting (5) in terms of copulas we obtain

1� u� uþ Cðu; uÞ1� u

> 1� u ð7Þ

where C is the copula of ðA;BÞ: If the random vector ðX1;X2Þ has tail dependence, then thecopula C has tail dependence. If ðX1;X2Þ is tail independent (as in the bivariate normaldistribution), then the copula C in (7) is the product copula, for large enough thresholds qi ineach margin [15].

According to (3), the left-hand side of (7) is the function lUðaÞ; a ¼ 1� u; whose limit asa! 0þ is the upper tail dependence coefficient lU: We investigate the behaviour of lUðaÞ for adecreasing sequence of values a;yy which we call ‘linkages at extreme levels’. Similarly, in thecase of joint losses, we examine the behaviour of lLðaÞ and its limit lL:

Asymmetric propagation of crisis may be captured by means of variations of the left-handside of (7)

1� u� vþ Cðu; vÞ1� v

ð8Þ

where u ¼ H1ðx1;pÞ and v ¼ H2ðx2;pÞ; for some (meaningful) choices of x1;p and x2;p on thesupport of A and B: We use the most popular risk measure in finance, the Value-at-Risk.

To explain the methodology for choosing u and v; we simplify the notation and consider arandom variable X with d.f. F ; the Value-at-Risk xp; that is, %FðxpÞ ¼ p; a threshold q; and theexcess Y ¼ X � q: Let H denote the conditional d.f. of Y ; that is, %HðyÞ ¼ PrfY > yjX > qg:Then

%Fðqþ yÞ ¼ %FðqÞ � %HðyÞ ð9Þ

where y > 0; and qþ y is the Value-at-Risk xp:

**Let X1;X2; . . . be a sequence of independent random variables with common distribution F ; and consider themaximumMn ¼ maxfX1; . . . ;Xng: The Fisher and Tippett theorem [25] gives the limit distribution of the (normalized)maximum Mn: This non-degenerate distribution is a member of the generalized extreme value distributions. If theFisher–Tippett theorem holds, then, for a large enough threshold q; the distribution of ðX � qÞ conditional on X > q;is approximately the generalized Pareto distribution (GPD). The result holds also for non-i.i.d. processes, see proof inReference [26]. We use here an extention of the GPD, the modified generalized Pareto distribution (MGPD) proposedin Reference [27]. The MGPD has survival function given by

%HðyÞ ¼1þ x

yy

c

� ��1=xif x=0

e�yy

c if x ¼ 0

8>>><>>>: ð6Þ

where y > 0; and c > 0 is a scale parameter.yyWe note that even tail-independent random variables may exhibit strong dependence at moderately extreme levels, seeReference [28].



In (9), %FðqÞ ¼ PrfX > qg may be estimated using the empirical survival function, that is,the ratio Tq=T � Rq; where Tq is the random number of exceedances of q; and T is the samplesize. In (9), the distribution H may be taken as the MGPD (6). By noting that %Fðqþ yÞ ¼ p; andusing the inverse function %H�1ð�Þ of (6) in (9), we obtain the EVT model-based Value-at-Risk xp

xp ¼ qþcx

p

Rq

� ��x�1

" # !1=y

if x=0 ð10Þ

and xp ¼ qþ ð�c logðp=RqÞÞ1=y if x ¼ 0: For details see Reference [29].

Back to the bivariate problem, let xi;pj denote the Value-at-Risk defined in (10) withexceedance probability pj in each margin i; i ¼ 1; 2:We fix p1 ¼ 0:01; p2 ¼ 0:005; and p3 ¼ 0:001;and let u ¼ H1ðx1;pj Þ and v ¼ H2ðx2;pj Þ; for any pj ; j 2 f1; 2; 3g: Expression (8) represents, forvarying values pj ; the probability that market 1 falls beyond its Value-at-Risk x1;pj ; given thatmarket 2 has broken through its x2;pj :

Let Aj denote the event ½A > x1;pj �; and Bj denote the event ½B > x2;pj �: To assess asymmetriesin contagion we compute the following probabilities:

