Vector copulas

Vector Copulas

\address{University of Washington and Penn State}

Introduction

The cornerstone of copula theory, known as Sklar's Theorem, from Sklar:59, states that (i) for any multivariate distribution function $F$ on $\mathbb{R}^{K}$, with univariate marginal distribution functions $ F_{1}$, ..., $F_{K}$, there exists a copula function $C$ such that

and (ii) given any copula function $C$ and any collection of univariate distribution functions $F_{1},\ldots ,F_{K}$, ((ref)) defines a multivariate distribution function with copula $C$ and the marginal distributions $F_{1},\ldots ,F_{K}$. When the marginal distributions are continuous, $C$ in part (i) of Sklar's Theorem is the unique copula associated with $F$ and it characterizes the rank dependence structure in $F$ . Moreover, (ii) provides a general approach to constructing multivariate distributions from univariate ones.

By virtue of Sklar's Theorem, copulas can be used to characterize nonlinear or rank dependence between random variables, as distinct from marginal distributional features, to compute bounds on parameters of multivariate distributions in problems with fixed marginals, and to construct parametric and semiparametric families of multivariate distributions from univariate ones. Applications to quantitative finance, particularly risk management, portfolio choice, derivative pricing, financial contagion and other areas, where precise measures of dependence are crucial, are well known and extensively reviewed, see for instance Embrechts:2009 and references therein.

Applications to economics, though fewer, have been no less expansive. First, the characterization of dependence, as distinct from marginal distributional features, has been instrumental in modeling the propagation of shocks in contagion models in BC:2013, in holding dependence fixed to measure partial distributional effects in Rothe:2012, in identifying and estimating auction models in HLP:2012 and He:2017. Second, the copula approach to problems with fixed marginals has allowed the computation of sharp bounds on various relevant parameters in treatment effects models with randomized treatment or treatment on observables, in CL:2019, FM:2017 and the many references therein. It has also been applied to problems of data combination, including ecological inference, see RM:2007 and FSS:2014 for instance. Third, the copula as a modelling and inference tool has been used to discipline multiple latent variables and multiple dimensions of unobserved heterogeneity. As such, the copula approach has been applied to sample selection models, in Smith:2003, to regime switching models in FW:2010 and CFW:2014, to simultaneous equations with binary outcomes in HV:2017, as well as the modeling of earnings dynamics in BR:2009 and the measure of intergenerational mobility in CHKS:2014. Fourth, semiparametric econometrics models including both time series and cross section models have been constructed using flexible parametric copulas to model contemporaneous dependence structures in multivariate models or time dependence in univariate time series models, see CF:2006 (CF:2006,CF:2006b), CFT:2006, Patton:2006, Beare:2010, CWY:2009, and CKZ:2009 for properties, estimation, and inference in such models. The list is surely not exhaustive.

In all these applications, the need for a notion of copula that links multivariate marginals arises naturally. In propagation models, MP:2017 highlight the need to distinguish within-group and between-group dependence. Models of treatment effects with multivariate potential outcomes of interest fall in the class of problems with given multivariate marginals, in cases where the latter are identified. Censored and limited dependent variables models with clustered latent variables call for hierarchical modeling, where a copula operates on vectors of latent variables, each of which can also be modeled with a traditional copula. In integrated risk management, modeling and measuring risks of portfolios of several groups of risks will also benefit from a copula-like tool for linking multivariate marginals, see EP:2006.

However, Sklar's Theorem, as stated above, requires that all the marginals be univariate. Indeed, GQR:95 shows that for two random vectors, if the function $C:\left[ 0,1\right] ^{2}\rightarrow \left[ 0,1\right] $ is such that $F\left( x_{1},x_{2}\right) =C\left( F_{1}\left( x_{1}\right) ,F_{2}\left( x_{2}\right) \right) $ defines a $\left( d_{1}+d_{2}\right) $ -dimensional distribution function with marginals $F_{1}$ with support in $ \mathbb{R}^{d_{1}\text{ }}$and $F_{2} $ with support in $\mathbb{R}^{d_{2} \text{ }}$for all $d_{1}$ and $d_{2}$ such that $d_{1}+d_{2}\geq 3$, and for all distribution functions $F_{1}$ and $F_{2}$, then $C\left( u_{1},u_{2}\right) =u_{1}u_{2}$. Hence, the only possible copula which works with non-overlapping multivariate marginals is the independence copula. Ressel:2019 generalizes this impossibility result to more than two random vectors.

The objective of the present work is to circumvent this impossibility theorem. The paper develops a vector copula that generalizes the traditional copula to model and characterize nonlinear or rank dependence between a finite number of random vectors of any finite dimensions. It relies on the combination of the theory of probability distribution with given overlapping marginals, particularly Vorobev:62 and Kellerer:64, with the theory of transport of probability distributions, particularly RR:90 , Brenier:91 and McCann:95. First, we introduce the concept of a vector copula and establish a vector version of Sklar's Theorem using vector ranks proposed in CGHH:2017 as multivariate probability transforms to remove marginal distributional features. Vector copulas and the vector Sklar theorem allow the construction of distributions with any given non overlapping multivariate marginals, thereby overcoming the weakness of traditional copulas identified in GQR:95. Second, we show that vector copulas are invariant to comonotonic transformations, where the multivariate notion of comonotonicity is borrowed from GH:2012 and EGH:2012, and we define comonotonic and countermonotonic vector copulas extending Fr\'{e}chet extremal copulas. Third, we construct flexible parametric families of vector copulas including Elliptical and Kendall vector copulas and provide algorithms for simulating from them. They reduce to the corresponding classical copulas when all the marginals are univariate. Using the Vector Sklar Theorem, we construct new families of multivariate distributions with any fixed non-overlapping multivariate marginals and Elliptical or Kendall vector copulas. To illustrate a possible use of vector copulas, we follow CF:2006b and MP:2017 and study financial contagion through the evolution of between-vector dependence before, during and after the 2008 financial crisis, for a collection of five aggregate stock indices.

