The Multivariate Generalized Autoregressive Conditional Heteroskedasticity model extends the univariate GARCH to multiple time series data. In a univariate setting, a GARCH(p,q) model assumes that the variance of the time series is influenced by past values of the squared returns (or innovations) and past variances. For a series of returns $\left\{r_t\right\}$ the GARCH model is defined as:

$\begin{array}{l}{r}_t=\mu +e_t\\ e_t=h_t^{1/2}{z}_t\\ h_t={\alpha }_0+\underset{i=1}{\overset{q}{{\displaystyle \sum }}}{\alpha }_i{e}_{t-i}^2+\underset{j=1}{\overset{p}{{\displaystyle \sum }}}{\beta }_j{h}_{t-j}\end{array}$ (1)

where $\mu$ represents the average of the asset’s log return, $e_t$ is the innovation at time $t$, $h_t$ stands for the variance, $z_t$ is a white noise process with a mean of zero and variance of one, ${\alpha }_0>0$, ${\alpha }_i\ge 0$, ${\beta }_j\ge 0$ and ${\alpha }_i+{\beta }_j\le 1$.

In order to generalize the univariate GARCH model to a multivariate framework, we consider a multivariate time series $r_t$, which represents the vector of log-returns of $d$ assets. This vector $r_t$ has a dimension of $d\times 1$ and is assumed to be conditionally heteroskedastic, meaning its variance changes over time depending on past information. Let ${ℰ}_{t-1}$ represent the information set created by the observed series $r_t$ up until time $t-1$. The standard MGARCH framework is given by:

$r_t={\mu }_t+{ϵ}_t$ (2)

where $r_t$ represents a $d\times 1$ vector of returns, ${\mu }_t$ denotes the mean vector, and ${ϵ}_t$ represents the error terms vector associated with time $t$;

${\epsilon }_t=H_t^{1/2}{\eta }_t$ (3)

where $H_t$ is an $d\times d$ conditional covariance matrix of $r_t$, $H_t^{1/2}$ is any $d\times d$ matrix at time $t$ that scales the innovations of ${\epsilon }_t$, obtainable via a Cholesky factorization of $H_t$ and ${\eta }_t$ represents a vector of iid errors with dimension $d\times 1$, such that $\mathbb{E}\left[{\eta }_t\right]=0$ and $\mathbb{E}\left[{\eta }_t{{\eta }^{\prime }}_t\right]=I$. There are two primary categories of MGARCH models: those that model $H_t$ directly and those that focus on conditional variances and correlations, decomposing the covariance matrix into correlations and standard deviations.

[26] , proposed CCC model, one of the simplest multivariate correlation models. Since the matrix of conditional covariances in this model doesn’t change over time, it can be written as:

$H_t=D_t{R}_t{D}_t$ (4)

where $R_t=\left[{\rho }_{ij}\right]$ is the $d\times d$ positive definite conditional correlation matrix, with ${\rho }_{ij}=1$ for $i=1,\cdots ,d$ and is time invariant such that $R_t=R$ and $D_t=\text{diag}\left(h_{1t}^{1/2},\cdots ,h_{dt}^{1/2}\right)$ is $d\times d$ matrix consisting of the standard deviations of the ${\epsilon }_t$ elements, with each $h_{it}$ being a univariate GARCH process. This suggests that the conditional covariance matrix’s off-diagonal components are provided by:

${\left[H_t\right]}_{ij}=h_i^{1/2}{h}_j^{1/2}{\rho }_{ij},\forall i\ne j$ (5)

where $h_{it}^{1/2}$ and $h_{jt}^{1/2}$ are the standard deviations of asset $i$ and $j$ at time $t$. Consequently, the vector form of $h_t$ can be given as:

$h_t=a+\underset{j=1}{\overset{q}{{\displaystyle \sum }}}{B}_j{ϵ}_{t-j}^{\left(2\right)}+\underset{j=1}{\overset{p}{{\displaystyle \sum }}}{C}_j{h}_{t-j}$ (6)

where $a$ represents a $d\times 1$ vector comprising constant values, while the matrices $B_j$ and $C_j$ are diagonal $d\times d$ matrices, and ${ϵ}_{t-j}^{\left(2\right)}={ϵ}_{t-j}\odot {ϵ}_{t-j}$. The formulation in Equation (6) enables each $h_{it}$ to be influenced by its historical variances and prior squared returns. For $H_t$ to maintain positive definiteness, $R$ needs to be positive definite, and the vectors $a$, along with the matrices $B_j$ and $C_j$, must also be positive.