PrðB2jA1Þ and PrðA2jB1Þ ð11Þ

where we investigate, for example, the probability that market 2 falls beyond its Value-at-Riskwith exceedance probability 0:005; given that market 1 has fallen beyond its x1;0:01; and comparethis to the probability that market 1 falls beyond its x1;0:005; given that market 2 has fallenbeyond its x2;0:01: PrðB2jA1Þ being greater than PrðA2jB1Þ; means that market 2 gets morecontaminated from market 1, than the opposite. We may say that market 2 is the ‘direction ofgreater contamination’, since it is more likely to spread the crisis. We also compare PrðB3jA2Þand PrðA3jB2Þ: Note that the excess probability attached to the conditioning event is alwaysgreater than the (excess) conditional probability.

We also examine the sequence of conditional probabilities fPrðB2jA1Þ;PrðB3jA2Þg andfPrðA2jB1Þ;PrðA3jB2Þg: This analysis would indicate if contamination increases with crisis, andalso the direction of faster propagation.

The dependence structure is in line with the structure of random variables positivelyassociated. According to Tawn [21], the strong stochastic ordering of Reference [30] must alsobe satisfied for the uniform data ðU;VÞ: It says that PrfU > ujV > vg is increasing in v for fixedu: This motivates the empirical analysis of fPrðB2jA1Þ;PrðB2jA2Þg and fPrðA2jB1Þ;PrðA2jB2Þg:The limit of these probabilities may be of great interest for markets participants.

The quantities exposed so far compared probabilities in different probability spaces, since theconditioning event was not fixed. Alternatively, we assess contagion by computing the sequenceof probabilities fPrðB2jA2Þ;PrðB3jA2Þg or fPrðA2jB2Þ;PrðA3jB2Þg; which measure tail events onthe same conditional probability space. These quantities have been previously named bypractitioners as the probabilities of flight-to-quality, in the case of joint gains, and fall-to-ruin,in the case of joint losses. They should reveal if a market amplifies (or shrinks) the situation inthe other one.

Finally, we investigate the probability of two markets crashing together, given that at least oneof them has experienced an extreme event (see also References [9, 31]). This means computing

1� u� vþ Cðu; vÞ1� Cðu; vÞ

for some fixed value u; say u ¼ 0:95; and increasing levels v; say v ¼ 0:95; 0:99; 0:999; 0:9999:



3. EMPIRICAL ANALYSIS

We analyse extreme co-movements of 21 pairs of stock market indexes from the 7 mostimportant emerging markets. Data consist of daily log returns from 3 July 1995 to 29 September2003. All series have length T ¼ 2154:

A critical issue is the selection of the thresholds defining the region in R2 to collect the jointdata. The limit distributional result given in footnote yy for the excesses beyond a threshold q;holds when q goes to the right end point of the support of X : In practice, we have a finite sample,and choosing a threshold implies a trade-off between bias and inefficiency of parameterestimates. We will observe either a large bias on the parameters estimates if the threshold is notlarge enough, or the estimates will possess a large variance if just a few observations are used.

Longin and Solnik [10] proposed a procedure for optimal threshold selection. However, thedistribution of the returns must be specified. Here, the 7 returns series present different scales,some of them are symmetric, some are left skewed, others present right asymmetry. Thethreshold selection should thus be data driven. After some experimentation, we decided tochoose as thresholds the return values separating 10% of the data in each tail. Our procedureallows for different threshold values for each series and each tail, thus adapting for market scaleand shape. We collect joint exceedances, both negative or positive, below and above thethresholds. The number of pairs of observations was in average 3–5% of the total number T ofobservations. We note that this is in agreement with Longin and Solnik [10] optimal rule, whichuses, in average, 4–5% of the total number of returns observations.zz

Parameters estimates of the univariate distributions and copulas are obtained by maximumlikelihood. Rather than maximizing the copula-based multivariate log-likelihood (log of (2)) inall the parameters together, we estimate the parameters from log-likelihoods associated with thedifferent MGPD marginal distributions, and then the copula parameters. This two-stepsprocedure is usually called inference functions for margins (IFM). Joe [15] argues that we canexpect the IFM method to be quite efficient because it is fully based on maximum likelihoodestimation. Efficiency may be assessed either by comparing the estimators asymptoticcovariance matrices, or by comparing their mean squared error from Monte Carlo simulations.For some models this efficiency has been established, see References [15, 33].