Related literature

Separate efforts have been carried out to develop dependence measures for random vectors robust to within-vector dependence on the one hand, and to construct specific multivariate distributions with given multivariate marginal distributions, on the other hand. For the former, MP:2017 propose a dependence measure between a finite number of random vectors that is robust to within-vector dependence and apply it to the study of contagion in financial markets, inter alia. GSS:2014 propose extensions of Spearman's rho and Kendall's tau for two random vectors and show that they are invariant to increasing transformations of each component of the random vector. As in the case of random variables, these global measures are insufficient to characterize the complete nonlinear dependence structure between random vectors for which analogues of copulas are needed. LSS:96 develop a a device to link several random vectors with given distributions, based on the Knothe-Rosenblatt transform ( Rosenblatt:52, Knothe:57). Related ideas are developed in Rusch:85 and Section 1.6 of Rusch:2010.

Notation and conventions

Let $(d_{1},\ldots ,d_{K})$ be a finite collection of integers and for each $ k\leq K$, let $\mu _{k}$ be the uniform distribution on $\mathcal{U} _{k}:=[0,1]^{d_{k}}$. Let $P_{X}$ stand for the distribution of the random vector $X$. Let $P$ denote a given distribution on $\mathbb{R}^{d_{1}}\times \ldots \times \mathbb{R}^{d_{K}}$ with marginals $P_{k}$ on $\mathbb{R} ^{d_{k}}$, each $k\leq K$. When stating generic results applying to all $ k\leq K$ such as vector quantiles and ranks, we omit the subscript $k $ from $d_{k},$ $\mu _{k}$, and $P_{k}$ unless stated otherwise. Following Villani:2003, we let $g\#\nu $ denote the image measure (or push-forward) of a measure $\nu$ by a measurable map $g:\mathbb{R} ^{d}\rightarrow \mathbb{R}^{d}$. Explicitly, for any Borel set $A$, $g\#\nu (A):=\nu (g^{-1}(A))$. Specifically, $(Id,g)\#\nu$ denotes the degenerate distribution of the vector $(X,g(X))$, where $X\sim\nu$, where the latter means that $X$ has distribution $\nu$, whereas $X\overset{d}{=}Y$ means that $X$ and $Y$ are identically distributed. The product measure of $\nu_1$ and $ \nu_2$ is denoted $\nu_1\otimes\nu_2$. The notation $\|x\|$ refers to the Euclidean norm. Denote $\mathbb{S}^{d}:=\{x\in \mathbb{R}^{d}:\;\Vert x\Vert \leq 1\}$ the unit ball, $\mathcal{S}^{d-1}:=\{x\in \mathbb{R}^{d}:\;\Vert x\Vert =1\}$ the unit sphere in $\mathbb{R}^{d}$. The symbol $\partial $ denotes the subdifferential, $\nabla$ the gradient, $D$ is the Jacobian and $ D^2$ the Hessian. The domain of a function $\psi$ is denoted dom$(\psi)$. The convex conjugate of a convex lower semicontinuous function $\psi$ is denoted $\psi^\ast$. We use the standard convention of calling weak monotononicity non-decreasing or non-increasing, as the case may be. If $ \Gamma: \mathbb{R}^d\rightrightarrows\mathbb{R}^d$ is a correspondence, then $\Gamma^{-1}(A):=\{x\in\mathbb{R}^d\;:\Gamma(x)=y, \mbox{ some }y\in A\}$. The transpose of a matrix $A$ is denoted $A^{\top } $. The collection of subsets of a set $S$ is denoted $2^S$.

Organization of the paper

Section (ref) introduces vector copulas and their properties. Section (ref) develops parametric families of vector copulas. Appendix (ref) reviews some notions in convex analysis, measure transport and vector quantiles and ranks. Appendix (ref) collects proofs of results in the main text.

Vector Copulas and Vector Sklar Theorem

The objective of this work is to provide a tool to analyze patterns of dependence between random vectors, separately from patterns of dependence within each vector and marginal information. This will be achieved in two steps. This section will be concerned with isolating dependence between random vectors from within dependence and marginal information, and characterizing the former with a mathematical object called vector copula. Section (ref) will be concerned with the development of parametric families of vector copulas for the purposes of fitting data, estimating dependence and constructing new probability distributions with given multivariate marginals.

The motivations and the details of the construction we propose are best illustrated with the manipulation of a real data set, namely a subset of the data analyzed in MP:2017 to highlight patterns of contagion between regional financial markets. We consider $5$ stock market indices, namely the Hang Seng Index (Hong Kong), the Nikkei 225 Index (Japan), the FTSE 100 Index (United Kingdom), the S&P 500 Index (United States) and the DAX 100 Index (Germany). We consider EGARCH(1,1) residuals from weekly returns for the period from May 2009 to January 2014 (post financial crisis) and compare with the periods January 2004-December 2007 (pre-crisis) and January 2008-April 2009 (crisis) respectively, whenever we find it useful for illustrative purposes. Unless otherwise mentioned, the names of the stock indices will refer to the data generating processes for the residuals in the post-crisis period. We shall index data points with $i$, and $n$ will denote the sample size, i.e., $209$, $65$ and $566$ for the pre-crisis, crisis and post-crisis periods respectively.

Copulas

Consider first a pair of indices, namely the Hang Seng Index and the FTSE 100 Index. Call $Y_1$ and $Y_2$ the random variables with continuous cumulative distribution functions $F_1$ and $F_2$, that generate the Hang Seng and FTSE residual data, respectively. In the scatterplot of the sample $ (Y_{1i},Y_{2i})_{i=1}^{n}$, shown in Figure (ref), patterns of dependence between $Y_1$ and $Y_2$ are blurred by the marginal features of each. To isolate patterns of dependence from marginal features, each random variable in the pair is mapped into a uniform through the probability integral transform. The resulting random vector is $(F_1(Y_1),F_2(Y_2))$. It has uniform marginals, and, since $F_1$ and $F_2$ are increasing transformations, rank orderings, hence rank dependence features, are undisturbed. The scatterplot of the sample of empirical ranks $(\hat F_1(Y_{1i}),\hat F_2(Y_{2i)})_{i=1}^{n}$, where $\hat F_j$ denotes the empirical cumulative distribution function of $Y_j$, $j=1,2$, now illustrates patterns of dependence, such as left and right tail dependence, isolated from marginal features in Figure (ref).