The Extended CCC (ECCC) model, which was introduced by [27] , relaxed the requirement that the matrices $B_j$ and $C_j$ are diagonal. The relaxation allows each conditional variance equation to incorporate the past variances and squared innovations from all series. For example, $h_{it}$ in the first order ECCC model is equivalent to

$h_{it}=c_i+a_{11}{\epsilon }_{1,t-1}^2+\cdots +a_{1d}{\epsilon }_{d,t-1}^2+b_{11}{\epsilon }_{1,t-1}^2+\cdots +b_{1d}{\epsilon }_{d,t-1}^2$ (7)

where $i=1,\cdots ,d$ and $a_{ij}$ and $b_{ij}$ are parameters that determine how past squared returns and past volatilities of asset $i$, respectively, influence the current volatility of asset $i$. An advantage of the ECCC over CCC model is that the autocorrelation structure for the squared returns is more flexible. Although the CCC and ECCC models’ parameter structures are relatively simple and computationally effective, the assumption that $R_t$ remains constant can be overly limiting. Therefore, it may be beneficial to generalize the CCC model by allowing $R_t$ to vary over time while preserving the previous decomposition.

The Dynamic Conditional Correlation model, introduced by [28] , extends the CCC model by permitting correlations to varying over time, which helps in capturing the dynamic co-volatility among assets. Using a proxy matrix $P_t$, the model breaks down $H_t$ into time-varying conditional standard deviations $D_t$ and the correlation matrix $R_t$, such that

$R_t=P_t^{\ast -1/2}{P}_t{P}_t^{\ast -1/2}$ (8)

$P_t=\left(1-a-b\right)S+a{\eta }_t{{\eta }^{\prime }}_t+b{P}_{t-1}$ (9)

where $P_t^{\ast }=\text{diag}\left(p_{1t},\cdots ,p_{dt}\right)$, while $S$ denotes standardized errors’ unconditional covariance matrix, such that

$S=\text{Cov}\left({\eta }_t{{\eta }^{\prime }}_t\right)=\frac{1}{T}{\displaystyle {\sum }_{t=1}^T{\eta }_t{{\eta }^{\prime }}_t}$

and parameters $a$ and $b$ represent scalars, where $a+b\le 1$, $a\ge 0$ and $b\ge 0$. While the aspect of DCC having two parameters can be seen as an advantage, it might also pose a challenge when many assets are considered, since all $\frac{d\left(d-1\right)}{2}$ processes are correlation are confined to the same structure. [29] , introduced a Quadratic Flexible DCC model which applies the BEKK structure to $R_t$. Thus, $P_t$ is given as:

$P_t=A^{\prime }SA+B^{\prime }{\eta }_{t-1}{{\eta }^{\prime }}_{t-1}B+C{P}_{t-1}{C}^{\prime }$ (10)

However, the Quadratic Flexible DCC has $\frac{3d\left(d+1\right)}{2}$ parameters, which

becomes impractical for large systems due to the curse of dimensionality. To reduce the parameters, assets can be grouped by characteristics such as industry or sector and a block diagonal structure applied to the coefficient matrices according to the grouping.

The Asymmetric DCC GARCH model by [30] incorporates asymmetry in the correlation, since the standard DCC model only permits asymmetries in the variance and not the correlations. Thus, $P_t$ is defined as:

$P_t=\left(S-BS{B}^{\prime }-CS{C}^{\prime }-G{S}^{-}{G}^{\prime }\right)+B^{\prime }{\eta }_{t-1}{{\eta }^{\prime }}_{t-1}B+C{P}_{t-1}{C}^{\prime }+G^{\prime }{\eta }_{t-1}^{-}{\eta }_{t-1}^{-}\text{'}G$ (11)

where the parameter matrices $B$, $C$, and $G$ are of dimension $d\times d$; ${\eta }_t^{-}=1_{\left\{{\eta }_t<0\right\}}\odot {\eta }_t$ denotes the element-wise product of ${\eta }_t$ and $S^{-}=\text{Cov}\left({\eta }_t^{-}{\eta }_t^{-}\text{'}\right)$.

