Is the Potential for International Diversiﬁcation Disappearing?A ...€¦ · Kris Jacobs...

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Is the Potential for InternationalDiversification Disappearing? A DynamicCopula Approach

Peter ChristoffersenUniversity of Toronto, Copenhagen Business School, and CREATES

Vihang ErrunzaDesautels Faculty of Management, McGill University

Kris JacobsUniversity of Houston and Tilburg University

Hugues LangloisDesautels Faculty of Management, McGill University

International equity markets are characterized by nonlinear dependence and asymmetries.We propose a new dynamic asymmetric copula model to capture long-run and short-rundependence, multivariate nonnormality, and asymmetries in large cross-sections. We findthat correlations have increased markedly in both developed markets (DMs) and emergingmarkets (EMs), but they are much lower in EMs than in DMs. Tail dependence hasalso increased, but its level is still relatively low in EMs. We propose new measuresof dynamic diversification benefits that take into account higher-order moments andnonlinear dependence. The benefits from international diversification have reduced overtime, drastically so for DMs. EMs still offer significant diversification benefits, especiallyduring large market downturns. (JEL G12)

Understanding and quantifying the evolution of security co-movements iscritical for asset pricing and portfolio allocation. The co-movements betweenequity markets in different countries determine how the diversification benefits

Christoffersen and Errunza gratefully acknowledge financial support from Institut de Finance Mathematique deMontreal and Social Sciences and Humanities Research Council of Canada. Errunza is also supported by theBank of Montreal Chair at McGill University, and Jacobs by the Bauer Chair at the University of Houston.Langlois is funded by Natural Sciences and Engineering Research Council of Canada, Centre Interuniversitairede Recherche en Economie Quantitative, and Institut de Finance Mathematique de Montreal. We are grateful tothe editor, Geert Bekaert, as well as two anonymous referees for very helpful comments. We also thank LievenBaele, Greg Bauer, Phelim Boyle, Ines Chaieb, Rob Engle, Frank de Jong, Rene Garcia, Sergei Sarkissian, ErnstSchaumburg, and seminar participants at the Bank of Canada, EDHEC, HEC Montreal, NYU Stern, SUNYBuffalo, Tilburg University, and WLU for helpful comments. Send correspondence to Kris Jacobs, C. T. BauerCollege of Business, University of Houston, 4800 Calhoun Road, Houston, TX 77204; telephone: 713-743-2826;E-mail: [email protected].

© The Author 2012. Published by Oxford University Press on behalf of The Society for Financial Studies.All rights reserved. For Permissions, please e-mail: [email protected]:10.1093/rfs/hhs104 Advance Access publication October 24, 2012

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The Review of Financial Studies / v 25 n 12 2012

of international investing have evolved over time. Measuring these benefitsrequires an answer to three distinct questions.

First, how has cross-country dependence changed through time? Most of theavailable evidence on the time variation in cross-country correlations is basedon factor models.1 Bekaert, Hodrick, and Zhang (2009) convincingly argue thatthe evidence from this literature is mixed at best and state (p. 2591): “It is fairto say that there is no definitive evidence that cross-country correlations aresignificantly and permanently higher now than they were, say, ten years ago.”They investigate international stock return co-movements for 23 DMs during1980–2005, and find an upward trend in return correlations only among thesubsample of European stock markets.

Second, is correlation a satisfactory measure of dependence in internationalmarkets, or do we need to consider different measures, notably those that focuson the dependence between tail events? This question is related to the analysisof correlation asymmetries, and changes in correlation as a function of businesscycle conditions or stock market performance. Following Longin and Solnik(2001), Ang and Bekaert (2002), and Ang and Chen (2002), the hypothesis thatcross-market correlations rise in periods of high volatility has been supplantedby the notion that correlations increase in down markets, but not in up markets.Longin and Solnik (2001) use extreme value theory in bivariate monthly modelsfor the United States with either the United Kingdom, France, Germany, orJapan during 1959–1996. Ang and Bekaert (2002) develop a regime-switchingdynamic asset allocation model, and estimate it for the United States, UK, andGermany over the period 1970–1997. Both papers estimate return extremesat predetermined threshold values, i.e., they define the tail observations exante, and then compute unconditional correlations for the tail for the developedmarkets above.

Third, over the last two decades much of the focus in international financehas shifted to the diversification benefits offered by emerging markets.2 Hence,it is important to investigate whether there are meaningful differences betweenemerging markets (EMs) and developed markets (DMs) in cross-countrydependence and tail dependence. Existing studies analyze tail dependence fora few DMs, and there is limited evidence on time variation in tail dependence.

1 King, Sentana, and Wadhwani (1994) do not find evidence of increasing cross-country correlations for 16developed markets (DMs) during the period 1970–1988, except around the market crash of 1987. Carrieri,Errunza, and Hogan (2007) do not find a common pattern in the correlation trend for eight emerging markets(EMs) during 1977–2000. Eiling and Gerard (2007) find an upward time trend in co-movements between 24DMs but not between 26 EMs over the period 1973–2005. Goetzmann, Li, and Rouwenhorst (2005) documentsubstantial changes in the correlation structure of world equity markets over the past 150 years. Baele andInghelbrecht (2009) report increasing correlations over the period 1973–2007 for their sample of 21 DMs. Seealso Karolyi and Stulz (1996), Forbes and Rigobon (2002), Brooks and Del Negro (2003), Lewis (2006), andRangel (2011).

2 For early studies documenting the benefits of international diversification, see Solnik (1974) for DMs and Errunza(1977) for EMs. For more recent evidence, see, for example, Erb, Harvey, and Viskanta (1994), DeSantis andGerard (1997), Errunza, Hogan, and Hung (1999), and Bekaert and Harvey (2000).

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Is the Potential for International Diversification Disappearing?

Moreover, with some exceptions, most notably the paper by Bekaert, Hodrick,and Zhang (2009), there is little regional analysis.

In this paper, we provide a comprehensive empirical study of the dynamicevolution of dependence and tail dependence for a large set of DMs and EMs, aswell as for regional subsets. We offer two methodological contributions. First,we propose a new model that allows for asymmetries, trends in dependence,and deviations from multivariate normality. In particular, we generalize theflexible dynamic conditional correlation (DCC) model of Engle (2002) andTse and Tsui (2002) in two ways: First, we do not model dependence as mean-reverting to a constant but instead allow it to mean-revert to a possibly nonlineartrend. Second, we do not model linear correlations, which are only sufficientunder multivariate normality; instead, we model the joint distribution usingtime-varying copulas to capture nonlinear dependence across markets. Oursecond methodological contribution is the development of a dynamic measureof diversification benefits that takes into account higher-order moments andnon-linear dependence. We also analyze this measure under the special case ofmultivariate normal returns.

We develop a novel dynamic asymmetric copula (DAC) model that allows forasymmetric and dynamic tail dependence in large portfolios. We implement thismodel relying on recent econometric innovations that overcome dimensionalityproblems, and that facilitate estimation using large numbers of countries andlong time series. Specifically, we rely on the numerically efficient compositelikelihood procedure proposed by Engle, Shephard, and Sheppard (2008). Thecomposite likelihood estimation procedure is essential for estimating dynamicdependence models on international equity data with large cross-sections andlong time series.

Using our new framework, we characterize time-varying dependence usingweekly returns during the 1973–2009 period for a large number of countries(either 13 or 17 EMs, 16 DMs, as well as combinations of the EM andDM samples). We also provide evidence on threshold correlations and otherindicators of asymmetric tail dependence. Our implementation is relativelystraightforward and computationally fast. We thus demonstrate that it is possibleto estimate dependence patterns in international markets using large numbers ofcountries and extensive time series. We extend existing results on dependenceto a more recent period characterized by significant liberalizations for the EMsample, as well as substantial market turmoil during 2007–2009, which helpsidentify tail dependence.

First and foremost, we find extremely robust evidence that internationalnonlinear dependence between stock markets, as measured by copulacorrelations, has been significantly trending upward for both DMs and EMs.3

3 Copula correlations are linear correlations of copula shocks. Copula shocks are nonlinear transformations of returnshocks. The nonlinear transformation depends on the marginal distribution assumed for individual returns, aswell as on the copula function used to link the marginal distributions.

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However, the dependence between DMs has been higher than the dependencebetween EMs at all times in our sample. For DMs, the average dependencewith other DMs is higher than the average dependence with EMs. For EMs,the dependence with DMs is generally somewhat higher than the dependencewith other EMs, but the differences are small. When dividing our sample intofour regions—European Union (EU), developed non-EU, Latin America, andEmerging Eurasia—we find that the dependence between all four regions hasgone up, and so has the average dependence within each region. While therange of dependence for DMs has narrowed around the increasing trend independence levels, this is not the case for EMs.

Second, we find overwhelming evidence that the assumption of multivariatenormality is inappropriate. Our parameter estimates for the dynamic copulamodels indicate substantial tail dependence, which furthermore appears to beasymmetric and increasing through time for both EMs and DMs.

Third, the most striking finding regarding tail dependence is that the levelof tail dependence is still very low at the end of the sample period for EMs ascompared to DMs. These findings on tail dependence suggest that EMs offerdiversification benefits during large market moves, and our new diversificationmeasure confirms that there are substantial benefits to adding EMs to a portfolio.The underlying intuition is that while financial crises in EMs are frequent, manyof them are country-specific.

Fourth, we demonstrate that the new DAC model can capture the empiricalasymmetries in threshold correlations. We document asymmetric thresholdcorrelation patterns for EMs, and find that they differ from those for DMs.Longin and Solnik (2001) and Ang and Bekaert (2002) document asymmetricthreshold correlation patterns among a select group of major DMs, but to thebest of our knowledge the literature does not contain evidence on EMs. Wedemonstrate that our multivariate asymmetric model can capture the thresholdcorrelation patterns observed in DMs and EMs.

Fifth, we use a regression analysis to link the time variation in dependenceto economic fundamentals, market characteristics, and measures of financialopenness. We also investigate the relationship between dependence andvolatility. Our model does not assume a factor structure, but we do find asignificant positive association between copula correlations and volatilities.We find that neither volatility nor other financial and macro variables are ableto drive out the trend in copula correlations.

The paper proceeds as follows. Section 1 introduces the new DAC modelwith dynamic copula correlations, allowing for dynamic tail dependence andasymmetries. We place special emphasis on the estimation of this model forlarge systems. Section 2 presents the data, as well as the empirical results ontime variation in copula correlations. Section 3 introduces a new non-linearconditional measure of diversification benefits that can take into account thenonlinear dependence, asymmetries, and nonnormalities in the DAC model. Wealso discuss empirical estimates of this measure. Section 4 discusses additional

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economic implications, including tail dependence, threshold correlations, anddependence over longer horizons. Section 4 also contains a regression analysisof the economic determinants of the dependence measures. Section 5 concludes.Technical details of the new DAC model are covered in Appendices A–C.