To each excess (univariate) data we fitted the full model MGPDðc; x; yÞ; and the three nestedmodels: the constrained MGPD distribution, where x ¼ 0; the GPD distribution, where y ¼ 1;and the unit exponential distribution, where x ¼ 0 and y ¼ 1: The standard likelihood ratiotest}} was used to discriminate between the nested models. We formally tested goodness of fit,and did not reject, for all univariate data sets, using the Kolmogorov test [34]. Goodness of fitswere also graphically checked by means of qq-plots. We note that serial dependence in theexceedances was investigated}and not found}using the sample auto-correlation function. Fora discussion on the issue of temporal dependence of financial return data see Reference [35].

zzWe also made use of the mean residual life plot [32], a graphical useful procedure which helps choosing the threshold.For all series, the graphical inspection indicated threshold values between 1.5 and 2.5, in agreement with the value 2selected by Coles [32] in Example 4.4.2, where he also uses log percentual daily index returns.

}}Let L1 and L2 represent the log-likelihood corresponding, respectively, to the full model (1) and a constrained model(2). The distribution of the test statistic 2ðL1 �L2Þ may be approximated by a chi-squared distribution with k degreesof freedom, where k ¼ k1 � k2; and ki is the number of parameters in model i; i ¼ 1; 2: Large values of the test statisticleads to the rejection of the null hypothesis, which chooses the constrained model.



The parameter estimates were then plugged in the corresponding MGPD distributionfunctions to transform the excesses into Uniformð0; 1Þ observations. The copulas presented inSection 2 were then fitted to the pairs of transformed data. To compare the different copula fitswe considered the log-likelihood value, the Akaike information criterion, and the distancebetween the fitted copulas and the empirical copula.}} We used the (discrete) L1 and L2 normscomputed on the lattice L:kk For other measures of goodness of fit based on the empiricalcopula, see Ane and Kharoubi [13]. To test goodness of fit we used a bivariate extention of theusual Pearson test, described in Reference [37]. For others tests for copula specification based onthe empirical copula, see Reference [38]. Finally, we test independence using the standardlikelihood ratio test (see footnote kk).

As pointed out in Reference [38], the reliability of the conclusions drawn from the analysesbased on (simultaneous, margins and copula) maximum likelihood estimates of these modelsdepends on the right specification of the margins and copula. In some situations the marginsspecification may be a very difficult task, for example, when dealing with daily returns. InReference [38], we find a simulation study where the potential impact of misspecified margins onthe estimation of the copula parameter is shown. Fortunately, here, the specification of themargins distribution is relatively easy, since the GPD family has proved (for example, Reference[32]) to provide a good fit for finite sample excesses over thresholds. In addition, theperformance of the GPD maximum likelihood estimators have been previously evaluated byHosking and Wallis [39], Castillo and Hadi [40], and Tajvidi [41].

The behaviour of the maximum likelihood estimators of copulas parameters were investigatedthrough simulations by Caperaa et al. [42] in the case of the Gumbel or logistic model, and byGenest [43] in the case of the Frank family. Genest [43] investigated the performance of fourestimators considering samples of size 10–50, and found that the method of moments estimatorappears to have smaller mean squared error than the maximum likelihood estimator. In thispaper, we report the results from a simulation study regarding the small-sample behaviour ofthe maximum likelihood estimators of the Joe–Clayton copula, since this copula was found torepresent the dependence structure of several pairs of markets analysed (see Table II). Acomprehensive simulation study would make this paper too long.