The parameters of the Multivariate DCC and CCC models are estimated in two step, where in step one the univariate conditional variance parameters ( ${\beta }_v$) are estimated while the correlation parameters ( ${\beta }_c$) are estimated in the step two, given ${\beta }_v$. When ${ϵ}_t$ follows a multivariate normal distribution with mean vector zero and covariance matrix $H_t$, the d-dimensional density function is:

$f\left({\epsilon }_t|H_t\right)=\frac{1}{{\left(2\pi \right)}^{d/2}{\left|H_t\right|}^{1/2}}\text{exp}\left(-\frac{1}{2}{{\epsilon }^{\prime }}_t{H}_t^{-1}{\epsilon }_t\right)$ (12)

The likelihood equation for $T$ observations is:

$L\left(\beta \right)=\underset{t=1}{\overset{T}{{\displaystyle \prod }}}\frac{1}{{\left(2\pi \right)}^{d/2}{\left|H_t\right|}^{1/2}}\text{exp}\left(-\frac{1}{2}{{ϵ}^{\prime }}_t{H}_t^{-1}{ϵ}_t\right)$ (13)

Thus, the log-likelihood equation is given as

$l\left(\beta \right)=-\frac{dT}{2}\log 2\pi -\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log \left|H_t\right|-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}{{\epsilon }^{\prime }}_t{H}_t^{-1}{\epsilon }_t$ (14)

where $\beta$ represents all parameters in the models such as elements of $A$, $B$ and $C$, $\left|H_t\right|$ and $H_t^{-1}$ represent the determinant and inverse of $H_t$, respectively, $d$ denotes total assets, and $T$ represents the total observations. If $H_t=D_t{R}_t{D}_t$, then the function of log-likelihood in Equation (14) can be expressed as

$\begin{array}{c}l\left(\beta \right)=-\frac{dT}{2}\log 2\pi -\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log \left|D_t{R}_t{D}_t\right|-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}{{\epsilon }^{\prime }}_t{D}_t^{-1}{R}_t^{-1}{D}_t^{-1}{\epsilon }_t\\ =-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(d\log 2\pi +2\log \left|D_t\right|+\log \left|R_t\right|+{{\epsilon }^{\prime }}_t{D}_t^{-1}{R}_t^{-1}{D}_t^{-1}{\epsilon }_t\right)\end{array}$ (15)

In step one, $R_t$ in Equation (15) is replaced by an identity matrix such that the $l\left(\beta \right)$ corresponds to a sum of $d$ univariate log-likelihood functions:

$\begin{array}{c}{l}_v\left(\beta \right)=-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(d\log \left(2\pi \right)+2\log \left|D_t\right|+\text{log}\left|I_n\right|+{{\epsilon }^{\prime }}_t{D}_t^{-1}{I}_n^{-1}{D}_t^{-1}{\epsilon }_t\right)\\ =-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(d\log \left(2\pi \right)+2\log \left|D_t\right|+{{\epsilon }^{\prime }}_t{D}_t^{-1}{I}_n{D}_t^{-1}{\epsilon }_t\right)\\ =-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\underset{i=1}{\overset{d}{{\displaystyle \sum }}}\left(d\log \left(2\pi \right)+\log \left(h_{i,t}\right)+\frac{{\epsilon }_{i,t}^2}{h_{i,t}}\right)\end{array}$ (16)

In step two, the log-likelihood function given ${\theta }_v$ in step one is

$l_c\left(\beta \right)=-\frac{1}{2}\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\left(\log \left|P_t\right|+{{\epsilon }^{\prime }}_t{P}_t^{-1}{\epsilon }_t-{{\epsilon }^{\prime }}_t{\epsilon }_t\right)$ (17)

Thus, ${\hat{\beta }}_v=\arg \max \left\{l_v\left(\beta \right)\right\}$ and ${\hat{\beta }}_c=\arg \max \left\{l_c\left(\beta \right)\right\}$.

A copula is a function that creates a joint distribution by coupling univariate marginal distributions. The concept of copula introduced by [6] , separates the effect of the dependency between variables from the impact of each marginal variable.

Definition 3.1. The function $C\left(u_1,\cdots ,u_d\right)$, which maps ${\left[0,1\right]}^d\to \left[0,1\right]$, is a copula and satisfied the following properties:

1) $C\left(u_1,\cdots ,u_d\right)$ is grounded, meaning that for any $C\left(u_1,\cdots ,u_d\right)=0$ if any $u_i=0$.

2) $C\left(u_1,\cdots ,u_d\right)=C\left(u_1,\cdots ,u_{i-1},u_{i+1},\cdots ,u_d\right)$ if any $u_i=1$.