1. Dynamic Dependence Models for Many Equity Markets

This section outlines the general model we use to capture dynamic dependenceacross equity markets. Our dynamic copula approach allows for multivariatenonnormalities, and models copula correlations as reverting to a long-run mean,which consists of a constant as well as a time-varying part. This model feature iscritical to capture dependencies that are potentially trending over time. We alsodescribe how this model can be reliably estimated using a large cross-sectionof assets.

1.1 The dynamic conditional copula approachOur objective is to characterize dependence in a general way using the largestpossible cross-section of international equity markets. In the existing literature,implementations of multivariate GARCH models have traditionally used alimited number of countries because of dimensionality problems.4 Recentmodeling innovations by Engle (2002) and Tse and Tsui (2002), combined withimplementation techniques discussed in Section 1.5 below, make it possible tostudy larger cross-sections of countries.

However, in characterizing international stock market dependence, a crucialissue is the use of the multivariate normal distribution, which is usuallyrelied upon to implement dynamic correlation models. The multivariate normaldistribution is the standard choice in the literature because it is convenient, andbecause quasi maximum likelihood results ensure that the dynamic correlationparameters will be estimated consistently even when the normal distributionassumption is incorrect, as long as the dynamic models are correctly specified.

While the multivariate normal distribution is a convenient statistical choice,the economic motivation for using it is more dubious. It is well known (e.g.,Longin and Solnik 2001; Ang and Bekaert 2002) that international equityreturns display threshold correlations not captured by the normal distribution:Large down moves in international equity markets are highly correlated, whichis of course crucial for assessing the benefits of diversification. The dynamiccorrelation models of Engle (2002) can generate more realistic thresholdcorrelations, but likely not to the degree required by the data. Moreover, theyare symmetric by design, and cannot accommodate Longin and Solnik’s (2001)finding that returns are more correlated in down markets. We therefore gobeyond the dynamic multivariate normal distributions used in Engle (2002)

4 See Kroner and Ng (1998) and Solnik and Roulet (2000) for a more elaborate discussion of the restrictionsimposed in the first generation of multivariate GARCH models.

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and Tse and Tsui (2002) and introduce dynamic copula models that have thepotential to generate empirically relevant levels of threshold correlations, aswell as asymmetric threshold correlations.

Copulas constitute an extremely convenient tool for building a multivariatedistribution for a set of assets from any choice of marginal distributions foreach individual asset.5 From Patton (2006), who relies on Sklar (1959), we candecompose the conditional multivariate density function of a vector of returnsfor N countries, ft (Rt ), into a conditional copula density function, ct , and theproduct of the conditional marginal distributions fi,t (Ri,t )

ft (Rt )=ct (F1,t (R1,t ),F2,t (R2,t ),...,FN,t (RN,t ))N∏i=1

fi,t (Ri,t ),

where Rt is a vector of returns at time t comprised of individual returns R1,t toRN,t , and Fi,t is the cumulative distribution function of Ri,t .

From this, the multivariate log-likelihood function can be constructed as

L=T∑

t=1

N∑i=1

log(fi,t (Ri,t ))+T∑

t=1

log(ct (F1,t (R1,t ),F2,t (R2,t ),...,FN,t (RN,t ))).

The upshot of this decomposition is that we can make assumptions aboutthe marginal densities that are independent of the assumptions made aboutthe copula function. We now discuss the modeling of the marginal densities,and subsequently we address how copula techniques can capture asymmetricthreshold correlations.

1.2 Modeling the marginal densityWhen modeling the marginal density, the critical issues are dynamic volatilityand the modeling of asymmetries. An important stylized fact is that a largenegative shock to an equity market increases volatility by much more than apositive shock of the same magnitude. Black (1976), Engle and Ng (1993), andBekaert and Wu (2000) document this so-called “leverage effect” for a varietyof different markets.

Existing studies usually find that even when including a leverage effect,model residuals continue to be skewed and fat tailed. We therefore allow forasymmetry in the marginal return distribution by modeling a leverage effect,but also by using an asymmetric marginal distribution for the return innovationin each country. To capture both effects as well as volatility persistence andheteroskedasticity, we assume that the return on equity market i at time t follows

5 McNeil, Frey, and Embrechts (2005) provide an authoritative review of the use of constant copulas in riskmanagement.

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an AR(2)-NGARCH(1,1) model of the form 6

Ri,t =μi,t +εi,t =μi,0 +μi,1Ri,t−1 +μi,2Ri,t−2 +σi,t zi,t (1)

σ 2i,t =ωi +αi (εi,t−1 −γiσi,t−1)2 +βiσ

2i,t−1, (2)

where we have used an autoregressive model of order two, AR(2), for eachmarket to pick up autocorrelation in returns. We assume that the distributionsof the innovations differ across assets, but are constant over time and followthe skewed t distribution of Hansen (1994).7 Country i’s residual conditionalskewness is driven by the parameter χ

i, and its degree of conditional kurtosis is

controlled by the parameter i . We write the cumulative distribution function as

ηi,t ≡Fi,t (zi,t )=Fχi,i

(zi,t ). (3)

Note that in our approach the individual return shock distributions are constantthrough time but the individual return distributions do vary through timebecause the return mean and variance are dynamic.

1.3 Allowing for multivariate nonnormality and asymmetryA useful model of international equity returns needs to account fortail dependence and asymmetries in threshold correlations mentioned inSection 1.1, which are well-established empirical facts. The asymmetriesdiscussed in Section 1.2 only address asymmetries in the marginal densities,and not the well-known multivariate asymmetries and asymmetric thresholdcorrelations. We rely on copula models to capture these multivariateasymmetries.

Within the class of copula models, the most widely applied copula functionis based on the multivariate normal distribution and referred to as the normalcopula. Though convenient to use, it is not flexible enough to capture thetail dependence in asset returns. While allowing for tail dependence, themore flexible t copula unfortunately implies symmetric threshold correlations.Asymmetry in the bivariate distribution of asset returns has generally beenmodeled using copulas from the Archimedean family, which include theClayton, the Gumbel, and the Joe-Clayton specifications (e.g., Patton 2004,2006, 2012). However, these models are not easily generalized to high-dimensional applications.

6 The nonlinear NGARCH model is developed in Engle and Ng (1993).

7 Alternatively, one can also use a nonparametric approach for modeling the marginals in copula modeling (e.g.,Chen and Fan 2006). We use Hansen’s (1994) skewed t distribution to ensure that the copula-based multivariatedistribution will be well specified, which allows us to conduct statistical inference, relying on the asymptotictheory discussed in Engle, Shephard, and Sheppard (2008), which requires a parametric approach. We verified thatour parameter estimates are similar to semiparametric estimates that rely on combining the empirical distributionfunction with a generalized Pareto distribution for the tails of the distribution. See McNeil (1999) and McNeiland Frey (2000) for more detail on this approach.

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We therefore consider the skewed t distribution discussed in ? and use theimplied skewed t copula whose cumulative distribution function, Ct , is givenby

Ct (η1,t ,η2,t ,...,ηN,t ;�,λ,ν)= t�,λ,ν(t−1λ,ν(η1,t ),t

−1λ,ν(η2,t ),...,t

−1λ,ν(ηN,t )),

where λ is an asymmetry parameter, ν is the degree of freedom parameter,t�,λ,ν is the multivariate skewed Student’s t density with correlation matrix �,and t−1

λ,ν is the inverse cumulative distribution function of the univariate skewedt distribution.

Note that the copula correlation matrix � is defined using the correlation ofthe copula shocks z∗

i,t ≡ t−1λ,ν(ηi,t ) and not of the return shocks zi,t . Notice also

that

z∗i,t ≡ t−1

λ,ν(ηi,t )= t−1λ,ν(Fχ

i,i

(zi,t )),

so that if the marginal distributions Fχi,i

are close to the tλ,ν distribution,then z∗

i,t will be close to zi,t . The skewed t copula is built from the skewedmultivariate t distribution and the symmetric t copula is nested when λ tends tozero. If the degrees of freedom tend to infinity, the normal copula is obtained.Appendix A provides the details needed to implement the skewed t copula.Note that the marginal model in Section 1.2 captures any univariate skewnesspresent in the equity returns. The λ parameter captures multivariate asymmetry.

The skewed t copula is parsimonious, tractable in high dimensions, andflexible, allowing us to model non-linear and asymmetric dependence withthe degree of freedom parameter, ν, and the asymmetry parameter, λ, whileretaining a copula correlation matrix, �, which can be made time-varying, as wewill see in the next section. For the sake of parsimony in our high-dimensionalapplications, we report on a version of the skewed t copula where the asymmetryparameter λ is a scalar. It is straightforward to develop a more general versionof the skewed t copula allowing for an N -dimensional vector of asymmetryparameters, but it is difficult to estimate such a model on a large number ofcountries.

1.4 Dynamic copula correlationsWe now build on the linear correlation techniques developed by Engle (2002)and Tse and Tsui (2002) to model dynamic copula correlations. As in thestandard dynamic conditional correlation (DCC) model, dynamic correlationsare driven by a multivariate GARCH process. However, the copula shocksz∗i,t ≡ t−1

λ,ν(ηi,t ) are used as the model’s building block instead of the returnshocks zi,t .8

8 Unless one uses the normal copula, the fractiles do not have zero mean and unit variance. We therefore standardizethe z∗

iin the dynamic copula dependence model.

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The copula correlation dynamic is driven by the trend matrix ϒt and thecross-products of the copula shocks:

�t =(1−β� −α�)[(1−ϕ�)�+ϕ�ϒt ]+β��t−1 +α�z∗t−1z

∗�t−1, (4)

where β� , α� , and ϕ� are scalars, and z∗t is an N -dimensional vector with typical

element z∗i,t =z∗

i,t

√�ii,t . 9 The conditional copula correlations are defined via

the normalization�ij,t =�ij,t /

√�ii,t�jj,t .

This normalization ensures that all copula correlations remain in the −1 to 1interval.

Note that the dynamic conditional copula correlation matrix mean-reverts attime t to a weighted average of a constant � and a time-varying matrix ϒt withweighting parameter ϕ� . We refer to this model as the dynamic asymmetriccopula (DAC) model. Its components are as follows.