The cumulative distribution function of the random vector $ (F_1(Y_1),F_2(Y_2))$ is the copula of $(Y_1,Y_2)$. Conditions for the claim that the copula characterizes dependence features of $(Y_1,Y_2)$ are uniqueness and invariance to transformations of the marginals that leave ranks unaffected. If $T_1$ and $T_2$ are increasing functions, then the copulas of $(Y_1,Y_2)$ and $(T_1(Y_1),T_2(Y_2))$ are identical.

Let $F$ be the cumulative distribution function of $(Y_1,Y_2)$. Assuming the marginals are continuous, it follows from Sklar's Theorem that there exists a unique copula function of $(Y_1,Y_2)$ or $F$. It is the distribution function of the pair of probability integral transforms\footnote{ For non-continuous marginals, one can use the distributional transform in Rusch:2009 to define a unique copula.} $(F_1(Y_1),F_2(Y_2))$, i.e.,

where $F_{1}^{-1}\left( u_{1}\right)$ and $F_{2}^{-1}\left( u_2\right)$ denote the quantile functions of $Y_{1}$ and $Y_2$. Uniqueness results from the fact that the probability integral transform is the unique increasing map pushing forward the marginals $F_{1}$ and $F_{2}$ to the uniform on $ \left[ 0,1\right] $. When $F$ is absolutely continuous with pdf $f$, the copula density function is given by

Much of the empirical success of copula theory stems from the ability to parsimoniously model this dependence separately from marginal features with parametric families of copulas, and to produce new families of distribution functions by fitting a parametric copula with any set of marginal distributions. A common procedure to construct parametric copula families is best illustrated with the Gaussian copula family. Let $\Phi_2(x,y;\rho)$ be the cumulative distribution function of the standard bivariate normal distribution with correlation coefficient $\rho$ and let $\Phi$ be the cumulative distribution function of the standard univariate normal distribution. Then,

is the bivariate Gaussian copula with parameter $\rho$. Popular alternatives include Student's $t$ and Archimedean copulas. The densities for the most common are given in Appendix (ref) for convenience.

Vector ranks

We now turn to the characterization of dependence between two random vectors. Take $Y_1$ to now be the bivariate random vector that is assumed to have generated the residuals from the Hang Seng and the Nikkei indices. Similarly, take $Y_2$ to now be the trivariate random vector that is assumed to have generated the residuals from the FTSE, S&P and DAX indices. The $ 5\times5$ scatterplot matrix in Figure (ref) shows patterns of dependence between components of $Y_1$ and $Y_2$ that are blurred by marginal features and within vector dependence, i.e., dependence between the Hang Seng and the Nikkei, and dependence between the FTSE, the S&P and the DAX.

We pursue the same three objectives as in the previous section: (1) separate between-vector dependence from within-vector dependence and marginal features with what we shall call a vector copula, (2) show uniqueness and invariance of the vector copula to transformations of the multivariate marginals that don't affect between-vector dependence, (3) develop new families of parametric models to fit multivariate data.

In direct analogy with the bivariate case described in Section (ref), we seek to transform $Y_1$ and $Y_2$ to uniform random vectors on $[0,1]^2$ and $[0,1]^3$ respectively, in a way that preserves the structure of dependence between $Y_1$ and $Y_2$. The notion of vector rank proposed in CGHH:2017, based on the theory of measure transportation, has the desired properties, as we now explain.

The probability integral transform turns a continuous random variable $W$ into its rank $F_W(W)$, i.e., a uniform random variable on $[0,1]$ with the same rank ordering of outcomes. It is the unique increasing transformation that turns $W$ into a uniform random variable on $[0,1]$. The monotonicity of the transformation is what preserves rank ordering. Similarly, Proposition (ref) below, a seminal result in the theory of measure transportation in McCann:95, states essential uniqueness of the gradient of a convex function (hence cyclically monotone map) $R_1$ (resp. $R_2$) that turns absolutely continuous random vector $Y_1$ (resp. $ Y_2$) to a uniform $R_1(Y_1)$ on $[0,1]^2$ (resp. $R_2(Y_2)$ on $[0,1]^3$). See Appendix (ref) for a primer on optimal measure transportation, cyclical monotonicity and vector ranks.

Proposition (ref) is an extension of Brenier:91 (see also RR:90). It removes the finite variance requirement, which is undesirable in our context. Proposition (ref) is the basis for the definition of vector quantiles and ranks in CGHH:2017. In our context, it is applied with uniform reference measure.\footnote{ This vector quantile notion was introduced in GH:2012 and EGH:2012 and called $\mu $-quantile.} In terms of our empirical illustration, distribution $P$ in Definition (ref) below stands for the distribution of $Y_1$ (resp. $Y_2$), $\mu$ stands for the uniform distribution on $[0,1]^2$ (resp. $[0,1]^3$), $d=2$ (resp. $d=3$) and $ \nabla\psi^\ast$ is $R_1$ (resp. $R_2$).

In case $d=1$, gradients of convex functions are nondecreasing functions, hence vector quantiles and ranks of Definition (ref) reduce to classical quantile and cumulative distribution functions. As the notation indicates, the function $\psi^\ast$ of Proposition (ref) and Definition (ref) is the convex conjugate of $\psi$. Here we only define vector ranks in the absolutely continuous case, and, hence, $ \nabla\psi^\ast=(\nabla\psi)^{-1}$, so that ranks and quantiles are inverses of each other. In case of absolutely continuous distributions $P$ on $ \mathbb{R}^d$ with finite variance, the vector rank function solves a quadratic optimal transport problem, i.e., vector rank $R$ minimizes, among all functions $T$ such that $T(Y)$ is uniform on $[0,1]^d$, the quantity $ \mathbb{E }\| Y-T(Y)\|^2$, where $Y\sim P $. This property, which underlies cyclical monotonicity of vector ranks and quantiles, as explained in Appendix (ref), also has important computational implications, most notably in allowing the computation of empirical vector ranks using linear programming.