3) $C\left(u_1,\cdots ,u_d\right)$ is d-increasing for any two points $l$ and $m$ in ${\left[0,1\right]}^n$, where $l\le m$ and the volume ( $V_C\left(\left[l,m\right]\right)$) of the box formed by the vertices $l$ and $m$ is non-negative. For instance, when $d=2$,

$V_C\left(\left[l,m\right]\right)=C\left(l_2,m_2\right)-C\left(l_1,m_2\right)-C\left(l_2,m_1\right)+C\left(l_1,m_1\right)\ge 0$

Theorem 3.1. (Sklar’s Theorem) Let $F_X\left(x_1,\cdots ,x_n\right)$ be the joint CDF of the random variables $X_1,\cdots ,X_d$ with marginal CDFs $F_1,\cdots ,F_d$. A d-dimensional copula $C$ exists such that the following relationship holds for every $x\in {ℝ}_d$:

$F_X\left(x_1,\cdots ,x_d\right)=C\left(F_1\left(x_1\right),\cdots ,F_d\left(x_d\right)\right)$ (18)

The copula $C$ is said to be unique if all the individual CDFs ( $F_1,\cdots ,F_d$) are continuous.

Theorem 3.2. Let $C$ be a multivariate Copula. For every $u_i$ in ${\left[0,1\right]}^d$, where $i=1,\cdots ,d$, the partial derivative $\frac{\partial C\left(u_1,\cdots ,u_d\right)}{\partial u_i}$ exist

$\frac{\partial C\left(u_1,\cdots ,u_d\right)}{\partial u_i}\ge 0$

Theorems 3.1 and 3.2 are proved in [31] .

Definition 3.2. (Copula density) For each $u_1,\cdots ,u_d$ in ${\left[0,1\right]}^d$, the copula density $c\left(u_1,\cdots ,u_d\right)$ is given by:

$c\left(u_1,\cdots ,u_d\right)=\frac{{\partial }^{n}C\left(u_1,\cdots ,u_d\right)}{\partial u_1\cdots u_d}$ (19)

Differentiating Equation (18) yields:

$f\left(x_1,\cdots ,x_d\right)=\underset{i=1}{\overset{d}{{\displaystyle \prod }}}{f}_i\left(x_i\right)\cdot c\left(F_1\left(x_1\right),\cdots ,F_d\left(x_d\right)\right)$ (20)

where $f_i$ is the probability density function of $X_i$ for $i=1,\cdots ,d$. Therefore, the copula density can now be given as the ratio of the joint density $f$ to the product of the individual densities:

$c\left(u_1,\cdots ,u_d\right)=\frac{f\left(u_1,\cdots ,u_d\right)}{{{\displaystyle \prod }}_{i=1}^d{f}_i\left(u_i\right)}$ (21)

Given two random variables $X_1$ and $X_2$, a bivariate copula function $C:{\left[0,1\right]}^2\to \left[0,1\right]$ couples the marginal CDFs of the two variables to form their joint distribution. The two main bivariate copula families are the Archimedean and elliptical Copulas.

The Gaussian (normal) and Student-t Copulas are mostly used for elliptical Copulas and they model symmetric dependence structures effectively. Based on the Sklar’s Theorem, elliptical copula models’ general equation is as follows:

$F_{X_1,X_2}\left(x_1,x_2\right)=C\left(F_{X_1}\left(x_1\right),F_{X_2}\left(x_2\right)\right)$ (22)

The Gaussian copula belongs to the normal distribution and its general form is:

$\begin{array}{l}{C}_{\rho }^{Gauss}\left(F_{X_1}\left(x_1\right),F_{X_2}\left(x_2\right)\right)={\Phi }_{\rho }\left(x_1,x_2\right)\\ ={\displaystyle {\int }_{-\infty }^{x_1}{\displaystyle {\int }_{-\infty }^{x_2}\frac{1}{2\pi {\left(1-{\rho }^2\right)}^{1/2}}\exp \left[-\frac{x_1^2-2\rho x_1{x}_2+x_2^2}{2\left(1-{\rho }^2\right)}\right]\text{d}{x}_1\text{d}{x}_2}}\end{array}$ (23)

where ${\Phi }_{\rho }\left(.,.\right)$ represents the CDF of bivariate standard normal and $\rho$ is the correlation coefficient between $x_1$ and $x_2$.

The student-t copula (or t-copula) belongs to the Student-t distribution expressed is:

$\begin{array}{l}{C}_{r,\rho }^t\left(F_{X_1}\left(x_1\right),F_{X_2}\left(x_2\right)\right)=t_{r,\rho }\left(x_1,x_2\right)\\ ={\displaystyle {\int }_{-\infty }^{x_1}{\displaystyle {\int }_{-\infty }^{x_2}\frac{1}{2\pi {\left(1-{\rho }^2\right)}^{1/2}}{\left[1+\frac{x_1^2-2\rho x_1{x}_2+x_2^2}{r\left(1-{\rho }^2\right)}\right]}^{-\frac{r+2}{2}}\text{d}{x}_1\text{d}{x}_2}}\end{array}$ (24)

where $t_{r,\rho }$ represents the CDF of the bivariate t-distribution with $r$ degrees of freedom and $\rho$ as the correlation coefficient.