Table I gives the results of 300 Monte Carlo simulations to assess the variability and bias ofthe maximum likelihood estimators bd; by; and clU of the Joe–Clayton copula. The empiricalanalyses (see Table II) suggested the sample sizes of 40 and 80 and the true parameters valuesðd ¼ 0:01; y ¼ 2Þ and ðd ¼ 0:5; y ¼ 1:5Þ: The models considered are given in the first and secondrows of Table I. The other rows show the average, the median, the standard deviation, and the95% percentile confidence interval based on the 300 simulations. To simulate from the

}}The notion of empirical copula was introduced by Deheuvels [36]. Let ðx1;t; x2;tÞ; t ¼ 1; . . . ;T ; denote a sample of Ti.i.d. bivariate observations, and let fx1;ðtÞ;x2;ðtÞg be the order statistics. Consider the lattice

L ¼t1

T;t2

T

� �: ti ¼ 0; . . . ;T ; i ¼ 1; 2

n oThe empirical copula *C is only defined on L; and it is given by

*Ct1

T;t2

T

� �¼

1

T

XTt¼1

I½x1;t4x1;ðt1 Þ ;x2;t4x¼2;ðt2 Þ �ð12Þ

where I½�� is the indicator function, and 14t14t24T :kkSee footnote kk. Here, we are testing the null hypothesis of independence, and L2 is zero.



Joe–Clayton copula we used the algorithm given in Reference [15] based on the conditionalcopula distribution.

As expected, we observe in Table I that the parameters distributions get more concentrated asthe sample size increases. Both estimators bd and by present small biases and variances. We mayconclude that the copula parameters maximum likelihood estimates may be considered reliable.

Table I. Results from 300 simulations of the Joe–Clayton copula. The table gives the average, the median,the standard deviation, and the 95% percentile confidence interval of the maximum likelihood

estimators bd; by; and clU:True d ¼ 0:010 y ¼ 2:000 lU ¼ 0:586 d ¼ 0:500 y ¼ 1:500 lU ¼ 0:413

Sample size ¼ 40Mean 0.1170 1.9868 0.5717 0.5443 1.5090 0.3836Median 0.0030 1.9500 0.5732 0.5280 1.4950 0.4101Std. Dev. 0.1788 0.2720 0.0686 0.2729 0.2780 0.147495% CI ½0:001; 0:540� ½1:58; 2:50� ½0:45; 0:68� ½0:01; 1:01� ½1:02; 2:00� ½0:03; 0:58�

Sample size ¼ 80Mean 0.0768 1.9881 0.5762 0.5357 1.5060 0.4107Median 0.0040 1.9950 0.5846 0.5214 1.4997 0.4100Std. Dev. 0.1221 0.2119 0.0539 0.2314 0.2015 0.102795% CI ½0:001; 0:380� ½1:64; 2:36� ½0:47; 0:66� ½0:35; 0:92� ½1:18; 1:82� ½0:24; 0:55�

Table II. Copula fits and estimates of dependence measures for negative and positive co-exceedances forall tail dependent pairs. Pairs of markets are ordered according to the strength of their tail

dependence coefficient.

Pairs Thresholds # Copula (param.) br crC lL ð:20Þ lL ð:10Þ lL ð:05Þ lL

Joint negative returnsArg. & Mex. �2:23 & �1:98 78 CJCð0:01;2:16Þ 0.44 0.79 0.632 0.624 0.622 0.621Arg. & Bra. �2:23 & �2:44 87 CJCð0:01;1:79Þ 0.49 0.41 0.551 0.534 0.529 0.527Bra. & Mex. �2:44 & �1:98 98 CJCð0:01;1:52Þ 0.55 0.64 0.468 0.438 0.428 0.422Arg. & Afr. �2:23 & �1:61 53 CJCð0:01;1:5Þ 0.16 0.51 0.461 0.430 0.419 0.413Tai. & Afr. �2:19 & �1:61 38 CJCð0:01;1:33Þ 0.15 0.33 0.392 0.346 0.328 0.316Mex. & Afr. �1:98 & �1:61 73 CAKSð0:60Þ 0.32 0.47 0.448 0.394 0.364 0.315Kor. & Chi. �2:98 & �1:63 30 CALMð2:63;:99;:30Þ 0.13 0.18 0.409 0.346 0.315 0.284Bra. & Afr. �2:44 & �1:61 60 CALMð3:09;:22;:24Þ 0.26 0.20 0.179 0.172 0.171 0.171Kor. & Afr. �2:98 & �1:61 45 CAKSð0:33Þ 0.20 0.32 0.351 0.274 0.228 0.122Kor. & Bra. �2:98 & �2:44 38 CALMð5:13;:04;:99Þ 0.16 0.06 0.229 0.134 0.087 0.040