Return to our illustrative bivariate data generating process $Y_1$ for the Hang Seng and Nikkei indices. The notion of vector rank $R_1(Y_1)$ is best illustrated with empirical vector ranks $(\hat R_1(Y_{1i}))_{i=1}^n$. The latter solve a discrete version of the optimal transport problem of the previous paragraph. Let $(u_i)_{i=1}^n$ be a set of regularly spaced points on $[0,1]^2$. Define the empirical vector ranks $(\hat R_1(Y_{1i}))_{i=1}^n$ as a permutation $(u_{\sigma(i)})_{i=1}^n$ of $(u_i)_{i=1}^n$ that solves

among all permutations $\sigma$ of $\{1,\ldots,n\}$. The latter is a discrete version of the optimal transport problem above, hence a linear programming problem. It is also an assignment problem, for which many efficient algorithms exist in the literature, most notably the Hungarian algorithm in Munkres:57 and the auction algorithm in Bertsekas:88. Efficient ready-to-use implementations abound.

Vector Copulas and Vector Sklar Theorem

The vector ranks $R_1$ and $R_2$ are cyclically monotone transformations that turn $Y_1$ and $Y_2$ into uniform random vectors $R_1(Y_1)$ and $ R_2(Y_2)$ on $[0,1]^2$ and $[0,1]^3$ respectively. Hence they play the role of the probability integral transform for random vectors. The resulting 5-variate random vector $(R_1(Y_1),R_2(Y_2))$ has uniform bivariate and trivariate marginals, so that the remaining dependence structure is purely between-vector dependence. Figure (ref) shows the $5\times5$ scatterplot matrix for the sample of empirical ranks $(\hat R_1(Y_{1i}),\hat R_2(Y_{2i}))_{i=1}^n$. The off diagonal scatterplots within the diagonal $ 2\times2$ and $3\times3$ blocks look uniform, as expected, given that the vector ranks $R_1(Y_1)$ and $R_2(Y_2)$ are uniform. The other off-diagonal scatterplots illustrate between-vector dependence, as desired.

By construction, the distribution of the vector $(R_1(Y_1),R_2(Y_2))$ is a probability distribution on $[0,1]^5$ with uniform multivariate marginals on $[0,1]^2$ and $[0,1]^3$ respectively. This is all we require to define a vector copula.

Definition (ref) implies that the class of vector copulas is a subclass of copulas of dimension $d$ having the special feature that the $K$ non-overlapping multivariate marginals are uniform. When $d_{1}=\ldots =d_{K}=1$, the class of vector copulas is the class of copulas of dimension $ K$.

Now that we have settled on a definition for vector copulas, we need to argue that they characterize between-vector dependence of random vectors with given multivariate marginals. To this end, we need to show that we can always associate a vector copula with any random vector, that such a vector copula is unique under suitable assumptions and that it is invariant to suitably defined monotonic transformations of the multivariate marginals. Existence and uniqueness are shown in the following multivariate version of Sklar's Theorem. Invariance to suitably defined monotonic transformations of the multivariate marginals will be shown in section (ref).

In terms of our empirical illustration, distribution $P$ in the theorem below stands for the distribution of vector $(Y_1,Y_2)$, $P_j$ stands for the distribution of $Y_j$, $j=1,2$. The vector quantiles are $ Q_1=\nabla\psi_1$, and $Q_2=\nabla\psi_2$, and the vector ranks are $ R_1=\nabla\psi_1^\ast$ and $R_2=\nabla\psi_2^\ast$, $K=2$, $d_1=2 $, and $ d_2=3$. Finally, $C$ is the cumulative distribution function of $ (R_1(Y_1),R_2(Y_2))$.

When $d_{1}=\ldots =d_{K}=1$, Theorem (ref) reduces to Sklar's theorem. The vector Sklar theorem plays the same role as Sklar's Theorem for multivariate marginals.

First, Part (1) is the corollary of an important result in the theory of probability measures with fixed overlapping multivariate marginals, based on combinatorial arguments: Proposition (ref) in the appendix, due to Vorobev:62 and Kellerer:64. In the illustrative case with two multivariate marginals only, the result could be proven more directly with an appeal to the theory of Markov chains. See for instance a result of Ionescu Tulcea's in Chapter V, Section 1 of Neveu:65. The idea runs as follows. Take a probability distribution $P$ on $\mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$ (the distribution of vector $(Y_1,Y_2)$). Assume its multivariate marginals are absolutely continuous for simplicity. Given the distribution $P$ on $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$ and given the degenerate probability distributions on $[0,1]^{d_1}\times\mathbb{R}^{d_1}$ and on $\mathbb{R}^{d_2}\times[0,1]^{d_2}$ (the distributions of $ (\nabla\psi^\ast_1(Y_1),Y_1)$ and $(Y_2,\nabla\psi^\ast_2(Y_2))$ respectively), there exists a joint distribution on $[0,1]^{d_1}\times \mathbb{R}^{d_1}\times\mathbb{R}^{d_2}\times[0,1]^{d_2}$ with the prescribed marginals. Then, the latter's marginal on $[0,1]^{d_1}\times[0,1]^{d_2}$ is the required distribution $P_C$ and its cumulative distribution function is the copula $C$.

Part (1) of Theorem (ref) shows existence of a vector copula associated with any random vector. This motivates the following definition:

In our definition of vector copulas associated to a distribution $P$, we only rely on the subdifferential $\partial\psi^\ast$, and the relation ((ref)), for each of the multivariate marginals. However, uniqueness of the vector copula is only guaranteed under absolute continuity of the multivariate marginals, as is the case for (classical) copulas.