Although elliptical copulas are simple to use and the simulations based on these models are simple to perform, they require numerous parameters to be estimated, generally lack closed-form expressions, and they assume symmetry, but real-world loss distributions are often skewed. These shortcomings highlight the need for more flexible alternatives such as Archimedean or vine copula constructions, which can better accommodate the complexities of real-world dependence patterns.

Archimedean copulas address the symmetric tail dependence limitation elliptical copula by modelling asymmetric dependence in either the left or right tail. Archimedean copulas also have closed-form expressions and are not constructed using Equation (18), but are instead associated with the Laplace transforms of bivariate distribution functions. These copulas, which include Clayton, Frank, Joe, and Gumbel Copulas, can also capture a wide range of dependence. The construction of the Archimedean copulas relies primarily on the generator function $\phi :\left[0,1\right]\to \left[0,\infty \right]$ and the pseudo-inverse of the generating function define by:

${\phi }^{\left[-1\right]}=\{\begin{array}{ll}{\phi }^{\left[-1\right]},\hfill & 0\le t\le \phi \left(0\right)\hfill \\ 0,\hfill & \phi \left(0\right)<t<\infty \hfill \end{array}$

The general Archimedean copula is given by:

$C_{\phi }\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)={\phi }^{\left[-1\right]}(\phi \left(F_1\left(x_1\right)\right)+\phi \left(F_2\left(x_2\right)\right)$ (25)

The Gumbel copula utilizes a generator function defined as $\phi ={\left(-\text{ln}\left(t\right)\right)}^{\theta }$ and is expressed as:

$C_{GU}\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)=\exp \left[-{\left({\left(-\ln F_1\left(x_1\right)\right)}^{\theta }+{\left(-\ln F_1\left(x_1\right)\right)}^{\theta }\right)}^{\frac{1}{\theta }}\right]$ (26)

with parameter $\theta$ limited to the interval $\left[1,\infty \right)$. It is asymmetric, does not account for negative dependence, exhibits upper tail-dependence ( ${\lambda }^{+}=2-2^{\frac{1}{\theta }}$) but contains no lower tail dependence.

The Clayton copula has a generating function given by $\phi =\frac{1}{\theta }\left(t^{-\theta -1}\right)$ and can be represented as:

$C_c\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)={\left(F_1{\left(x_1\right)}^{-\theta +F_1{\left(x_1\right)}^{-\theta }}-1\right)}^{-1/\theta }$ (27)

with $\theta$ the dependence parameter limited in the interval $\left(0,\infty \right)$. The Clayton Copula is asymmetric, ${\lambda }^{-}=2^{-\frac{1}{\theta }}$, ${\lambda }^{+}=0$ and cannot account for negative dependence.

The Frank copula generator function corresponds to $\phi =-\text{ln}\frac{\text{exp}\left(-\theta t\right)-1}{\text{exp}\left(-\theta \right)-1}$ and its equation is given as:

$C_F\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)=-{\theta }^{-1}\ln \left(1+\frac{\exp \left(-\theta F_1\left(x_1\right)\right)\cdot \exp \left(-\theta F_2\left(x_2\right)\right)}{\exp \left(-\theta \right)-1}\right)$ (28)

with single dependency parameter $\theta \in \left(0,\infty \right)$. Similar to elliptical copulas, the Frank copula is symmetrical and ${\lambda }^{-}=0$.

The Joe copula is characterized by the generating function $\phi =-\text{ln}\left[1-{\left(1-t\right)}^{\theta }\right]$ and is define as follows:

$\begin{array}{l}{C}_J\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)\\ =1-\left({\left(1-F_1\left(x_1\right)\right)}^{\theta }+{\left(1-F_2\left(x_2\right)\right)}^{\theta }-{\left(1-F_1\left(x_1\right)\right)}^{\theta }{\left(1-F_2\left(x_2\right)\right)}^{\theta }\right)\end{array}$ (29)

with parameter $\theta \ge 1$. The Joe copula is asymmetric, ${\lambda }^{+}=2-2^{\frac{1}{\theta }}$ and ${\lambda }^{-}=0$.

The drawback of the Archimedean and Student-t copulas in tail dependence modelling is their reliance on a single parameter. This single parameter constrains them to symmetric dependence or to specific types of tail dependence that cannot be independently adjusted for upper and lower tails. As a result, they may fail to capture the asymmetric tail dependence often observed in financial markets, where extreme losses and gains can exhibit different dependency structures. This limitation reduces their effectiveness in accurately modelling risk during market stress and in constructing robust portfolios.