Pairs Thresholds # Copula (param.) br brC lU ð:20Þ lU ð:10Þ lU ð:05Þ lU

Joint positive returnsArg. & Mex. 2.11 & 1.98 69 CJCð0:01;1:51Þ 0.44 0.64 0.464 0.434 0.423 0.417Arg. & Bra. 2.11 & 2.37 76 CJCð0:01;1:41Þ 0.49 0.45 0.426 0.388 0.374 0.365Bra. & Mex. 2.37 & 1.98 70 CALMð1:73;:37;:99Þ 0.55 0.55 0.398 0.333 0.301 0.269Tai. & Chi. 2.29 & 1.70 31 CALMð2:06;:25;:99Þ 0.11 0.29 0.363 0.292 0.257 0.222Arg. & Tai. 2.11 & 2.29 28 CALMð3:88;:19;:99Þ 0.04 0.28 0.338 0.263 0.226 0.190Mex. & Afr. 1.98 & 1.63 46 CAKSð0:10Þ 0.32 0.08 0.250 0.154 0.100 0.001



We now give the results from the empirical analyses for the dependent co-exceedances.Approximately half of joint extreme losses (10 out of 21) are dependent, but most of jointextreme gains (15 out of 21) are independent. All asymmetric cases are based on the CALM

copula. The results are summarized in Tables II–IV. In the upper part of all tables we presentthe results for bear markets (in the lower part of these tables we provide the results for bullmarkets).

Table II gives in columns 1, 2 and 3, respectively, the dependent pairs, the threshold valuesdefining the excess data, and corresponding sample sizes. In column 4 we give the maximumlikelihood copula parameters estimates. In columns 5 and 6 we provide the values of the samplecorrelation coefficients, based on the entire data set (br), and based just on the co-exceedances(brC). Columns 7–9 show the values of the extreme linkages, the function lLðaÞ in the case of jointlosses (and lUðaÞ in the case of joint gains), for a equals to 0:20; 0:10; and 0:05: Markets areordered according to the strength of their lower (upper) tail dependence coefficients, given in thelast column. Note that crC would provide a different markets ranking.

Table III. Estimated probabilities of shocks transmission for the asymmetric dependent pairs.

Asymmetric copulas

Pairs A–B PðB2jA1Þ PðB2jA2Þ PðB3jA2Þ PðA2jB1Þ PðA2jB2Þ PðA3jB2Þ

Joint negative returnsKor.–Chi. 0.0702 0.3076 0.0377 0.1426 0.3076 0.1071Bra.–Afr. 0.0298 0.1519 0.0178 0.0284 0.1519 0.0164Kor.–Bra. 0.0550 0.0585 0.0415 0.0231 0.0585 0.0068

Joint positive returnsBra.–Mex. 0.0104 0.1004 0.0129 0.0097 0.1004 0.0056Tai.–Chi. 0.1631 0.2611 0.1958 0.0799 0.2611 0.1195Arg.–Tai. 0.0803 0.2017 0.0665 0.0311 0.2017 0.0146

Table IV. Probability of both markets crashing (booming) together given that at leastone crashed (boomed).

For fixed u ¼ 0:95 For fixed v ¼ 0:95v value u value

Pairs A–B 0.95 0.99 0.999 0.9999 0.95 0.99 0.999 0.9999

Asymmetric copulasJoint negative returns

Korea–China 0.1868 0.0587 0.0066 0.0007 0.1868 0.1603 0.0197 0.0020Korea–Brazil 0.0455 0.0422 0.0196 0.0020 0.0455 0.0149 0.0017 0.0002Brazil–Africa 0.1171 0.0479 0.0055 0.0006 0.1171 0.0446 0.0051 0.0005