First, � is a constant copula correlation matrix. Therefore, by setting ϕ� to 0,we obtain the DCC approach of Engle (2002) as applied to copula correlations.Second, the matrix ϒt contains information about time trends and explanatoryvariables. While this matrix can take on a very general form, as described inAppendix B, we consider here a version with only a time trend. For simplicity,we further constrain the time trend parameters to be equal across all developedmarkets and also across all emerging markets. The resulting matrix ϒt is givenby

ϒt =

[ϒDM,t ϒDM,EM,t

ϒ�DM,EM,t ϒEM,t

], (5)

where ϒDM,t is an NDMxNDM correlation matrix with all off-diagonal elements

equal toδ2DM

t2

1+δ2DM

t2 , ϒEM,t is an NEMxNEM correlation matrix with all off-

diagonal elements equal toδ2EM

t2

1+δ2EM

t2 , and ϒDM,EM,t is an NDMxNEM matrix

with all elements equal to δDMδEMt2

1+δDMδEMt2 , where δDM and δEM are parameters to

be estimated.10

The conditional copula correlation matrix can be seen as the weightedaverage of a slowly varying component, (1−ϕ�)�+ϕ�ϒt , the laggedconditional correlation matrix, �t−1, and the lagged cross-product of thestandardized copula shocks, z∗

t−1z∗�t−1. Note that a negative pair of copula shocks

impacts correlation in the same way as does a positive pair of copula shocksof the same magnitude. We also investigate an alternative specification using

9 The transformation from z∗i,t

to z∗i,t

is motivated by Aielli (2009), who shows that it ensures a consistent estimate

of � when using moment matching. Moment matching is imperative in high-dimensional applications, and wediscuss it in detail in Appendix C.

10 The parameterization of our trend correlation matrix is motivated by the approach in Marsaglia and Olkin (1984).

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the cross-products of the recentered copula shocks,(z∗t−1 −ξ�ι

)(z∗t−1 −ξ�ι

)�,

where ξ� is a scalar and ι is an N -dimensional vector of ones. This specificationintroduces additional multivariate asymmetry into the model. However, ourresults show that the estimates of parameter ξ� are not statistically significant,and the more general model does not yield a better fit than the model in (4).We therefore conclude that the skewed t distribution adequately captures themultivariate asymmetries in the data.

1.5 EstimationIf N denotes the number of equity markets under study, then the DAC model hasN (N −1)/2+7 parameters to be estimated. Below, we will study up to 17 EMsand 16 DMs, thus N =33 and so the DAC model will have 535 parameters. Itis well recognized in the literature that it is impossible to estimate so manyparameters reliably using numerical optimization techniques.11 In order tooperationalize estimation, we use the average level of the copula correlationsover the entire sample to fix the time-invariant parameters:

�=1T

∑Tt=1 z

∗t z

∗�t −ϕ�

1T

∑Tt=1ϒt

1−ϕ�

. (6)

The numerical optimizer now only has to search in seven dimensionscorresponding to the parameter vector θ =(α�,β�,ϕ�,δDM,δEM,ν,λ)�, ratherthan in the original 535 dimensions. Note that this implementation also ensuresthat the estimated DAC model yields a positive semi-definite correlation matrix,because z∗

t z∗�t and thus � is positive semi-definite by construction. Appendix

C contains more details on the estimation of � in the DAC model.Even in parsimonious models estimation is cumbersome with many assets

due to the need to invert the NxN correlation matrix, �t , for every observationin the sample. Furthermore, the likelihood must be evaluated many times inthe numerical optimization routine. More importantly, Engle, Shephard, andSheppard (2008) find that in large-scale estimation problems, the parametersα� and β� that drive the correlation dynamics are estimated with bias whenusing conventional estimation techniques. They propose an ingenious solutionbased on the composite likelihood which, in our context, is defined as

CL(θ )=T∑

t=1

N∑i=1

∑j>i

lnct (ηi,t ,ηj,t ;θ ), (7)

where ct (ηi,t ,ηj,t ;θ ) denotes the bivariate copula distribution of asset pair i

and j .The composite log-likelihood is thus based on summing the log-likelihoods

of pairs of assets. Each pair yields a valid (but inefficient) likelihood for θ ,

11 See, for instance, Solnik and Roulet (2000) for a discussion.

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but summing over all pairs produces an estimator that is relatively efficient,numerically fast, and free of bias even in large-scale problems. We use thecomposite log-likelihood in all our estimations below. We have found it to bevery reliable and robust, effectively turning a numerically impossible task intoa manageable one. The composite likelihood procedure allows us to estimatedynamic copula correlations in larger systems of international equity data usinglonger time series of returns than previously reported in the literature. This isimportant because long time series on large sets of countries are needed for theidentification of variance and covariance dynamics.

2. Empirical Dependence Analysis

This section contains our empirical findings on dependence patterns. We firstdescribe the different data sets and briefly discuss the univariate results. Wethen analyze the time variation in copula correlations and dispersion in copulacorrelations across pairs of assets at each point in time and check if thisdispersion has changed over time.

2.1 Data and univariate modelsWe employ three data sets. First, from DataStream we collect weekly closingU.S. dollar returns for 16 DMs: Australia, Austria, Belgium, Canada, Denmark,France, Germany, Hong Kong, Ireland, Italy, Japan, Netherlands, Singapore,Switzerland, UK, and United States. This data set contains 1,901 weeklyobservations from January 12, 1973, through June 12, 2009.

Second, from Standard and Poor’s we collect the IFCG weekly closing U.S.dollar returns for 13 EMs: Argentina, Brazil, Chile, Colombia, India, Jordan,Korea, Malaysia, Mexico, Philippines, Taiwan, Thailand, and Turkey. This dataset contains 1,021 weekly observations from January 6, 1989, through July 25,2008.

Third, from Standard and Poor’s we collect the weekly closing investableIFCI U.S. dollar returns for 17 EMs: Argentina, Brazil, Chile, China, Hungary,India, Indonesia, Korea, Malaysia, Mexico, Peru, Philippines, Poland, SouthAfrica, Taiwan, Thailand, and Turkey. This data set contains 728 weekly returnsfrom July 7, 1995, through June 12, 2009.

We use two EM data sets because each has distinct advantages. The IFCG dataset spans a longer time period, and represents a broad measure of EM returns,but is not available after July 25, 2008. The IFCI data set tracks returns on aportfolio of EM securities that are legally and practically available to foreigninvestors. The index construction takes into account portfolio flow restrictions,liquidity, size, and float. It continues to be updated, but the sample period isshorter, which is a disadvantage in model estimation and of course in assessinglong-term trends in correlation.

The first step in our empirical analysis is to estimate the univariate skewed t

AR(2)-NGARCH(1,1) model in (1)–(2) for each country and perform various

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model diagnostics. Our results show that the skewed t AR(2)-NGARCH(1,1)model is successful in delivering white-noise residuals, which are requiredto obtain unbiased estimates of the dynamic copula correlations below. Theunivariate model estimates and diagnostics are fairly standard, and are availablefrom the authors upon request. On average, EMs have higher expected returns,standard deviations, and kurtosis. Skewness is slightly positive in EMs andslightly negative in DMs.

2.2 Copula correlation patterns over timeTable 1 reports the parameter estimates and composite log-likelihood valuesfor the DAC model. We report results for the three data sets introduced above.For each set of countries we estimate two versions of each model: oneversion allowing for full copula correlation dynamics, and another labeled“Trend Only” where the stochastic copula correlation dynamics are shut down,and thus α� =β� =0. A conventional likelihood ratio test would suggest thatthe last model is rejected for all sets of countries, but unfortunately thestandard chi-squared asymptotics are not available for composite likelihoods.The dependence persistence (α� +β�) is close to one in all models, implyingvery slow mean-reversion in copula correlations, despite the presence of adeterministic trend.

Table 1 also reports standard errors for all parameter estimates using thetechnique in Engle, Shephard, and Sheppard (2008).12 They also show thatthe composite likelihood parameter estimates are asymptotically normallydistributed, which enables us to test for parameter significance. Rather thanreporting the copula degree of freedom parameter ν, we report its inverse,which is zero in the benchmark normal copula. Using the argument in Patton(2012), we can conduct a one-sided test against the null hypothesis that 1/ν =0.Table 1 shows that 1/ν is significantly greater than zero in all 14 cases. Thecopula asymmetry parameter λ is significant in 11 of 14 cases, thus typicallyrejecting the symmetric t copula in favor of the skewed t copula. All estimatesof α� and β� are significant.

The total increase in long-run average correlation depends on the time trendparameters, δDM and δEM , as well as on the weighting parameter ϕ� . We reportthe increase in copula correlation for DMs over the sample that is due to thetime trend component using (4) and (5) as follows:

{ϕ�ϒDM,t

}t=T

−{ϕ�ϒDM,t

}t=0 =ϕ�

δ2DMT 2

1+δ2DMT 2

=ϕ�ϒDM,T .

12 Asymptotic standard errors are computed using each bivariate element of the composite log-likelihood inEquation (7). The standard errors are a function of the gradient and the Hessian of the copula parameters,the gradient of the moment estimator in Equation (6), the gradient of the AR-NGARCH models, and theircross-derivatives. See Engle, Shephard, and Sheppard (2008) for further detail.

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Table 1Parameter estimates for dynamic asymmetric t copula (DAC) models

Compositeα� β� Persistence ϕ�ϒDM,T ϕ�ϒEM,T ν−1 λ Likelihood

A: Weekly Returns, January 26, 1973 to June 12, 2009

16 Developed 0.026∗∗ 0.963∗∗ 0.988 0.3877∗ – 0.0658∗∗ −0.38∗∗ 31,349Markets (0.004) (0.008) (0.1950) (0.0106) (0.12)

Trend Only – – – 0.3026∗∗ – 0.1185∗∗ −0.33∗∗ 28,631(0.0392) (0.0000) (0.11)

B: Weekly IFCG Returns, January 20, 1989 to July 25, 2008

16 Developed 0.030∗∗ 0.954∗∗ 0.984 0.3018∗ – 0.0567∗∗ −0.48∗∗ 22,131Markets (0.008) (0.026) (0.1245) (0.0003) (0.10)

Trend Only – – – 0.2522∗∗ – 0.0927∗∗ −0.26 20,872(0.0526) (0.0001) (0.15)

13 Emerging 0.019∗∗ 0.935∗∗ 0.953 – 0.3743∗∗ 0.0447∗∗ −0.49∗ 3,192Markets (0.005) (0.030) (0.0553) (0.0000) (0.22)

Trend Only – – – – 0.3765∗∗ 0.0512∗∗ −0.30∗∗ 3,061(0.0334) (0.0040) (0.09)

All 29 0.026∗∗ 0.937∗∗ 0.963 0.2862∗∗ 0.3009∗∗ 0.0458∗∗ −0.41∗∗ 36,390Markets (0.007) (0.041) (0.0557) (0.0708) (0.0000) (0.14)

Trend Only – – – 0.2620∗∗ 0.2628∗∗ 0.0531∗∗ −0.50∗∗ 34,248(0.0184) (0.0260) (0.0004) (0.00)

C: Weekly IFCI Returns, July 21, 1995 to June 12, 2009

16 Developed 0.040∗∗ 0.925∗∗ 0.965 0.2679∗ – 0.0537∗∗ −0.56 21,213Markets (0.009) (0.018) (0.1206) (0.0000) (0.40)

Trend Only – – – 0.2829∗∗ – 0.0616∗∗ −0.73∗∗ 20,535(0.0895) (0.0118) (0.05)

17 Emerging 0.025∗∗ 0.920∗∗ 0.945 – 0.2891∗∗ 0.0781∗∗ −0.45∗∗ 9,181Markets (0.006) (0.025) (0.0537) (0.0001) (0.10)

Trend Only – – – – 0.3170∗∗ 0.0521∗∗ −0.45 8,844(0.0458) (0.0004) (0.27)

All 33 0.026∗ 0.924∗∗ 0.950 0.3200∗∗ 0.2985∗∗ 0.0823∗∗ −0.40∗∗ 52,143Markets (0.011) (0.101) (0.0551) (0.1571) (0.0003) (0.11)

Trend Only – – – 0.3814∗∗ 0.2890∗∗ 0.0726∗∗ −0.61∗∗ 50,593(0.0814) (0.0944) (0.0137) (0.09)

We report parameter estimates for the DAC models for the 16 developed markets, 13 emerging markets (IFCG),17 emerging markets (IFCI), and all markets for different sample periods. We also report the special case ofconstant copula correlation (β�=0 and α�=0). The standard errors in parentheses are computed using the methodin Engle, Shephard, and Sheppard (2008). Significance at the 5% and 1% levels is denoted by * and **.