Second, Part (2) shows that a vector copula associated with $P$ does indeed measure the between-dependence structure in $P$. To see this, let $ Y=(Y_{1},\ldots ,Y_{K})$ be a random vector with distribution $P$ and let each $Y_{k}$, $k\leq K$ follow the multivariate marginal distribution $P_{k}$ . For each $k\leq K$, let $\nabla \psi _{k}$ be the vector quantile associated with $P_{k}$. Suppose that for each $k\leq K$, $P_{k}$ is absolutely continuous on $\mathbb{R}^{d_{k}}$ with support in a convex set. Then, from Definition (ref), $\nabla \psi _{k}^{\ast }\#P_{k}=\mu _{k} $\ for each $k\leq K$. Since the reference measure $\mu _{k}$ is an independence measure for each $k\leq K$, the (classical) copula function of $ \nabla \psi _{k}^{\ast }\left( Y_{k}\right) $ is the independence copula and hence the joint distribution of $\left( \nabla \psi _{1}^{\ast }\left( Y_{1}\right) ,\ldots ,\nabla \psi _{K}^{\ast }\left( Y_{K}\right) \right) $, i.e., the vector copula associated with $P$, measures the between-dependence structure in $P$.

Third, Part (3) of the vector Sklar theorem, or ((ref)), provides a general approach to computing vector copulas of multivariate distributions. In fact, for absolutely continuous marginals $P_{k}$ with density $f_{k}$, the Monge Amp\`{e}re Equation ((ref)) gives for each $k\leq K$ ,

We therefore obtain the following expression for the vector copula density $ c $, in terms of the density $f$ of distribution $P$ (when it exists):

Expression ((ref)) extends the copula density in ((ref)) to multivariate marginals with the vector quantile $\nabla\psi_{k}$ replacing $ F_{k}^{-1}$ in ((ref)).

Finally, Part (4) of the vector Sklar theorem provides a way of constructing distributions with given non-overlapping marginal distributions of any finite dimensions. Specifically, it states that for any distributions $P_{k}$ on $\mathbb{R}^{d_{k}}$ with vector quantiles $\nabla\psi_{k}$, each $k\leq K $,

defines a distribution $P$ on $\mathbb{R}^{d_{1}}\times \ldots \times \mathbb{R}^{d_{K}}$\ with marginals $P_{k}$, $k\leq K$, where $C$ is any vector copula. When $P_{C}$ and $P_{k}$ for each $k\leq K$ are absolutely continuous, the density function $f$ associated with $P$ is given by

which is a direct extension of the density decomposition of copula-based density functions in the univariate case. The above expresses the multivariate density function as the product of the copula density function evaluated at the vector ranks and the density function of $K$ independent random vectors with marginals $P_{1},\ldots ,P_{K}$. This can be used to construct both MLE and two-step estimators of vector copula-based models in exactly the same way as copula-based models, see FP:2014 and references therein.

Comonotonic Invariance

Return to our bivariate illustration, where $Y_1$ and $Y_2$ are random variables. The claim that the copula characterizes dependence relies on the fact that it is not affected by transformations of the marginals that leave rank ordering intact. In other words, $(T_1(Y_1),T_2(Y_2))$ has the same copula as $(Y_1,Y_2)$ if $T_1$ and $T_2$ are increasing functions. Now $ \tilde Y_1:=T_1(Y_1) $ and $Y_1$ are called comonotonic when $T_1$ is increasing. Equivalently, $Y_1$ and $\tilde Y_1$ are comonotonic if they are both increasing transformations of the same uniform random variable $U$ on $ [0,1]$. In this case, letting $U:=F_1(Y_1)$, we have indeed $Y_1=F_1^{-1}(U)$ and $\tilde Y_1=T_1(F_1^{-1}(U))$. So $Y_1$ and $\tilde Y_1$ are comonotonic if and only if they have the same ranks. We call the copula comonotonic invariant because vectors $(Y_1,Y_2)$ and $(\tilde Y_1,\tilde Y_2)$ have the same copula when $Y_1$ and $\tilde Y_1$ are comonotonic, and $ Y_2$ and $\tilde Y_2$ are comonotonic. This follows directly from the fact that $Y_1$ and $\tilde Y_1$ (resp. $Y_2$ and $\tilde Y_2$) have identical ranks.

Now, in the multivariate case, where $Y_1$ and $Y_2$ are random vectors, we obtain the same invariance property relative to a suitable extension of the notion of comonotonicity, where quantile functions $F_{1}^{-1}$ and $ F_{2}^{-1}$ are replaced with vector quantiles $Q_1$ and $Q_2$. The following definition is due to GH:2012 and EGH:2012, where it is called $\mu$-comonotonicity.

A related notion, namely $c$-comonotonicity, was proposed by PS:2010 . According to their definition, two random vectors $Y$ and $\tilde Y$ are $ c $-comonotonic if $\tilde Y=\nabla\psi(Y)$ for some convex function $\psi$. The two notions of multivariate comonotonicity both reduce to traditional comonotonicity in the univariate case, but they differ in the multivariate case. For instance, two Gaussian random vectors $Y$ and $\tilde Y=\Sigma^{1/2}Y$ may not be comonotonic according to Definition (ref) , so that the invariance result does not apply to them.

We now state properties of vector copulas that relate to vector comonotonicity. Since two vectors are comonotonic if they have identical vector ranks, comonotonic invariance is indeed an invariance property of vector copulas to transformations that leave ranks unchanged, as desired.

When $d_{k}=1$ for all $k\leq K$, comonotonic continuous random variables are increasing transformations of each other and Theorem (ref) reduces to the well-known invariance property of copulas. Theorem (ref) shows the importance of deriving vector copulas to flexibly model between-vector dependence. Indeed, existing parametric multivariate distribution families cannot model between-vector dependence without constraining the within vector dependence and marginals. Conversely, a change in the within-vector dependence, such as replacing $Y$ by $\tilde{Y} =\Sigma ^{1/2}Y$, also changes the between-vector dependence as characterized by the vector copula. If $\tilde{Y}:=a+bY$, where $a\in R^{d}$ and $b>0$ is a scalar, then $Y$ and $\tilde{Y}$ are comonotonic. Indeed, if $\psi$ is a convex function, then $a+b\nabla\psi$ is the gradient of a convex function. Hence, Theorem (ref) applies to location transformations.

Antitone Transformations

For two vectors, we can also entertain a notion of countermonotonicity as a multivariate extension of decreasing transformations of two random variables.