Joint positive returnsArgentina–Taiwan 0.1276 0.1534 0.0198 0.0020 0.1276 0.0398 0.0045 0.0005Brazil–Mexico 0.1773 0.1258 0.0185 0.0020 0.1773 0.0658 0.0078 0.0008Taiwan–China 0.1474 0.1283 0.0192 0.0020 0.1474 0.0493 0.0057 0.0006



The scatter plot of positive co-exceedances of Brazil and Mexico (not shown here) reveals anasymmetric relationship, as well as an extreme point, ð14:36; 14:26Þ; which may have had a largeinfluence on the estimate 0.55 of rC:Without this point, we obtain brC ¼ 0:31; and the orderingimplied by blU would coincide with the one implied by brC: The CALM copula provided a good fitfor this data, see Figure 2.

All pairs presented lL different from lU; the first type of asymmetry we are looking for. Thismeans that emerging markets relate differently during bear and bull markets. Moreover, allpairs (excepted Taiwan and China, and Argentina and Taiwan) presented lL larger than lU: Thepairs presenting stronger asymptotic dependence during bear and bull markets are Argentinaand Mexico, Argentina and Brazil, and Brazil and Mexico (first three positions). The pairsoccupying the following two positions during bear markets, Argentina and Africa and Taiwanand Africa, are independent with respect to extreme joint gains.

The functions lLðaÞ and lUðaÞ show that the linkage at extreme levels of daily returns may bestrong. For example, the joint gains from Mexico and Africa presented lU ¼ 0:001 (the smallertail dependence among all dependent co-exceedances), but lUð0:20Þ ¼ 0:250: Note theconvergence of lLðaÞ (and lUðaÞ) to their corresponding tail dependence coefficient, as a! 0;including in the case of the asymmetric logistic model copula.

Tables III and IV gather some other results for the asymmetric cases. The results for thesymmetric cases are available from the author on request. First aspect analysed in Table III arethe comparisons of columns 2 and 4, and 3 and 5. The asymmetry reveals that the Korea marketmagnifies what is happening in China during bear markets. This means that Korea is thedirection of faster propagation of crisis. In fact, the probability that Korea falls beyond itsValue-at-Risk with exceedance probability 0:005 given that China has observed its x2;0:01 is0:1426; whereas PrfB2jA1g is approximately half of that, 0:0702: In the case of positive co-exceedances of Taiwan and China, China is the direction of faster propagation, and benefitsmost from the contagion during bull markets.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.19

0.39

0.390.59

0.79

Figure 2. ALM copula fit and empirical copula on positive co-exceedances from Brazil and Mexico.



The asymmetric fits involving China and Korea in the case of joint losses, and China andTaiwan in the case of joint gains, also show that the probability of fall-to-ruin is three timesgreater for Korea (0:1071) than for China (0:0377). The probability of flight-to-quality is greaterfor China (0:1958) than for Taiwan (0:1195).

Still in Table III we examine the sequence of probabilities in columns 2 and 3, and 4 and 5.We observe that the probability of shocks transmission does not increase as the conditioningevent gets more extreme, except, perhaps, for the symmetric cases (not shown in Table III)Argentina and Mexico, and Taiwan and Africa. The stochastic ordering of Reference [30] wasobserved for all pairs and can be seen for the asymmetric cases. For example, in the sym-metric case of Argentina and Mexico, with copula CJC; we obtained PrfA2jB1g ¼ 0:1345 andPrfA2jB2g ¼ 0:6902:

Finally, Table IV shows the probabilities of both markets crashing (booming) together. Forthe symmetric cases the limit probability is independent of the order of the markets, and themost impressive cases were those involving Argentina. Again, the asymmetric fit of Korea andChina provides a good example, with a limit probability of 0.2%, showing again that the Koreamarket amplifies what is happening in China.

4. CONCLUSIONS

This article measured interdependencies during extreme events by fitting copulas to pairwiseobservations collected beyond high thresholds, which were modelled using the modifiedgeneralized Pareto distribution. We briefly reviewed definitions and properties of copulas, andselected a set of five copulas to model the co-exceedances. Our main interest was in copulassuitable for modelling two types of asymmetries. The asymmetric copulas reveal the direction offaster propagation of crisis, and the upper and lower tail dependence coefficients allow fordifferentiating the strength of asymptotic dependence during crisis. We derived the expression ofthe upper tail dependence coefficient for the asymmetric copula associated to the asymmetriclogistic model.