The degree to which variables $X$ and $Y$ are related is indicated by a measure of dependence. The dependence coefficients lie from −1 and 1, where the value of −1 means a perfect negative dependency, 0 means no dependency and 1 indicates a perfect positive dependency. The Pearson’s correlation coefficient is commonly used measure of dependency; however, it only focuses on linear dependencies between variables, it is sensitive to outliers, and assumes the data is homoscedastic, meaning it may be inaccurate if the variability of one variable changes with respect to the other. Therefore, copula-based dependency measures, particularly Kendall’s tau and tail dependency, offer more flexibility and accuracy by capturing non-linear dependencies and the probability of extreme co-movements. The dependence coefficients lie from −1 and 1, where the value of −1 means a perfect negative dependency, 0 means no dependency and 1 indicates a perfect positive dependency.

Kendall’s Tau is a non-parametric measure of concordance between two pairs. If there is a tendency for high values of a variable to be linked to high values of the other, then the two variables are concordant; the same is true for low values. Kendall’s tau always lies in the interval [−1, 1] and is equal to 0 if the variables are independent. The likelihood that the variables will move in the same direction is greater (lower) than the likelihood that they will move in opposite ways if Kendall’s tau is positive (negative).

Definition 3.3. (Concordance, Discordance) Consider two observations from a continuous vector $\left(X,Y\right)$: $\left(x_i,y_i\right)$ and $\left(x_j,y_j\right)$. Both pairs are considered concordant and discordant if $\left(x_i-x_j\right)\left(y_i-y_j\right)>0$ and $\left(x_i-x_j\right)\left(y_i-y_j\right)<0$, respectively.

Definition 3.4. (Kendall’s $\tau$) Let $\left(x_1,y_1\right),\left(x_2,y_2\right),\cdots ,\left(x_n,y_n\right)$ be a sample of n observations from a vector $\left(X,Y\right)$ of continuous random variables, with at least two distinct pairs of observations, each pair can either be concordant or discordant. Assume that the concordant and discordant pairs are denoted by $c$ and $d$, respectively, $\tau$ is calculated as:

$\tau =\frac{c-d}{c+d}$ (30)

Alternatively, $\tau$ is given as the probability of concordance less discordance as:

$\begin{array}{c}\tau =P\left(\left(X_1-X_2\right)\left(Y_1-Y_2\right)>0\right)-P\left(\left(X_1-X_2\right)\left(Y_1-Y_2\right)<0\right)\\ =2P\left(\left(X_1-X_2\right)\left(Y_1-Y_2\right)>0\right)-1\end{array}$ (31)

where $\left(X_1,Y_1\right)$ and $\left(X_2,Y_2\right)$ are iid random vectors with a joint distribution $F$. In terms of the copula $C$, $\tau$ is represented as follows:

$\tau =4{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{1}C}}\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)\text{d}C\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)-1$ (32)

Table 1 presents formula and the range of $\tau$ for different bivariate copulas, where $\rho =\frac{\mathrm{cov}\left(XY\right)}{{\sigma }_x{\sigma }_y}$ is the correlation coefficient between $x$ and $y$, $\theta$ represents the parameter in each copula, $D_1\left(\theta \right)=\frac{1}{\theta }{\displaystyle {\int }_0^{\theta }\frac{t}{\text{exp}\left(t\right)-1}\text{d}t}$ is the order one Debye function and $DG$ is the digamma function such that $DG\left(z\right)=\frac{\text{d}}{\text{d}z}\left(\ln \Gamma \left(z\right)\right)$.

Kendall’s tau correlation is an effective metric for assessing the direction and strength of the relationship between two variables. However, it may not be the best measure of dependence, particularly when the variables’ relationship is not monotonic, there are tied ranks, or if outliers are present in the data. Additionally, while Kendall’s Tau captures overall dependence, it doesn’t specifically measure the likelihood of joint extreme values.

Tail dependence measures, such as upper and lower tail dependence coefficients, directly assess the probability that one variable will experience an extreme value given that another does. The tail dependence coefficients measure the strength of dependency in the joint upper and lower tails of bivariate or multivariate distributions and are derived using conditional probability, hence their values are between 0 and 1.