The expression for EMs is similar. The increases in long-run copula correlationranges from 0.25 to 0.39 for DMs and 0.29 to 0.38 for EMs and are all positiveand always significant, as reported in Table 1.13

Figure 1 presents time series of averages of the dynamic copula correlationsacross countries for several samples. The left panels in Figure 1 present resultsfor 29 DMs and EMs for the sample period from January 20, 1989, to July

13 We compute asymptotic standard errors of the correlation trend increase using the delta method.

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Figure 1Average dynamic copula correlations for developed, emerging, and all marketsWe report dynamic copula correlations, long-run copula correlations, and constant copula correlation averagedacross countries, along with a 90% bootstrap confidence interval around the constant correlation. The copulacorrelations are computed from the DAC model.

25, 2008. We refer to this sample as the 1989–2008 sample. As explained inSection 2.1, 16 of these markets are DMs and 13 are EMs.

The right panels in Figure 1 present results for 33 DMs and EMs for thesample period July 21, 1995, to June 12, 2009. This sample contains the same16 DMs and 17 EMs. There is considerable overlap between this sample ofEMs and the one used in the left panels of Figure 1. Section 2.1 discusses thedifferences. We refer to this sample as the 1995–2009 sample.

The top panel in Figure 2 contains results for the group of 16 DMs betweenJanuary 26, 1973, and June 12, 2009. We refer to this sample as the 1973–2009sample. Figure 2 also shows results for the 1989–2008 and the 1995–2009 datafor comparison.

Figures 1 and 2 contain some of the main messages of our paper. The dynamiccopula correlations fluctuate considerably from year to year, but have been onan upward trend since the beginning of the sample. Figure 2 shows that for the16 DMs, the average copula correlation increased from approximately 0.2 in

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Figure 2Average dynamic copula correlations for developed marketsWe report dynamic copula correlations, long run copula correlations and constant copula correlation averagedacross developed markets, along with a 90% bootstrap confidence interval. The copula correlations are computedfrom the DAC model.

the mid-1970s to around 0.8 in 2009. Figure 1 indicates that over the 1989–2009period, the copula correlations between EMs are lower than those between DMs,but that they have also been trending upward, from approximately 0.1–0.2 in theearly 1990s to over 0.5 in 2009. Figure 1 also indicates that the model-impliedtrend, indicated by the dashed line, is roughly linear for the EMs, while theincrease in copula correlations has somewhat slowed in recent years for theDMs.

When estimated on all markets, recall that our DAC model has two differenttime trend parameters; one for DMs and one for EMs. The last row in Figure 1presents the resulting time trend for cross-correlation between DMs and EMs,which depends on these two parameters. While the first three rows of graphsindicate that average correlations have increased within DMs, within EMs

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and on average across all markets, the bottom graphs confirm that averagecorrelations between DMs and EMs have also increased.

To illustrate further the importance of the evolution of the general levelof dependence, we also report in Figures 1 and 2 the average constant copulacorrelation estimated on each sample, along with bootstrapped 90% confidenceintervals (dash-dot line and gray area).14 In all cases, both the dynamic and thelong-run copula correlations are significantly lower than the constant copulacorrelation at the start of the sample, and higher at the end.

Because the model allows for dynamic copula correlations with a long-run trend, one may wonder whether the choice of sample period stronglyaffects inference on dependence estimates at a particular point in time. Figure 2addresses this issue by reporting estimates for the 16 DMs for three differentsample periods. Whereas there are some differences, the copula correlationestimate at a particular point in time is remarkably robust to the sampleperiod used, and the conclusion that copula correlations have been trendingupward clearly does not depend on the sample period used. Comparing theleft and right panels of Figure 1, it can be seen that a similar conclusionobtains for EMs, even though this comparison is more tenuous, as the samplecomposition and the return data used for the EMs are somewhat different acrosspanels.

2.3 Cross-sectional differences in copula correlationsThe average copula correlations indicate that dependence has increased overour sample. The next question is how much cross-sectional heterogeneitythere is in the copula correlations, and if the increases in dependence aredifferent across countries and regions. Reporting on all the time-varyingpairwise copula correlation paths is not feasible, and we have to aggregatethe correlation information in some way. Figures 3–5 provide an overview ofthe results.15

Figure 3 uses the 1989–2008 sample to report, for each of the 29 countriesin the sample, the average of its copula correlations with all other countriesusing light gray lines. Figure 3A contains the 16 DMs, and Figure 3B containsthe 13 EMs. While these paths are averages, they give a good indication of thedifferences between individual countries, and they are also informative of thedifferences between DMs and EMs. In order to further study these differences,each figure also gives the average of the market’s copula correlations with

14 In the bootstrap we generate 10,000 samples by randomly drawing with replacement from ηt defined inEquation (3). For each bootstrap sample, we compute the average pairwise copula correlation. Using the 10,000average copula correlations, we then form a 90% confidence interval around the constant copula correlation.Alternatively, we can construct confidence intervals around the time-varying copula correlations, but this iscomputationally much more expensive.

15 Throughout the paper, we report equal-weighted averages of the pairwise copula correlations. Value-weightedcorrelations (not reported here) also display an increasing pattern over our samples.

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Figure 3ACopula correlations for each developed marketWe report dynamic copula correlations for 16 developed markets for the period January 20, 1989, to July 21, 2008.For each country, at each point in time we report the average copula correlations with the 15 other developedmarkets (black line), with the 13 emerging markets (dark gray line), and with all the 28 other markets (light grayline). The copula correlations are computed from the DAC model.

all (other) DMs using black lines and all (other) EMs using dark gray lines.Figures 3A and 3B yield some very interesting conclusions. First, the copulacorrelation paths display an upward trend for all 29 countries, except perhapsJordan. Second, for DMs the average copula correlation with other DMs ishigher than the average copula correlation with EMs at virtually each point intime for virtually all markets. Third, for EMs the copula correlation with DMsis generally higher than the copula correlation with other EMs. However, thedifference between the two copula correlation paths is much smaller than in thecase of DMs, and in several cases the average paths are very similar. Note that inFigure 3A the trend patterns for European countries are also not very differentfrom those for other DMs. Notice that, even if their level is still somewhatlower, the correlations for Japan and the United States have increased just asfor the European countries during the last decade. Inspection of the pairwiseDAC paths, which are not reported because of space constraints, reveals thatthe trend patterns are remarkably consistent for almost all pairs of countries,

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Figure 3BCopula correlations for each emerging marketWe report dynamic copula correlations for 13 emerging markets for the period January 20, 1989, to July 25, 2008.For each country, at each point in time we report the average copula correlations with 16 developed markets(black line), with the 12 other emerging markets (dark gray line), and with all the 28 other markets (light grayline). The copula correlations are computed from the DAC model.

and there is no meaningful difference between European countries and otherDMs.

Figure 3 reports the averages of the copula correlations between each marketand all other markets. It could be argued that instead the correlation betweeneach market and a portfolio of the other markets ought to be considered. Wehave computed these correlations as well, but we do not show them in order tosave space. While the correlation with the aggregate portfolio return is nearlyalways higher than the average correlation from Figure 3, the conclusion thatthe correlations are trending upward is not affected.

Figure 4 uses the correlation paths from the DAC model to assess regionalpatterns in correlation dynamics. We divide the 16 DMs into two regions (EUand non-EU) and the 13 EMs into another two EM regions: Latin America and

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Figure 4Regional copula correlation patternsWe use the DAC model to plot the average copula correlation within and across four regions. We also plotthe constant copula correlation along with a 90% bootstrap confidence interval (dashed line in gray area). TheEuropean Union (EU) includesAustria, Belgium, Denmark, France, Germany, Ireland, Italy, Netherlands, and theUK. Developed Non-EU includes Australia, Canada, Hong Kong, Japan, Singapore, Switzerland, and the UnitedStates. LatinAmerica includesArgentina, Brazil, Chile, Colombia, and Mexico. Emerging Eurasia includes India,Jordan, Korea, Malaysia, Philippines, Taiwan, Thailand, and Turkey.

Emerging Eurasia.16 We report in Figure 4 the average copula correlation withinand across the four regions, based on the model’s country-specific correlationpaths. Strikingly, Figure 4 shows that the increasing dependence patterns areevident within each of the four regions and also across all the six possible pairsof regions. The highest levels of copula correlations are found in the upper-left panel, which shows the intra-EU copula correlations. The lowest levelsare found in the bottom-right panel, which shows the intra Emerging Eurasiacopula correlations. Emerging Eurasia in the right-most column generally hasthe lowest interregional copula correlations.

Figures 3 and 4 do not tell the entire story, because we have toresort to reporting copula correlation averages due to space constraints.Figure 5 provides additional perspective by providing copula correlationdispersions for the DMs, EMs, and all markets, respectively. In particular, at

16 The European Union (EU) includes Austria, Belgium, Denmark, France, Germany, Ireland, Italy, Netherlands,and the UK. Developed Non-EU includes Australia, Canada, Hong Kong, Japan, Singapore, Switzerland, andthe United States. Latin America includes Argentina, Brazil, Chile, Colombia, and Mexico. Emerging Eurasiaincludes India, Jordan, Korea, Malaysia, Philippines, Taiwan, Thailand, and Turkey.