The lemma below shows that vector copulas for random vectors with comonotonic and countermonotonic subvectors are related in simple, predictable ways.

Extremal Vector Copulas

A first step towards modeling dependence with copulas is to model extremes. We present vector copulas that characterize independence on the one end, and maximal dependence on the other end.

We will denote $C^{I}$, $\overline{C}$ and $\underline{C}$ the independence, comonotonic and countermonotonic vector copulas respectively.

Hence, for any collection of Borel measurable subsets $A_{k}\subset \lbrack 0,1]^{d}$, $k\leq K,$ the probability distribution associated with the comonotonic vector copula takes values $P_{\overline{C}}(A_{1}\times\ldots \times A_{K})=\mu \left( \cap _{k\leq K}A_{k}\right) $ and the countermonotonic vector copula takes values $P_{\underline{C}} (A_{1}\times A_{2})=\mu \left( A_{1}\cap (1_{d}-A_{2})\right) .$ Letting $A_{k}=(0,u_{k}]$ with $u_{k}\in \lbrack 0,1]^{d}$ for each $k\leq K$, we obtain

When $d=1$, $\overline{C}\left( u_{1},\ldots ,u_{K}\right) =\min\{u_1,\ldots,u_K\}$, the Fr\'{e}chet upper bound copula. However $ \overline{C}$ differs from the Fr\'{e}chet upper bound when $d>1$. Since it is a copula, $\overline{C}\left( u_{1},\ldots ,u_{K}\right) \leq \min_{j\leq d,k\leq K}u_{kj}$ and strict inequality holds for some $\left( u_{1},\ldots ,u_{K}\right) \in \lbrack 0,1]^{dK}$. Similarly, when $d=1$,

which is the Fr\'{e}chet lower bound copula. When $d>1$, $\underline{C} (u_{1},u_{2})$ is still a copula although the Fr\'{e}chet lower bound is not.

It follows from Theorem (ref) that for any collection of absolutely continuous distributions $P_{k}$ on $\mathbb{R}^{d}$ with vector quantiles $\nabla\psi_{k} $, each $k\leq K$, the expression below defines a distribution $\overline{P}$ on $\mathbb{R}^{d}\times \ldots \times \mathbb{R} ^{d}$ with marginals $P_{k}$, $k\leq K $:

The distribution $\overline{P}$ characterizes comonotonic random vectors with absolutely continuous marginals $P_{k}$, $k\leq K$. To show this, first let $\left( Y_{1},\ldots ,Y_{K}\right) \sim \overline{P}$. Then $\left( \nabla\psi_{1}^{\ast}(Y_{1}),\ldots ,\nabla\psi_{K}^{\ast}(Y_{K})\right) \sim P_{\overline{C}}$ and hence there exists a uniform random vector $U$ on $[0,1]^d$ such that $\nabla\psi_{k}^{\ast}(Y_{k})=U$ for all $k\leq K$. Conversely, let $\left( Y_{1},\ldots ,Y_{K}\right) $ be comonotonic random vectors with absolutely continuous marginals $P_{k}$, $k\leq K$. By definition, there exists a uniform random vector $U$ on $[0,1]^d$ such that $ \nabla\psi_{k}^{\ast}(Y_{k})=U$ for all $k\leq K$. Hence

Specifically for $d=1,A_{k}=(-\infty ,x_{k}]$ with $x_{k}\in \mathbb{R}$ for each $k\leq K$, we have

which is the well-known Fr\'{e}chet upper bound distribution. However, it is well known that for $d>1$, the Fr\'{e}chet upper bound $\min_{1\leq k\leq K}F_{k}\left( x_{k}\right) $ is not a distribution function except for very specific marginals, see Proposition 5.3 in Rusch:2010. In sharp contrast, the comonotonic vector copula $\overline{C}$ always defines a distribution function for any marginals through ((ref)).

Dependence modeling

As noted above, the success of copula theory is partly due to the ability to parsimoniously model dependence in random vectors with parametric copula families, and to produce flexible new families of multivariate distributions by fitting parametric copulas with any given marginals. We show that the same objective can be achieved with vector copulas, thanks to the vector Sklar Theorem.

Theorem (ref) implies that given a vector copula $C$, and given a set of absolutely continuous multivariate marginal distributions $P_{k}$ on $ \mathbb{R}^{d_{k}}$ with associated vector quantile $\nabla\psi_{k}$, the distribution $P$ defined for Borel sets $A_{1},\ldots ,A_{K}$, by

is a multivariate distribution with vector copula $C$ and non-overlapping marginals $P_{k}$. Furthermore, if vector copulas $C$ admits density $c$, ( (ref)) implies that

is the density of a multivariate distribution with vector copula $C$ and marginal distributions $P_{k}$ for $k\leq K$. Given a parametric vector copula family $\{C(.;\theta): \theta\in\Theta\}$, but without parameterizing the marginal distributions, the above expression results in a semiparametric multivariate distribution, with finite dimensional vector copula parameter $ \theta$ (which measures the between vector dependence) and infinite dimensional marginal parameters $f_{k}$, all $k\leq K $. Combined with traditional copula modeling of the multivariate marginals to further reduce dimensionality, vector copulas developed here also allow a flexible hierarchical approach to multivariate modeling.

Our first family of vector copulas is obtained using Theorem (ref) from a Gaussian vector, whose multivariate marginals are standard Gaussian vectors.

To illustrate the vector copula approach to multivariate modeling, we fit a Gaussian vector copula to our 5-dimensional random vector $(Y_1,Y_2)$ of residuals. Assume $(Y_1,Y_2)$ have Gaussian vector copula $C^{\mbox{\scriptsize Ga} }(u_1,u_2;\Omega)$ as in Example (ref). The covariance matrix $ \Omega$ is estimated from the empirical ranks $(\hat R_1(Y_{1i}),\hat R_2(Y_{2i}))_{i=1}^n$ using the method of moments and Figure (ref) shows a $2\times3$ off diagonal scatterplot matrix from a sample of $n$ data points simulated independently from the estimated Gaussian vector copula. Figure (ref) replicates the top right off-diagonal block from the vector rank scatterplot matrix in Figure (ref).