We applied the methodology to daily log returns from the seven most important emergingstock markets. We analysed the 21 pairs of joint excess losses, and joint excess gains. For allpairs we were able to find very good univariate and copula fits, as verified by several goodness offit tests. A small simulation experiment was carried out to verify the bias and variability of themaximum likelihood of copula parameters.

Linkages at (joint negative and positive) extreme levels were assessed by examining thebehaviour of the dependence functions lLðaÞ and lUðaÞ for a decreasing sequence of values a;whose limits are the upper and lower tail dependence coefficients, lL and lU: These measures ofasymptotic non-linear dependence were compared to the Pearson linear correlation coefficientand its conditional version. In many cases we could observe the lack of robustness of thecorrelation coefficient.

We computed alternative definitions of interdependencies and shocks propagation. Ouranalysis may help answering questions raised by risk managers, such as whether or not thecontagion increases as the contaminant market crisis worsens, and if the spillovers aresymmetric across markets. They are not, if the dependence structure is modelled by aasymmetric copula, an interesting example being Korea and China. Typically, dependence is



stronger during bear markets, with some markets, though dependent during crisis, beingindependent during bull markets.

The methodology applied here may be used for selecting stocks for composing portfolios, forquantifying joint credit risks, for risk classification, and so on. Further investigations include theanalysis of the effect of lagged observations.

ACKNOWLEDGEMENTS

The author is thankful to the Editor-in-Chief and to an anonymous referee for their helpful comments andsuggestions that have improved the paper. The author was partially supported by CNPq-Brazil and byConvenio MCT/CNPq/FAPERJ}Edital 2004 Copulas: Permutabilidade, Assimetrias, Inferencia Esta-tıstica.

REFERENCES

1. Lin W-L, Engle RF, Ito T. Do bulls and bears move across borders? International transmission of stock returns andvolatility. The Review of Financial Studies 1994; 7:507–538.

2. Longin F, Solnik B. Is the correlation in international equity returns constant: 1960–1990? Journal of InternationalMoney and Finance 1995; 14:3–26.

3. Ramchmand L, Susmel R. Volatility and cross correlation across major stock markets. Journal of Empirical Finance1998; 5:397–416.

4. Ang A, Chen J. Asymmetric correlations of equity portfolios. Working Paper, G.S.B., Stanford, CA, 2000.5. Embrechts P, McNeil A, Straumann D. Correlation and dependency in risk management: properties and pitfalls.

Risk Management: Value at Risk and Beyond, Dempster M, Moffat HK (eds). Cambridge University Press:Cambridge, 2001.

6. Forbes K, Rigobon R. No contagion, only interdependence: measuring stock market co-movements. WorkingPaper, M.I.T.-Sloan School of Management, 2000.

7. Boyer B, Gibson M, Mulder C. Pitfalls in tests for changes in correlations. International Finance Discussion Paper,597, Board of Governors of the Federal Reserve System, 1999.

8. Straetmans S. Extreme financial returns and their comovements. Erasmus University Rotterdam’s Thesis, TinbergenInstitute Research Series, vol. 181, 1999.

9. Longin F, Solnik B. Extreme correlation of international equity markets. Journal of Finance 2001; 56(2):649–676.10. Malevergne Y, Sornette D. Minimizing extremes. RISK 2002; November, 129–133.11. Schmidt R. Tail dependence for elliptically contoured distributions. Mathematical Methods of Operation Research

2002; 55(2):301–327.12. Ane T, Kharoubi C. Dependence structure and risk measure. Journal of Business 2003; 76(3):411–438.13. Breymann W, Dias A, Embrechts P. Dependence structures for multivariate high-frequency data in finance.