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Figure 5Copula correlation range (90th and 10th percentiles)The shaded areas show the DAC copula correlation range between the 90th and 10th percentiles.

each point in time, the shaded areas in Figure 5 show the range between the 10thand 90th percentiles based on all pairwise copula correlations between groups ofcountries. The top panel considers the 16 developed countries. The middle panelin Figure 5 reports the same statistics for the EMs for the 1989–2008 sample,and the bottom panel shows all 29 markets together. While the increasing levelof dependence is evident, the range seems to have narrowed for DMs, wideneda bit for EMs, and the range width seems to have stayed roughly constant forall markets taken together.

3. Conditional Diversification Benefits

If the level of dependence is changing over time, then the benefits of portfoliodiversification are likely changing as well. We therefore need to develop adynamic measure of diversification benefits that takes into account higher-ordermoments and nonlinear dependence. Motivated by the analysis in Basak andShapiro (2001), we develop a dynamic measure based on expected shortfall.

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We first discuss this measure in detail. We then provide more intuition byconsidering this measure under the special case of multivariate normality, andwe report our empirical estimates.

3.1 A conditional measure of diversification benefitsOur approach is based on the expected shortfall measure defined as

ESqt (Ri,t )=−E

[Ri,t |Ri,t ≤F−1

i,t (q)],

where F−1i,t (q) is the inverse cumulative distribution function of asset i at time t ,

and q is a probability commonly set to 5% or 1%.Artzner et al. (1999) show that, unlike value-at-risk, expected shortfall

satisfies the sub-additivity property so that

ESqt (wt )≤

N∑i=1

wi,tESqt (Ri,t ), for all wt,

where ESqt (wt ) is the expected shortfall for a portfolio with weights wt .

Consequently, the weighted average of the assets’individual expected shortfallsconstitutes an upper bound on the portfolio ES:

ESq

t (wt )≡N∑i=1

wi,tESqt (Ri,t ).

This corresponds to the case of no diversification benefits. A lower bound onexpected shortfall is

ESqt (wt )≡−F−1

t (wt,q),

where F−1t (wt,q) is the inverse cumulative distribution function for the

portfolio with weights wt . This lower bound corresponds to the extreme casewhere the portfolio never loses more than its qth distribution quantile.

Using these two extreme cases, we define the conditional diversificationbenefit (CDB) measure by

CDBt (wt,q)≡ ESq

t (wt )−ESqt (wt )

ESq

t (wt )−ESqt (wt )

, (8)

where ESqt (wt ) denotes the expected shortfall of the portfolio at hand. This

measure is designed to take values on the [0,1] interval, and is increasing in thelevel of diversification benefit. Note also that it does not depend on the levelof expected returns.

Below we report on the case where q is 5% and we choose wt to maximizeCDBt (wt,q) subject to the weights being non-negative and summing to one.17

The ESqt (wt ) measure is not available in closed form for our DAC model and

so we need the following steps for computing CDBt (wt,q):

17 The results for different values of q are qualitatively similar and available from the authors upon request.

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• Use the multivariate DAC model including the univariate skewed t AR-NGARCH to simulate 10,000 returns for each asset in week t conditionalon returns, volatility, and correlations available at the end of week t −1.

• Maximize CDBt (wt,q) over wt subject to the weights being non-negative and summing to one. For each candidate wt chosen by thenumerical optimizer, we compute ES

q

t (wt ), ESqt (wt ), and ES

qt (wt ) using

the 10,000 simulated returns.• Report the CDBt (wt,q) value corresponding to the optimal set of

weights. Proceed to simulation and optimization for week t +1.

To assess how much of the conditional diversification benefit derives fromactive asset allocation, we also construct a CDBEW

t measure for an equal-weighted portfolio where wi,t =1/N for all i. By definition CDBEW

t will beless than or equal to its optimal counterpart CDBt at any point in time. Thedifference between these two measures indicates whether or not equal-weightedportfolios are close to optimal, and measures to what extent changing volatilitiesand dependence can be exploited via dynamic asset allocation.

3.2 The special case of normalityOur CDBt measure is designed to capture nonnormalities and simplifies greatlywhen returns are multivariate normally distributed. Under normality, we have

ESq =−μ+σφ(�−1(q)

)q

, (9)

where μ and σ are the mean and standard deviation of returns, and φ and� are the standard normal density and cumulative distribution functions,respectively.18 Substituting (9) into Equation (8), we obtain

CDBt (wt,q)=w�

t σt −σP,t

w�t σt +σP,t

�−1(q)qφ(�−1(q))

, (10)

where σt denotes the vector of individual asset volatilities at time t and σP,t

is portfolio volatility. Note that in the special case of q =50%, the measurereduces to

VolCDBt (wt )≡CDBt (wt,0.5)=w�

t σt −σP,t

w�t σt

=1−√

w�t �twt

w�t σt

, (11)

where �t denotes the variance-covariance matrix of returns.19 The denominatorw�

t σt represents portfolio volatility in the extreme case of no diversification

18 In copula models, the implied univariate portfolio returns generally do not have a location-scale structure. Weare grateful to a referee for pointing this out.

19 Ferreira and Gama (2005) study related measures in the context of international factor models with dynamicvolatility.

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Figure 6Conditional diversification benefits (CDB) using the DAC modelEach week and for each set of countries, we use the DAC and AR-NGARCH models to compute the conditionaldiversification benefit (CDB) in Equation (8) in the left panels. In the right panels we use the volatility-basedmeasure conditional diversification benefit (VolCDB) measure from Equation (11). The superscript “EW” refersto equal-weighted portfolios. We also plot the 90% bootstrap confidence interval around constant diversificationbenefits (dash-dot line in gray area).

benefits, and VolCDBt measures conditional diversification benefits as thedegree of portfolio volatility reduction from this upper bound. We reportVolCDB estimates in our empirical analysis below, along with the more generalCDB estimates.

3.3 Diversification benefits

Figure 6 plots the conditional diversification benefit measures developed inEquations (8) and (11) for developed, emerging, and all markets, evaluatedusing our estimates of the DAC model. The left-hand panels present resultsfor the general measure in (8), and the right-hand panels present results for thespecial case in (10), where only the volatilities and linear correlations from themodel are used.

Figure 6 includes 90% bootstrapped confidence bands computed arounda constant level of diversification benefits, which is computed assuming thereturns data are independently distributed over time. The bootstrap proceeds

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as follows:

• Generate a sample of T returns drawn randomly with replacement fromthe historical return data set.

• Compute the portfolio weights that maximize the constant CDB(w,q).• Store the CDB(w,q) value corresponding to the optimal weights, draw

a new random sample of returns, and repeat 10,000 times.• Report the 5% and 95% quantiles of the bootstrapped distribution of

CDB(w,q) (gray area in Figure 6), as well as its average value (dash-dotline in Figure 6).

The top-left panel of Figure 6 shows a significantly decreasing trend indiversification benefits in DMs for both the optimal (black) and equal-weighted(gray) allocations: Dependence has been rising rapidly and the benefits ofdiversification have been decreasing during the last ten years. Diversificationbenefits have also decreased significantly in EMs (middle left panel), but thelevel of benefit is still higher than in DMs. When combining the DMs and EMs(bottom left panel), the diversification benefits are declining as well but thelevel is again much higher than when considering DMs alone. EMs thus offersubstantial diversification benefits to investors.

In the case of the volatility-based measure in the right-side panels in Figure 6,similar conclusions obtain, but there are some interesting differences. First, theconfidence intervals for VolCDB are much tighter than for CDB due to increaseduncertainty from computing tail expectations for CDB. The declines in VolCDBare therefore clearly significant. Second, compared to the general CDB case,the declining VolCDB trend seems more pronounced for EMs. We thereforeconclude that the diversification benefits offered by EMs are partly due to thediversification of large market downturns.

For both the general CDB and the VolCDB measures, the differences betweenthe optimal and the equally weighted portfolio are nonzero, but not verylarge. The differences are somewhat larger for the volatility-based measurein the right panels and largest when considering EMs and DMs jointly inthe bottom-right panel. Overall, the relatively modest differences betweenoptimal and equal-weighted diversification benefits suggest that the “1/N”style portfolios recently advocated in a normal setting may work relativelywell in our nonnormal context as well.20

4. Further Economic Implications

The declining diversification benefits constitute an important implication ofthe rising dependence across international equity markets. We now exploresome further economic implications of our empirical estimates. We document

20 In the multivariate normal case, DeMiguel, Garlappi, and Uppal (2009) and Tu and Zhou (2011) analyze therelative performance of equal-weighted versus optimally weighted portfolios in an unconditional setting.

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the models’ implications for tail dependence, and investigate if the modelscan match well-known stylized facts regarding threshold correlations. We alsoderive the implications of our estimates for copula correlations over longerhorizons. Finally, we use regression analysis to assess how equity marketcorrelations are related to market openness, as well as to financial and macrovariables more generally.

4.1 Tail dependenceTables 2 and 3 present additional estimation results for two nested copulamodels: the dynamic symmetric t copula (DSC) and the dynamic normal copula(DNC). The significance of 1/v in Table 2 shows that the normal copula isstrongly rejected in favor of the symmetric t copula. Note that the copulacorrelation persistence is, as was the case in Table 1, very close to one in allmodels. To further illustrate the differences between the DAC model and thetwo nested cases, we now discuss model-implied tail dependence coefficients,while the next section discusses threshold correlations.

The DAC model generalizes the DNC model by allowing for non-zerodependence in the tails, and it generalizes the DSC model by allowing fordifferent dependence in the lower and upper tails. One way to measure thelower tail dependence is via the probability limit:

τLi,j,t = lim

ζ→0Pr[ηi,t ≤ζ |ηj,t ≤ζ ]= lim

ζ→0

Ct (ζ,ζ )

ζ, (12)

where ζ is the tail probability. The upper tail dependence at time t can similarlybe defined by

τUi,j,t = lim

ζ→1Pr[ηi,t ≥ζ |ηj,t ≥ζ ]= lim

ζ→1

1−2ζ +Ct (ζ,ζ )

1−ζ. (13)

The normal copula has the empirically questionable property that its taildependence is zero. Tail dependence is positive in the DAC and DSC models wedevelop, and the tail dependence measure depends on the degree of freedom,ν, the copula correlation, �i,j , and the copula skewness parameter, λ.21

In the DSC model, the lower and upper tail dependence is identical, thatis, τL

i,j,t =τUi,j,t . Based on the work by Longin and Solnik (2001) and Ang

and Bekaert (2002), we suspect that such symmetry does not characterizeinternational equity index returns, thus we investigate the difference betweenupper and lower tail dependence using the DAC model.

Figure 7 plots the dynamic measure of tail dependence in Equations (12)and (13) for the DAC model. We report the average of the bivariate tail

21 See Patton (2006) for an application of the tail dependence measure to exchange rates. See also Poon, Rockinger,and Tawn (2004).