Suppose $X_1$ and $X_2$ are random variables with $F_1$ and $F_2$ as their marginal distributions. The dependence coefficient in the lower tail is defined as

${\lambda }^{-}=\underset{q\to 0}{\text{lim}}P\left(X_2\le F_2^{-1}\left(q\right)|X_1\le F_1^{-1}\left(q\right)\right)$ (33)

provided the limit ${\lambda }^{-}\in \left[0,1\right]$ exist and $q$ is the quantile. Thus, dependency in the lower tail is the likelihood that $X_2$ does not exceed its $q^{th}$-quantile given that $X_1$ does not exceed its $q^{th}$-quantile. The lower tail dependency is present if ${\lambda }^{-}\in \left(0,1\right]$, and $X_1$ and $X_2$ are asymptotically independent if ${\lambda }^{-}=0$. Similarly, the upper tail dependence coefficient is given as

${\lambda }^{+}=\underset{q\to 1}{\text{lim}}P\left(X_2\ge F_2^{-1}\left(q\right)|X_1\ge F_1^{-1}\left(q\right)\right)$ (34)

provided ${\lambda }^{+}\in \left[0,1\right]$ exist. Thus, given that $X_1$ exceeds its $q^{th}$-quantile, the likelihood that $X_2$ exceeds its $q^{th}$-quantile is the upper tail dependency, also known as the extremal dependence. When $F_1$ and $F_2$ are continuous, ${\lambda }^{-}$ and ${\lambda }^{+}$ can be represented in terms of a bivariate copula $C$ as follows:

$\begin{array}{c}{\lambda }^{-}=\underset{q\to 0}{\text{lim}}\frac{P\left[X_1\le F_1^{-1}\left(q\right),X_2\le F_2^{-1}\left(q\right)\right]}{P\left[X_2\le F_2^{-1}\left(q\right)\right]}\\ =\underset{q\to 0}{\text{lim}}\frac{C\left(q,q\right)}{q}\end{array}$ (35)

The bivariate copula shows dependency in the lower tail, if ${\lambda }^{-}\in \left(0,1\right]$ and independence if ${\lambda }^{-}=0$.

$\begin{array}{c}{\lambda }^{+}=\underset{q\to 1}{\text{lim}}\frac{P\left[X_1\ge F_1^{-1}\left(q\right),X_2\ge F_2^{-1}\left(q\right)\right]}{P\left[X_2>F_2^{-1}\left(q\right)\right]}\\ =\underset{q\to 1}{\text{lim}}\frac{S_C\left(q,q\right)}{1-q}=\underset{q\to 1}{\text{lim}}\frac{1-2q-C\left(q,q\right)}{1-q}\end{array}$ (36)

where $S_C$ represents the survival copula. The copula $C$ has dependency in the upper tail if ${\lambda }^{+}\in \left(0,1\right]$ and independent ${\lambda }^{+}=0$. Table 2 summarizes the tail dependence characteristics of various bivariate copulas, providing an overview of each copula’s upper and lower tail dependencies.

[11] , presented the pair-copula decompositions graphically using a series of nested trees with undirected edges, referred to as vine trees. Each vine copula is structured according to a Regular Vine framework, and the two specific varieties of R-vine are C-Vine (Canonical Vine) and D-Vine (Drawable Vine).

Definition 3.5. (Tree) $T=\left(N,E\right)$ has N nodes and E is the edge connecting these nodes. A node’s degree, represented as $d\left(v\right)$, is the number of neighbouring nodes it has, $v\in N$.

Definition 3.6. (Regular Vine (R-Vine)) A regular vine tree sequence on $d$ elements consists of a set of trees $V=\left(T_1,\cdots ,T_{d-1}\right)$, satisfying the following conditions:

1) For each $i=1,\cdots ,d-1$, the trees $T_i=\left(N_i,E_i\right)$ must be connected.

2) The tree $T_1$ has a node set $N_1=1,\cdots ,d$ and an edge set $E_1$.

3) For $i\ge 2$, the tree $T_i$ has a node set $N_i=E_{i-1}$ and an edge set $E_j$.

4) For $i=2,\cdots ,d-1$ and $\left\{a,c\right\}\in E_i$, it must hold that $\left|a\cup c\right|=1$ (proximity condition).

As long as the relevant edges in the preceding tree, $T_{i-1}$, have at least one shared node, the proximity condition guarantees that two nodes in tree $T_i$, where $i\ge 2$, can be connected.