Parametric Vector Copula Families

This section introduces two general classes of parametric vector copulas. Similarly to Gaussian vector copulas constructed in the previous section, the first class of vector copulas is constructed via the multivariate analogue of the inversion method from elliptical distributions. However, since vector ranks of general elliptical distributions do not have closed form expressions, we introduce compositions of a finite number of McCann's measure transport maps and call them composition measure transports. These maps will be used to construct parametric vector copulas from elliptical distributions. The same principle can be extended to other parametric families of distributions. The second class of vector copulas we develop here, called Kendall vector copulas is derived from a stochastic representation of vector copulas in Proposition (ref).

Composition Measure Transport

We first introduce a class of maps that are compositions of gradients of convex functions and push the uniform forward to arbitrary distributions.

Existence of composition measure transports (hereafter composition MT) is guaranteed by Proposition (ref). When $L=1$, the maps $T$ and $ T^{\ast}$ in Definition (ref) reduce to vector quantiles and ranks of Definition (ref). By allowing $L$\ to be larger than $1$, we are able to choose convex functions $\psi _{l},$ $l\leq L$\ such that the composition MT maps have explicit expressions. Composition MT maps are the tools we use to map multivariate marginal distributions into multivariate uniform distribution to remove all within vector dependence and marginal information. This is achieved with the following proposition, whose proof only requires a very minor variation on the proof of Theorem (ref).

When $T_{k}$ is the vector quantile of $P_{k}$ for each $k\leq K$, the $ \left( T_{1},...,T_{K}\right) $-vector copula derived from $P$ is the vector copula associated with $P$ of Definition (ref). When at least one of the composition MT maps $T_{k}$, $k\leq K$ is not a vector quantile, the $ \left( T_{1},...,T_{K}\right) $-vector copula derived from $P$ is not the vector copula associated with $P$, but can be used to construct multivariate distributions $P^{\prime }$ via part (4) of the vector Sklar Theorem (ref) such that the vector copula associated with $P^{\prime }$ is the prespecified $\left( T_{1},...,T_{K}\right) $-vector copula derived from $P$ . Since compositions of gradients of convex functions are in general not gradients of convex functions themselves, a $\left( T_{1},...,T_{K}\right) $ -vector copula derived from $P$ does not characterize rank dependence between random vectors $\left( Y_{1},...,Y_{K}\right)$ with joint distribution $P$, unless all the maps $T_{1},...,T_{K}$ happen to be gradients of convex functions.

Proposition (ref) presents a general approach to constructing parametric families of vector copulas from parametric families of multivariate distributions $P$ with absolutely continuous marginals $P_{k}$ on $\mathbb{R}^{d_{k}}$, $k\leq K$. A critical step in this approach is to derive composition MT map $T_{k}$ of the marginal distribution $P_{k}$, $ k\leq K$ such that $T_{k}$ has an explicit expression. Below we illustrate this approach for general elliptical distributions to construct elliptical $ \left( T_{1},...,T_{K}\right) $-vector copulas.

Elliptical $\left( T_{1},...,T_{K}\right) $-Vector Copulas

We first present explicit expressions for composition MT maps of general elliptical distributions and then present explicit expressions for $\left( T_{1},...,T_{K}\right) $-vector copulas derived from elliptical distributions.

Examples of elliptical distributions include the following (see e.g., Chapter 3 of NFE:2005):

Explicit expressions for elliptical $\left( T_{1},...,T_{K}\right) $-vector copulas rely on explicit expressions for the composition MT maps $\left( T_{1},...,T_{K}\right) $. To construct these maps, we rely on Lemma (ref) in Appendix (ref).

We now combine Lemma (ref) and Proposition (ref) to characterize elliptical $\left( T_{1},...,T_{K}\right) $-vector copulas.

Explicit expressions for the composition MT maps $T_{k}$ of elliptical distributions $P_{k}$ can be derived from Lemma (ref) and the corresponding elliptical copulas can be obtained from Lemma (ref). The latter also provides a generic procedure to simulate a random vector distributed according to a prescribed elliptical $\left( T_{1},...,T_{K}\right) $-vector copula:

Step 1. Generate a random vector $Y=(Y_1,\ldots,Y_K)$ from the prescribed elliptical distribution;

Step 2. Let $U=(T_1^\ast(Y_1),\ldots,T_K^\ast(Y_K))$, where $T_k$ is given in Lemma (ref). Vector $U$ is distributed according to the desired elliptical $\left( T_{1},...,T_{K}\right) $-vector copula.

We present two examples of elliptical $\left( T_{1},...,T_{K}\right) $ -vector copulas below.

Kendall Vector Copulas

A second method to construct parametric vector copulas, inspired by hierarchical Kendall copulas in Brechmann:2014, is to make use of the stochastic representation of a vector copula we establish below. We illustrate this method by introducing a new class of parametric vector copulas. It turns out to be the subclass of hierarchical Kendall copulas with independence cluster copulas. We thus call them Kendall vector copulas.

Since $U_{1}^{\left( d_{1}\right) },...,U_{K}^{\left( d_{K}\right) }$ are mutually independent, the dependence structure in a Kendall vector copula denoted as $C_{KV}$ is characterized by the nesting copula $C_{0}$. We show in Appendix (ref) that the class of Kendall vector copulas defined in Definition (ref) is the subclass of hierarchical Kendall copulas introduced in Brechmann:2014 with independence cluster copulas. When $C_{0}$ is absolutely continuous, this allows us to derive the following closed form expression for the density of Kendall vector copulas:

where $c_{0}$ is the density of $C_{0}$.

A special class of Kendall vector copulas is obtained when the nesting copula $C_{0}$ is Archimedean (see Appendix (ref)). By choosing Clayton, Gumbel, and Frank copula generators, we obtain different classes of Kendall vector copulas with different between vector dependence structure.

Finally we present an algorithm to simulate from a Kendall vector copula with nesting copula $C_{0}$ based on the stochastic representation of a Kendall vector copula in Proposition (ref). It is similar to Algorithms 14 and 20 in Brechmann:2014.