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Table 2Parameter estimates for dynamic symmetric t copula (DSC) models

Compositeα� β� Persistence ϕ�ϒDM,T ϕ�ϒEM,T ν−1 Likelihood


16 Developed 0.025∗∗ 0.965∗∗ 0.990 0.4078∗∗ – 0.0740∗∗ 31,076Markets (0.006) (0.018) (0.1436) (0.0099)

Trend Only – – – 0.3044∗∗ – 0.1043∗∗ 28,448(0.0375) (0.0065)


16 Developed 0.028∗∗ 0.952∗∗ 0.980 0.2895∗ – 0.0683∗∗ 21,984Markets (0.010) (0.041) (0.1223) (0.0094)

Trend Only – – – 0.2898∗∗ – 0.0905∗∗ 20,687(0.0729) (0.0170)

13 Emerging 0.018∗∗ 0.937∗∗ 0.955 – 0.3971∗∗ 0.0458∗∗ 3,148Markets (0.004) (0.021) (0.0513) (0.0075)

Trend Only – – – – 0.3819∗∗ 0.0454∗∗ 3,021(0.0415) (0.0074)

All 29 Markets 0.026∗∗ 0.951∗∗ 0.977 0.3349∗∗ 0.2986∗∗ 0.0466∗∗ 36,151(0.004) (0.020) (0.0745) (0.1241) (0.0088)

Trend Only – – – 0.3069∗∗ 0.2758 0.0400 33,907(0.0464) (0.0921) (0.0362)


16 Developed 0.037∗∗ 0.922∗∗ 0.958 0.2631∗ – 0.0877∗∗ 21,057Markets (0.008) (0.027) (0.1119) (0.0140)

Trend Only – – – 0.3217∗∗ – 0.1158∗∗ 20,355(0.0713) (0.0132)

17 Emerging 0.025∗∗ 0.917∗∗ 0.942 – 0.3134∗∗ 0.0756∗∗ 8,920Markets (0.006) (0.023) (0.0556) (0.0100)

Trend Only – – – – 0.3152∗∗ 0.0272∗∗ 8,565(0.0466) (0.0042)

All 33 Markets 0.027∗∗ 0.932∗∗ 0.959 0.3320∗∗ 0.2772∗∗ 0.0653∗∗ 51,316(0.006) (0.023) (0.0330) (0.0063) (0.0206)

Trend Only – – – 0.1753∗∗ 0.1478∗∗ 0.0395∗∗ 48,040(0.0516) (0.0490) (0.0088)

We report parameter estimates for the DSC models for the 16 developed markets, 13 emerging markets (IFCG),17 emerging markets (IFCI), and all markets for different sample periods. We also report the special case ofconstant copula correlations (β� =0 and α� =0). The standard errors in parentheses are computed using themethod in Engle, Shephard, and Sheppard (2008). Significance at the 5% and 1% levels is denoted by * and **.

dependence across all pairs of countries.22 In each graph, the dark line depictsthe evolution of the lower tail dependence, while the gray line is for the uppertail dependence.23 Figure 7 shows quite dramatic differences across markets.

22 The tail dependence concept introduced above is inherently bivariate and not easily generalized to higherdimensions. In order to convey the empirical evolution of tail dependence for many countries, we report theaverage of the bivariate tail dependence across all pairs of countries.

23 To the best of our knowledge, a closed form solution is not available for the tail dependence measure in the caseof the skewed t copula. We therefore approximate it by numerical integration using ζ =0.001.

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Table 3Parameter estimates for dynamic normal copula (DNC) models

Compositeα� β� Persistence ϕ�ϒDM,T ϕ�ϒEM,T Likelihood


16 Developed Markets 0.023∗∗ 0.967∗∗ 0.990 0.5085∗∗ – 30,248(0.005) (0.010) (0.0669)

Trend Only – – – 0.3281∗∗ – 27,549(0.0365)


16 Developed Markets 0.030∗ 0.960∗∗ 0.990 0.3099∗ – 21,651(0.012) (0.024) (0.1313)

Trend Only – – – 0.2795∗∗ – 20,241(0.0367)

13 Emerging Markets 0.017∗∗ 0.938∗∗ 0.955 – 0.3973∗∗ 3,064(0.004) (0.022) (0.0506)

Trend Only – – – – 0.3815∗∗ 2,938(0.0386)

All 29 Markets 0.021∗∗ 0.962∗∗ 0.983 0.2764∗∗ 0.2797∗ 35,574(0.004) (0.017) (0.1024) (0.1193)

Trend Only – – – 0.2797∗∗ 0.2834∗∗ 33,213(0.0423) (0.0648)


16 Developed Markets 0.034∗∗ 0.929∗∗ 0.963 0.3032∗∗ – 20,741(0.007) (0.023) (0.1163)

Trend Only – – – 0.3097∗∗ – 19,977(0.0774)

17 Emerging Markets 0.024∗∗ 0.911∗∗ 0.935 – 0.3093∗∗ 8,643(0.006) (0.025) (0.0542)

Trend Only – – – – 0.3019∗∗ 8,386(0.0456)

All 33 Markets 0.023∗∗ 0.940∗∗ 0.962 0.4502∗∗ 0.3771∗∗ 50,291(0.005) (0.018) (0.0423) (0.0356)

Trend Only – – – 0.4142∗ 0.4351∗∗ 47,218(0.1688) (0.1326)

We report parameter estimates for the DNC models for the 16 developed markets, 13 emerging markets (IFCG),17 emerging markets (IFCI), and all markets for different sample periods. We also report the special case ofconstant copula dynamic correlation (β� =0 and α� =0). The standard errors in parentheses are computed usingthe method in Engle, Shephard, and Sheppard (2008). Significance at the 5% and 1% levels is denoted by* and **.

The tail dependence in DMs has risen markedly during the last 20 years.Remarkably, while the EM tail dependence measures in the middle panel ofFigure 7 have also increased, they remain very low compared to DMs. Whenconsidering all markets in the bottom panel of Figure 7, we find that while thetail dependence is rising, it is still much lower than for the DMs alone. From thisperspective, the diversification benefits from adding EMs to a portfolio appearto be large compared to those offered by DMs alone, even if these benefitshave become smaller over time. In all cases, lower tail dependence is higher

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1 9 9 0 1 9 9 5 2 0 0 0 2 0 0 5 2 0 1 00

0 . 1

0 . 2

0 . 3

0 . 4

1 6 D ev elo p ed M ark et s

L o w er T ai l

U p p er T ai l

1 9 9 0 1 9 9 5 2 0 0 0 2 0 0 5 2 0 1 00

0 . 1

0 . 2

0 . 3

0 . 4

1 3 E m erg in g M ark ets

1 9 9 0 1 9 9 5 2 0 0 0 2 0 0 5 2 0 1 00

0 . 1

0 . 2

0 . 3

0 . 4

A l l 2 9 M ark ets

Figure 7Dynamic average bivariate tail dependence for the DAC modelWe report the average bivariate tail dependence for the DAC model. The black line denotes lower tail dependence,and the gray line denotes upper tail dependence.

than upper tail dependence, suggesting substantial negative skewness in themultivariate distribution of international equity returns.

The relatively modest tail dependence found for EMs in Figure 7 helpsexplain the diversification benefits offered by EMs in the left panels of Figure 6where the general CDB measure is used.

4.2 Threshold correlationsFor each pair of countries, we compute threshold correlations from returnshocks, as follows:

ρπ (zi,zj )=

{Corr

(zi,zj |zi ≤π,zj ≤π

)if π ≤0,

Corr(zi,zj |zi >π,zj >π

)if π >0,

where zi are shocks for country i. We also compute similar thresholdcorrelations based on standardized returns. Figure 8 plots the pairwise empirical

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Figure 8Empirical threshold correlationsThe left column reports the average pairwise threshold correlations computed on returns, which are standardizedby their unconditional means and standard deviations. The right column presents the average pairwise thresholdcorrelations computed on shocks, which are returns standardized using the AR-NGARCH model with skewedt innovations. In each case, we compare the empirical threshold correlation (solid line) to the one implied by anormal distribution (dashed line) using the average linear correlation estimated from the data.

threshold correlations averaged across countries. The left panels report thethreshold correlations corresponding to returns, which are standardized by theirunconditional mean and standard deviations, and the right panels report thethreshold correlations computed on shocks, which are returns standardized bythe AR-NGARCH model with skewed t innovations discussed in Section 1.2.The correlations are computed for a grid of thresholds π (denoted in standarddeviations), provided that at least 20 observations are available. The dashedline reports threshold correlations implied by a normal distribution usingthe average linear correlation estimated from the data. As is well known,asymmetric threshold correlations cannot be captured using a multivariatenormal distribution: threshold correlations in the normal distribution aresymmetric, and also decrease rather quickly in the tails.

When considering monthly returns in the United States versus other DMs,Longin and Solnik (2001) found that the downside threshold correlations weremuch larger than their upside counterparts. The solid black line in the top leftpanel of Figure 8 confirms the findings of Longin and Solnik (2001): Whencomputing the average of all possible pairwise threshold correlations for weekly

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Figure 9Model threshold correlationsWe compute threshold correlations using the empirical AR-NGARCH residuals as well as threshold correlationsbased on generated data from three models: the multivariate Gaussian distribution, the DSC model, and the DACmodel. The DAC model with λ=−1 calibrates the λ parameter rather than using the empirical estimate.

returns in 16 DMs, we find that the downside threshold correlations indeed aremuch larger than upside threshold correlations regardless of how returns arestandardized.

The bottom left panel of Figure 8 shows the average threshold correlationsfor EMs, which has not yet been documented in the literature. Thresholdcorrelations are also asymmetric for EMs. As in the case of DMs, the downsidethreshold correlations are larger than upside threshold correlations. Whencomparing the top and bottom panels of Figure 8, we see that the downsidethreshold correlations are higher for DMs than for EMs, which is consistentwith our earlier findings.

The comparison of the left-side panels in Figure 8 based on returns and theright-side panels based on shocks is very instructive to illustrate the advantagesof copula modeling, and of the DAC model in particular. The asymmetries andnonnormalities built into the model for the marginals do generate multivariate

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asymmetries and nonnormalities. If these asymmetries and nonnormalities aresufficient to capture the stylized facts in the data, we should not observeasymmetries in the right-side panels. This is clearly not the case. In fact, theright-side panels are very similar to the left-side panels, indicating the limitedrole that the modeling of the marginals can play in capturing multivariateasymmetries, and emphasizing the need for asymmetric copula models.