Quantitative Finance 2003; 3(1):1–16.14. Joe H. Multivariate Models and Dependence Concepts. Chapman & Hall: London, 1997.15. Li DX. On default correlation: a copula function approach. Journal of Fixed Income 2000; 9:43–54.16. Embrechts P, Lindskog F, Mc Neil A. Modelling dependence with copulas and applications to risk management. In

Handbook of Hevy Tailed Distributions in Finance. Rachev ST (ed.). Elsevier: Amsterdam, 2003.17. Costinot A, Roncalli T, Teiletche J. Revisiting the dependence between financial markets with copulas. Working

Paper, 2000.18. Nelsen RB. An Introduction to Copulas. Lecture Notes in Statistics, vol. 139. Springer: New York, 1999.19. Bouye E, Durrleman V, Nikeghbali A, Riboulet G, Roncalli T. Copulas for finance, a reading guide and some

applications. Working Paper. City University Business School}Financial Econometrics Research Centre, Londres,2000.

20. Tawn J. Bivariate extreme value theory: models and estimation. Biometrika 1988; 75(3):397–415.21. Sklar A. Fonctions de Repartition a n Dimensions et Leurs Marges. Publications de l’Institut de Statistiquede

l’Universite de Paris, vol. 8, 1959; 229–231.22. Frees EW, Valdez E. Understanding relationships using copulas. North American Actuarial Journal 1998; 2(1):1–25.23. Ghoudi K, Khoudraji A, Rivest LP. Proprietes statistiques des copules de valeurs extremes bidimensionneles.

Canadian Journal of Statistics 1998; 26:187–197.



24. Fisher RA, Tippett LHC. Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of aSample. Proceedings of the Cambridge Philosophical Society, 1928; 24:180–190.

25. Leadbetter M, Lindgren G, Rootz H. Extremes and Related Properties of Random Sequences and Processes.Springer: New York, 1983.

26. Anderson CW, Dancy GP. The severity of extreme events. Research Report 92/593, Department of Probability andStatistic, University of Sheffield, 1992; 24.

27. Coles SG, Heffernan J, Tawn JA. Dependence measures for multivariate extremes. Extremes 1999; 2:339–365.28. Embrechts P, Kluppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. Springer: Berlin,

1997.29. Lehmann EL. Some concepts of dependence. Annals of Mathematical Statistics 1966; 37:1137–1153.30. Coles S. An Introduction to Statistical Modeling of Extreme Values, Springer Series in Statistics. Springer: Berlin,

2001.31. Xu JJ. Statistical modeling and inference for multivariate and longitudinal discrete response data. Ph.D. Thesis,

Department of Statistics, University of British Columbia, 1996.32. Bickel JP, Doksum KA. Mathematical Statistics: Basic Ideas and Selected Topics, Holden-day. Inc: U.S.A., 1977.33. Alexander C. Market Models: A Guide to Financial Data Analysis. Wiley: New York, 2001.34. Deheuvels P. La funcion de dependance empirique et ses proprietes. Un test non parametrique d’ independance.

Academic Royale de Belgique, Bulletin Classe des Sciences 1979; 5(65):247–292.35. Genest C, Rivest L. Statistical inference procedures for bivariate Archimedean copulas. Journal of the American

Statistical Association 1993; 88:1034–1043.36. Fermanian JD, Scaillet O. Some statistical pitfalls in copula modeling for financial applications. Technical Report,

2004.37. Hosking JRM, Wallis JR. Parameter and quantile estimation for the generalized pareto distribution. Technometrics

1987; 29:339–349.38. Castillo E, Hadi AS. Fitting the generalized Pareto distribution to data. Journal of the American Statistical

Association 1997; 92(440):1609–1620.39. Tajvidi N. Characterization and some statistical aspects of univariate and multivariate generalized Pareto

distributions. Ph.D. Thesis. Department of Mathematics, Goteborg, 1996.40. Caperaa P, Fougeres AL, Genest C. A nonparametric estimation procedure for bivariate extreme value copulas.

Biometrika 1997; 84(3):567–577.41. Genest C. Frank’s family of bivariate distributions. Biometrika 1987; 74(3):549–555.



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Asymmetric extreme interdependence in emerging equity markets