Step 1. Generate $\left( V_{1},...,V_{K}\right) $ from $C_{0}.$

Step 2. Let $R_{k}=F_{R_{k}}^{-1}\left( V_{k}\right) $ for each $k\leq K$.

Step 3. Generate mutually independent $U_{k}^{\left( d_{k}\right)}$ from the uniform distribution on the unit simplex on $\mathbb{R}^{d_{k}}$ for $k\leq K $.

Step 4. Let $U_{k}=\left( U_{k1},\ldots ,U_{kd_{k}}\right) $, with $ U_{kj}=\exp \left( R_{k}U_{kj}^{\left( d_{k}\right) }\right) $ for $ j=1,...,d_{k}$ and $k\leq K$. Then $U=\left( U_{1},...,U_{K}\right) $ follows the Kendall vector copula $C_{KV}$ with nesting copula $C_{0}.$

To illustrate the Kendall vector copula construction and its possible uses, we fit Kendall copulas with Clayton, Frank, Gaussian or Gumbel nesting copulas and investigate the effect of the financial crisis on between-vector dependence in our vector of five international stock indices. As before, $ R_{1}(Y_{1})$ and $R_{2}(Y_{2})$ are the population vector ranks of $Y_{1}$ (Hang Seng and Nikkei) and $Y_{2}$ (FTSE, S&P and DAX) respectively. Assume $(Y_{1},Y_{2})$ have Kendall vector copula given in ((ref)). The nesting copula $C_{0}$, i.e., the copula of $(R_{1},R_{2})$, is either Clayton, Frank, Gaussian or Gumbel. Formulas for the latter are given in Appendix (ref) for convenience. Estimation is straightforward, as the densities of the Kendall copula with Clayton, Frank, Gaussian, and Gumbel nesting copulas are given in closed form.

Figure (ref) shows bivariate scatterplots for the empirical version $(\hat{V}_{1},\hat{V}_{2})$ of $ (V_{1},V_{2}):=(F_{R_{1}}(R_{1}),F_{R_{2}}(R_{2}))$ for the pre-crisis, crisis and post-crisis periods respectively. Appendix B.9 shows that $V_{1}=K_{2}(U_{1}U_{2})$ and $V_{2}=K_{3}(U_{3}U_{4}U_{5})$ , where $(U_{1},U_{2})$ and $(U_{3},U_{4},U_{5})$ denote the components of the ranks $R_{1}(Y_{1})$ and $ R_{2}(Y_{2})$ respectively and $K_{k}$ denotes the Kendall distribution of the independence copula, whose expression is given in ((ref)). The sample $(\hat{V}_{1i},\hat{V}_{2i})_{i=1}^{n}$ is computed from the sample of empirical vector ranks via $\hat{V}_{1i}:=K_{2}( \hat{U}_{1i}\hat{U}_{2i})$ and $\hat{V}_{2i}:=K_{3}(\hat{U}_{3i}\hat{U}_{4i} \hat{U}_{5i})$, where $(\hat{U}_{1i},\hat{U}_{2i})$ and $(\hat{U}_{3i},\hat{U }_{4i},\hat{U}_{5i})$ denote the components of the empirical ranks $\hat{R} _{1}(Y_{1i})$ and $\hat{R}_{2}(Y_{2i})$ respectively. Clayton, Gaussian, Frank and Gumbel bivariate copulas are fitted to the sample $(\hat{V}_{1i}, \hat{V}_{2i})_{i=1}^{n}$ and estimated parameters are transformed to the corresponding value of Kendall's $\tau $ for comparison. Results are given in Table (ref). The vector copula analysis of these five stock indices thereby yields an insight into the financial contagion that accompanied the 2008 financial crisis. Kendall's $\tau $ increased during the crisis, then declined again, but remained weakly higher in the post-crisis than in the pre-crisis periods.

Concluding remarks

We have proposed a framework to characterize dependence between random vectors, as distinct from within vector dependence in the same way copulas characterize dependence between random variables as distinct from marginal information. The basic building block is what we call a vector copula, which is simply a multivariate distribution with uniform multivariate marginals. Hence, the class of vector copulas is merely a subclass of the class of copulas, with the added constraint that the multivariate marginals are uniform. The contribution of the paper was to associate one such vector copula with any multivariate distribution with given non overlapping marginals and to show that this vector copula does indeed characterize between vector dependence for such a distribution. By characterization of between vector dependence, we mean that the original distribution can be recovered from the vector copula and the multivariate marginals, the vector copula is unique in case of absolutely continuous multivariate marginals, and finally, that the vector copula is invariant to the class of transformations that leave between vector dependence unchanged.

The main device we used to derive the vector copula associated with a distribution is the multivariate analogue of the probability integral transform, or vector ranks, introduced in CGHH:2017 and based on the theory of measure transportation. However, since vector ranks are rarely available in explicit form, parametric families of vector copulas cannot be readily constructed from existing parametric families of multivariate distributions. We therefore proposed a couple of strategies to construct parametric families of vector copulas. One is based on ideas inherited from the literature on hierarchical copula models, in which the lower level copulas are independence copulas. Another relies on measure transport theory to turn parametric families of multivariate distributions into vector copulas by transforming the multivariate marginals into uniforms. To obtain explicit forms, we rely on compositions of optimal transport maps. However, since the composition of optimal transport maps is not, in general, an optimal transport map itself, the parametric family of vector copulas thus obtained is not, in general, the family of vector copulas associated with the multivariate distribution used to derive it. In other words, it does not characterize between vector dependence in the latter. It is nonetheless a useful tool to construct new families of multivariate distributions in a way that properly distinguishes between vector dependence and within vector dependence, a goal that cannot be achieved with existing parametric families of multivariate distributions.

Throughout the paper, we have illustrated the use of semiparametric vector copula based models to study financial contagion through the evolution of between-vector dependence before, during and after the 2008 financial crisis for a collection of five aggregate stock indices. Developing formal estimation and inference methods for such models is of utmost importance and is an on-going project of the authors. Once developed, we anticipate many applications of this new tool, that mirror applications of traditional copulas in quantitative finance and econometrics.