To see how well our models fare, we follow the empirical setup in Ang andBekaert (2002) and compare the pattern in empirical threshold correlationswith threshold correlations from simulated data generated using the models.Figure 9 presents the threshold correlations for return shocks implied by theDAC and DSC models. Note that the DSC model (marked by “+”) is successfulin producing higher threshold correlations than the Gaussian distribution, but bydesign these correlations are also symmetric. The DAC model (marked by “©”)produces an asymmetric pattern in threshold correlations, with substantiallyhigher downside threshold correlations.

The DAC model does not capture the threshold correlation patternsperfectly. This is not surprising, because its parameters optimize the compositelikelihood and not the distance between model-based threshold correlations andtheir empirical counterpart. Moreover, the averaging of empirical thresholdcorrelations and model parameters may induce biases in the empiricalcomparison. To demonstrate the model’s potential, we simulate data usinga bivariate DAC model with the average values for all parameter estimates,except for λ, which we set equal to minus one to roughly match the thresholdasymmetry in the data. The threshold correlation (marked by “×”) shows thatif the objective is to match the average level of threshold correlation, the DACmodel is able to do so.

4.3 Implied copula correlations for longer horizonsWe have used weekly returns in order to be able to capture volatility anddependence dynamics while avoiding non-synchronicity and other problemswith daily data. However, not all investors have a weekly horizon. Hence,we now consider the implications of our findings for longer horizons. Theleft panels in Figure 10 present monthly copula correlations averaged acrosscountries, while the right panels report annual copula correlations. Each month(year), we simulate returns from the AR-NGARCH and DAC models over thenext 4 (52) weeks, compound them, and compute the copula correlations ofthe resulting returns. Not surprisingly, the long horizon copula correlationsare smoother than the shorter horizon correlations. Further, the dependence isclearly increasing over time for DMs as well as EMs and confirms our earlierfinding that the dependence between EMs is lower than between DMs.

4.4 Correlation regressionsIn this section we ask how the copula correlations plotted in Figure 3 arerelated to volatility, market openness, and financial and macro variables more

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Figure 10Monthly and annual model-based copula correlationsWe report monthly and annual copula correlations averaged across countries simulated from the AR-NGARCHand DAC models. The left panels report monthly copula correlations, and the right panels report annual copulacorrelations.

generally. We run country-level panel regressions on the copula correlationsfrom the DAC model. Because some of the variables used in the regressionsare generated using models and thus estimated with error, we must of course becautious regarding the inference on the regression parameters. Table 4 reportsthe results in four panels. The regressand in Panel A is the natural logarithmof the average DM copula correlation with other DMs. In Panel B it is thenatural logarithm of the average DM copula correlation with all EMs. PanelsC and D report on the natural logarithm of the average EM copula correlationswith all DMs and all other EMs, respectively. Due to macro data availability,the data frequency is quarterly, and our sample is restricted between the thirdquarter of 1995 and the second quarter of 2008. We include a quadratic trend inall specifications and assess if the explanatory variables are able to render thecorrelation trend insignificant. The specifications labeled (i) in Table 4 containonly the trend. Note that it is significant throughout Panels A–D.

Factor models typically imply a positive relationship between correlationand volatility. Because we obtain our results without the aid of a factor model,

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it is worth investigating whether or not this positive relationship is confirmed.Specification (ii) in Table 4 reports the results in each of the panels. We regresscorrelation on the average log GARCH volatility across EM or DM countriesand find a positive relationship throughout Panels A–D. Comparing the resultsof specification (ii) with those of specification (i), we see that includingvolatility in the regression does not affect the trend coefficient estimates northe statistical significance of the estimated coefficients much.24 We concludethat the documented trend in correlation cannot be explained by volatilitymovements.

We now investigate the relationship between financial development anddependence in EMs. In order to measure financial development, we rely onBekaert (1995) and Edison and Warnock (2003), who use a direct measureof de jure market openness. Their measure is defined as the ratio of themarket capitalizations of the investable (IFCI) and global (IFCG) indexes fromS&P/IFC, and we denote it by “MCR.” When the MCR measure is one, themarket capitalization of the investable index is equal to that of the market-wide index. We report the regression results in specification (iii) of PanelsC and D of Table 4. The relationship between the openness indicator MCR

and dependence is statistically significant at the 5% level in both cases, andthe estimate is positive as expected. However, the estimates of the time trendcoefficients are little affected and still statistically significant.

Finally, we regress copula correlations on the following financial variables:the volatility risk premium defined as the VIX less a moving 22-day standarddeviation on daily index returns; the 3-month U.S. T-bill; the U.S. term spread(10-year less 1-year); the U.S. credit spread; and the average of the real interestrate in the G7 economies. We also include MCAP defined as the market valuedivided by the preceding year’s GDP. Finally, we include turnover, which ismeasured as trading value divided by market capitalization, and serves as aliquidity proxy. In addition we use the following macro variables: the worldreal GDP growth rate, and the ratio of the sum of exports and imports of goodsand services to the gross domestic product denoted by “T rade.”25

We have investigated all permutations of the financial and macroeconomicvariables, but for brevity we report the most general specification including allvariables. The results are in specification (iii) for Panels A and B of Table 4, andin specification (iv) for Panels C and D. We find that it is possible to identifyfinancial variables that affect international return dependence, even with thesesmall samples. The volatility risk premium coefficient is significantly positivein all cases. The U.S. 3-month T-bill rate coefficient is significantly negative in

24 Following Petersen (2009), we compute White standard errors adjusted for within-clusters (country and quarter)correlation.

25 Our variable selection is motivated by the work of Schwert (1989), Engle, Ghysels, and Sohn (2008), Engle andRangel (2008), Baele, Bekaert, and Inghelbrecht (2010), Boyer, Mitton, and Vorkink (2010), Ghysels, Plazzi,and Valkanov (2011), Bekaert et al. (2011), and Carrieri, Chaieb, and Errunza (2012).

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Tabl

e4

Cor

rela

tion

regr

essi

ons

1995

–200

8

Pane

lA:L

ogof

Ave

rage

Qua

rter

lyPa

nelB

:Log

ofA

vera

geQ

uart

erly

Pane

lC:L

ogof

Ave

rage

Qua

rter

lyE

MPa

nelD

:Log

ofA

vera

geQ

uart

erly

EM

DM

Cor

rela

tion

With

Oth

erD

Ms

DM

Cor

rela

tion

With

EM

sC

orre

latio

nW

ithD

Ms

Cor

rela

tion

With

Oth

erE

Ms

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

Con

stan

t−1

.256

4∗∗

−0.7

339∗

∗−0

.973

6∗∗

−1.9

773∗

∗−0

.822

0∗∗

−1.0

921∗

∗−1

.984

1∗∗

−0.8

776∗

∗−2

.201

5∗∗

−0.5

885

−1.9

708∗

∗−0

.507

9∗∗

−2.1

378∗

∗−0

.127

7(0

.063

8)(0

.095

9)(0

.167

4)(0

.087

2)(0

.206

3)(0

.274

0)(0

.109

3)(0

.269

7)(0

.085

5)(0

.411

8)(0

.082

9)(0

.173

2)(0

.079

1)(0

.194

9)

Tim

eT

rend

0.12

69∗∗

0.11

89∗∗

0.11

60∗∗

0.18

43∗∗

0.17

75∗∗

0.17

42∗∗

0.18

47∗∗

0.16

78∗∗

0.17

28∗∗

0.17

60∗∗

0.16

33∗∗

0.15

46∗∗

0.15

41∗∗

0.15

86∗∗

(0.0

016)

(0.0

013)

(0.0

025)

(0.0

036)

(0.0

034)

(0.0

036)

(0.0

041)

(0.0

038)

(0.0

050)

(0.0

067)

(0.0

037)

(0.0

034)

(0.0

038)

(0.0

043)

Tim

eT

rend

Squa

red

−0.0

012∗

∗−0

.001

0∗∗

−0.0

012∗

∗−0

.001

6∗∗

−0.0

013∗

∗−0

.001

7∗∗

−0.0

016∗

∗−0

.001

3∗∗

−0.0

015∗

∗−0

.001

6∗∗

−0.0

013∗

∗−0

.001

0∗∗

−0.0

012∗

∗−0

.001

3∗∗

(0.0

000)

(0.0

000)

(0.0

000)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

Ave

rage

Log

Vol

DM

0.13

67∗∗

0.07

83∗∗

0.28

94∗∗

0.27

52∗∗

(0.0

206)

(0.0

206)

(0.0

714)

(0.0

726)

Ave

rage

Log

Vol

EM

0.36

74∗∗

0.22

68∗∗

0.46

52∗∗

0.41

04∗∗

(0.0

626)

(0.0

697)

(0.0

463)

(0.0

308)

MC

R0.

3284

∗0.

2168

0.25

22∗

0.01

87(0

.162

4)(0

.219

6)(0

.107

5)(0

.151

6)

Vol

Ris

kPr

emiu

m0.

0108

∗∗0.

0093

∗∗0.

0139

∗∗0.

0041

∗∗(0

.001

4)(0

.002

9)(0

.003

3)(0

.001

5)

3-M

onth

T-bi

ll−0

.033

6∗∗

−0.0

816∗

∗−0

.090

7∗∗

−0.0

809∗

∗(0

.010

4)(0

.026

2)(0

.028

0)(0

.015

6)

Term

Spre

ad−0

.025

2−0

.104

2∗∗

−0.1

317∗

∗−0

.110

6∗∗

(0.0

129)

(0.0

321)

(0.0

363)

(0.0

215)

Cre

ditS

prea

d0.

0163

0.02

19−0

.024

0−0

.103

4∗∗

(0.0

244)

(0.0

565)

(0.0

768)

(0.0

353)

(con

tinu

ed)

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Tabl

e4

Con

tinu

ed

Pane

lA:L

ogof

Ave

rage

Qua

rter

lyPa

nelB

:Log

ofA

vera

geQ

uart

erly

Pane

lC:L

ogof

Ave

rage

Qua

rter

lyE

MPa

nelD

:Log

ofA

vera

geQ

uart

erly

EM

DM

Cor

rela

tion

With

Oth

erD

Ms

DM

Cor

rela

tion

With

EM

sC

orre

latio

nW

ithD

Ms

Cor

rela

tion

With

Oth

erE

Ms

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(i)

(ii)

(iii)

(iv)

(i)

(ii)

(iii)

(iv)

G7

Rea

lRat

e1.

5373

∗∗2.

4068

1.78

941.

5423

∗(0

.412

3)(1

.625

3)(1

.545

1)(0

.599

4)

MC

AP

−0.0

129

0.02

210.

1276

0.06

52(0

.035

5)(0

.031

9)(0

.163

1)(0

.084

7)

Tur

nove

r0.

4485

∗0.

4394

∗∗−0

.195

3−0

.131

8(0

.186

0)(0

.149

0)(0

.160

6)(0

.122

3)

Wor

ldG

DP

Gro

wth

0.00

95∗∗

0.01

30∗

0.01

210.

0079

∗(0

.002

9)(0

.005

9)(0

.009

6)(0

.003

7)

Tra

de0.

0194

0.04

82−0

.156

9−0

.097

3(0

.079

9)(0

.058

2)(0

.093

9)(0

.055

0)

Adj

uste

dR

20.

6385

0.64

360.

6737

0.75

040.

7719

0.79

940.

6304

0.64

000.

6554

0.68

290.

7123

0.74

980.

7316

0.80

36

We

regr

ess

the

aver

age

DA

Cco

pula

corr

elat

ion

for

each

coun

try

from

Figu

re3

ona

quad

ratic

time

tren

d,av

erag

elo

gN

GA

RC

Hvo

latil

ity,m

arke

tcap

itali

zato

nra

tio(M

CR

),th

edi

ffer

ence

betw

een

VIX

and

real

ized

vola

tility

(Vol

Ris

kPr

emiu

m),

the

U.S

.cre

dits

prea

d,th

e3-

mon

thU

.S.T

-bill

rate

,the

U.S

.ter

msp

read

,the

aver

age

ofth

ere

alin

tere

stra

tein

the

G7

econ

omie

s,th

em

arke

tval

uedi

vide

dby

the

prec

edin

gye

ar’s

GD

P(M

CA

P),t

rade

dva

lue

divi

ded

bym

arke

tcap

italiz

atio

n(T

urno

ver)

,the

wor

ldre

alG

DP

grow

thra

te,a

ndth

era

tioof

the

sum

ofex

port

san

dim

port

sof

good

san

dse

rvic

esto

the

gros

sdo

mes

ticpr

oduc

t(T

rade

).Fo

llow

ing

Pete

rsen

(200

9),w

eco

mpu

teW

hite

stan

dard

erro

rsad

just

edfo

rw

ithi

n-cl

uste

rs(c

ount

ryan

dqu

arte

r)co

rrel

atio

n.Pa

nels

A–D

use

asre

gres

sand

log

aver

age

copu

laco

rrel

atio

nsfo

rea

chD

Mw

ithot

her

DM

s,ea

chD

Mw

ithE

Ms,

each

EM

with

DM

s,an

dea

chE

Mw

ithot

her

EM

s,re

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all cases. The point estimate on the U.S. term spread is negative in all cases,and it is statistically significant in three of four cases. The credit spread issignificant in Panel D only. The G7 real rate positively impacts dependencebut it is only significant in two of four cases. MCAP is not significant, andturnover is significantly positive in Panels A and B only. In terms of the macrovariables, the world GDP growth is positive in all cases and significant in threeout of four. The trade variable is not significant.

Note that in Table 4, although the various variables we have consideredare often significant, the time trend coefficients remain highly significant.We therefore conclude that the long-run dynamics in dependence cannot beexplained by standard macroeconomic and financial variables. Clearly, theseresults raise important questions as to which factors drive the dynamics incross-country dependence.

5. Summary and Conclusion

We characterize time-varying dependence using long samples of weekly returnsfor a large number of countries. We propose a new dynamic asymmetriccopula model that can capture dynamic dependence while accommodatingmultivariate nonnormality, asymmetries, and trends in dependence. Weovercome econometric complications arising from the dimensionality problemusing the composite likelihood procedure.

Our results are extremely robust and suggest that dependence hassignificantly trended upward for both DMs and EMs. Copula correlationsbetween DMs have exceeded copula correlations between EMs throughoutthe 1989–2009 period. Moreover, for DMs, the average dependence with otherDMs is higher than the average dependence with EMs. For EMs, the dependencewith DMs is generally somewhat higher than the dependence with the otherEMs, but the differences are small. While the range of copula correlations forDMs has narrowed around the increasing trend in correlation levels, this is notthe case for EMs.

We find substantial evidence of asymmetric tail dependence with lowertail dependence being larger than upper tail dependence. Moreover, taildependence as computed from the DAC model is large and increasing throughtime for DMs, but remains low for EMs. We propose a new measure ofdiversification benefits that takes into account tail dependence, and find thatwhereas diversification benefits have largely disappeared for DMs, EMs stilloffer substantial diversification benefits. From this perspective, our resultssuggest that although diversification benefits have lessened in the case of DMs,the case for EMs remains strong. This is due to the fact that while equity marketcrises in EMs are frequent, many of them are country-specific.

These results have very important implications for portfolio management,and it may prove interesting to further explore them in future work. It mayalso prove useful to investigate the robustness of our findings to allowing for

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multiple regimes, or to the inclusion of multiple stochastic components, as forexample in the model of Colacito, Engle, and Ghysels (2011).

Appendix

A. The Skewed t CopulaThe skewed t distribution discussed in Demarta and McNeil (2005) has the stochastic representation

X=√

WZ+λW , (A1)

where λ is the asymmetry parameter, W is an inverse gamma variable W ∼IG(

ν2 , ν

2

), Z is a normal

variable Z∼N (0N ,�), and Z and W are independent. The skewed t distribution generalizes thet distribution by adding a second term related to the same inverse gamma random variable, whichis scaled by a vector λ of asymmetry parameters. Note that we constrain the copula to have thesame asymmetry parameter across all assets, that is, λi =λj for all i, j .

The probability density function of the skewed t copula defined from the skewed t distributionis given by

c(η;λ,ν,�) =2

(ν−2)(N−1)2 K ν+N

2

(√(ν+z∗��−1z∗)λ��−1λ

)ez∗��−1λ

�(

ν2

)1−N |�| 12(√(

ν+z∗��−1z∗)λ��−1λ)− ν+N

2 (1+ 1

νz∗��−1z∗) ν+N

2

×N∏

j=1

(√(ν+(z∗

j )2)λ2

j

)− ν+12(

1+ 1ν

(z∗j

)2) ν+1

2

K ν+12

(√(ν+(z∗

j )2)λ2

j

)ez∗jλj

, (A2)

where K(·) is the modified Bessel function of the third kind, and the copula shocks z∗i = t−1

λ,ν (ηi )are defined from the skewed univariate Student t density defined by

tλ,ν (ηi )=

ηi∫−∞

21− ν+12 K ν+1

2

(√(ν+x2

)λ2

i

)exλi

�(

ν2

)√πν

(√(ν+x2

)λ2

i

)− ν+12 (

1+ x2ν

) ν+12

dx. (A3)

The skewed Student t quantile function, t−1λ,ν (ηi ), is not available in closed form but can be well

approximated by simulating 100,000 replications of Equation (A1).

The moment of order i of the W variable is given by mi =νi/(∏i

j=1(ν−2j ))

, and from the

normal mixture structure of the distribution, we can derive the expected value:

E [X]=E (E [X|W ])=E(W )λ=ν

ν−2λ

and the variance-covariance matrix:

Cov(X) = E (Var(X|W ))+Var(E [X|W ])

=ν

ν−2� +

2ν2λ2

(ν−2)2(ν−4). (A4)

Notice that the covariances are finite if ν >4. These moments provide the required link betweenthe multivariate skewed t distribution and the copula correlation matrix �.

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We also provide results for the conventional symmetric t distribution, which is nested when λ

tends to zero. The symmetric N -dimensional t distribution has the stochastic representation

X=√

WZ, (A5)

where W is an inverse gamma variable W ∼IG(

ν2 , ν

2

), Z is a normal variable Z∼N (0N ,�), and

Z and W are independent.The probability density function of the t copula defined from the t distribution is given by

c(η;ν,�)=�(

ν+N2

)|�| 1

2 �(

ν2

)(

�(

ν2

)�(

ν+12

))N (1+ 1

νz∗��−1z∗)− ν+N

2

∏Nj=1

(1+

z∗2jν

)− ν+12

,

where z∗i = t−1

ν (ηi ) and tν (ηi ) is the univariate Student’s t density function given by

tν (ηi )=

ηi∫−∞

�(

ν+12

)√

πν�(

ν2

) (1+x2

ν

)− ν+12

dx.

The normal copula is further nested as ν →∞. The probability density function of the normalcopula defined from the normal distribution is given by

c(η;�)=1

|�| 12

e− 1

2 z∗�(�−1−IN

)z∗

,

where z∗i =�−1(ηi ) and �(ηi ) is the univariate normal density function given by

�(ηi )=

ηi∫−∞

1√2π

e− 1

2 x2dx.

B. The General Form of ϒt

Let N denote the number of assets, and NX the number of explanatory variables. Define A as anNx(N +NX) matrix whose first N columns are an identity matrix. The last NX columns are theHadamard product of an NxNX matrix of coefficients θ and a matrix of explanatory regressors Xt

A=

⎡⎢⎢⎢⎣1 0 ... 0 θ11X11,t ... θ1NX

X1NX,t

0 1 ... 0 θ21X21,t ... θ2NXX2NX,t

.

.

....

. . . 0...

.

.

....

0 0 ... 1 θNX1XNX1,t ... θNXNXXNXNX,t

⎤⎥⎥⎥⎦.

The matrix A is standardized by dividing each element by its row’s root mean square:

Ai,j =Ai,j√∑N+NX

k=1 A2i,k

=Ai,j√

1+∑N+NX

k=N+1 A2i,k

,

and then ϒt is given byϒt = AA�.

These restrictions ensure that ϒt is a proper correlation matrix. Up to a normalizing constant, thismatrix will have as a typical off-diagonal element:

ϒij,t =NX∑k=1

θikθjkXik,tXjk,t , i �=j.

This specification can easily accommodate for a time trend (Xik,t = t), a common positive factor(Xik,t =Xjk,t =

√Ft ), and asset-specific variables, for example, volatilities Xik,t =σi,t .

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C. Copula Correlation Moment MatchingEquating the elements of � in the DAC model to the sample average via

�=1T

∑Tt=1 z∗

t z∗�t −ϕ�

1T

∑Tt=1ϒt

1−ϕ�

allows us to drastically reduce the number of parameters estimated via numerical optimization ofthe likelihood function.

Recall that the DAC recursion is given by

�t =(1−β� −α�)[(1−ϕ�)�+ϕ�ϒt

]+β��t−1 +α�z∗

t−1z∗�t−1 , (A6)

where z∗i,t =z∗

i,t

√�ii,t .Acircularity problem is apparent because we need �ii,t to estimate �, which

in turn is required to compute the time series of �ii,t . Note however that � is a copula correlationmatrix, so that �ii =1, for all i, and note also that only the diagonal elements of �t are neededto compute z∗

t . Aielli (2009) therefore proposes to first compute Equation (A6) for the diagonalelements only, that is,

�ii,t =(1−α� −β�)+β��ii,t−1 +α�(z∗ii,t−1)2

for all i and t . Having computed the �ii,t , the sample correlation matrix of the z∗i,t can be obtained,

which in turn yields �, and the recursion in (A6) can now be implemented replacing � by �.

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