Figure 1 illustrates an example of 5-dimensional R-vine with 5 variable, 10 edges and 4 trees. The unconditional bivariate copula models all pairs of the variables in $T_1$, while at the lower trees ( $T_2,\cdots ,T_4$) dependency is modelled using the conditional bivariate copulas. For example, given the third variable, the conditional bivariate copula $c_{25|3}$ represents the dependency between the second and fifth variables. This 5-dimensional R-vine’s joint density is as follows:

$\begin{array}{c}{f}_{12345}\left(x\right)=c_{41}\left(F_4\left(x_4\right),F_1\left(x_1\right)\right)\cdot c_{13}\left(F_1\left(x_1\right),F_3\left(x_3\right)\right)\cdot c_{32}\left(F_3\left(x_3\right),F_2\left(x_2\right)\right)\\ \cdot c_{35}\left(F_3\left(x_3\right),F_5\left(x_5\right)\right)\cdot c_{43|1}\left(F\left(x_4|x_1\right),F\left(x_3|x_1\right)\right)\\ \cdot c_{12|3}\left(F\left(x_1|x_3\right),F\left(x_2|x_3\right)\right)\cdot c_{25|3}\left(F\left(x_2|x_3\right),F\left(x_5|x_3\right)\right)\\ \cdot c_{42|13}\left(F\left(x_4|x_1,x_3\right),F\left(x_2|x_1,x_3\right)\right)\cdot c_{15|23}\left(F\left(x_1|x_2,x_3\right),F\left(x_5|x_2,x_3\right)\right)\\ \cdot c_{45|132}\left(F\left(x_4|x_1,x_3,x_2\right),F\left(x_5|x_{21},x_3,x_2\right)\right)\cdot \underset{i=1}{\overset{5}{{\displaystyle \prod }}}{f}_i\left(x_i\right)\end{array}$ (37)

Definition 3.7. An R-vine can be categorized into two specific types based on its structure:

1) C-Vine, if every tree $T_i$ contains a distinct node connected by $d-i$ edges. The node in tree $T_1$ with $d-1$ edges is referred to as the root node.

2) D-Vine, if $T_1$ nodes’ have a maximum of two edges.

Canonical vines (C-vines)

A d-dimensional C-vine copula has $\frac{d!}{2}$ possible tree structures. For example,

for a 5-dimensional, there are 60 decompositions to order the variables.

Figure 2 shows a C-vine copula with each node representing a variable and each edge corresponds to unconditional or conditional bivariate copula family. It has 4 trees, $T_i,i=1,2,3,4$, each tree has $5-i$ edges and $6-i$ nodes and the root node is variable 4. This 5-dimensional C-vine has the joint density expressed as:

$\begin{array}{c}{f}_{12345}\left(x\right)=c_{41}\left(F_4\left(x_4\right),F_1\left(x_1\right)\right)\cdot c_{42}\left(F_4\left(x_4\right),F_2\left(x_2\right)\right)\cdot c_{43}\left(F_4\left(x_4\right),F_3\left(x_3\right)\right)\\ \cdot c_{45}\left(F_4\left(x_4\right),F_5\left(x_5\right)\right)\cdot c_{12|4}\left(F\left(x_1|x_4\right),F\left(x_2|x_4\right)\right)\\ \cdot c_{13|4}\left(F\left(x_1|x_4\right),F\left(x_3|x_4\right)\right)\cdot c_{15|4}\left(F\left(x_1|x_4\right),F\left(x_5|x_4\right)\right)\\ \cdot c_{23|41}\left(F\left(x_2|x_4,x_1\right),F\left(x_3|x_4,x_4\right)\right)\cdot c_{25|41}\left(F\left(x_2|x_4,x_1\right),F\left(x_5|x_4,x_1\right)\right)\\ \cdot c_{35|412}\left(F\left(x_3|x_4,x_1,x_2\right),F\left(x_5|x_4,x_1,x_2\right)\right)\cdot \underset{i=1}{\overset{5}{{\displaystyle \prod }}}{f}_i\left(x_i\right)\end{array}$ (38)

Drawable vines (D-vines)

Similarly to C-vines, there are $\frac{d!}{2}$ possible D-vine decompositions. However,

unlike C-vines, the D-vine copulas structure does not require a root variable. The variables are linked sequentially, one following the other, with unconditional bivariate copulas used to model dependency between variables in the tree one and conditional pair-copulas employed in the subsequent trees where the conditioning sets in the D-vine structure are different.

Figure 3 represents a five-dimensional drawable vine copula with four trees, fourteen nodes, and ten edges. This five-dimensional D-vine copula’s joint PDF is as follows:

$\begin{array}{c}{f}_{12345}\left(x\right)=c_{42}\left(F_4\left(x_4\right),F_2\left(x_2\right)\right)\cdot c_{23}\left(F_2\left(x_2\right),F_3\left(x_3\right)\right)\cdot c_{35}\left(F_3\left(x_3\right),F_5\left(x_5\right)\right)\\ \cdot c_{51}\left(F_5\left(x_5\right),F_1\left(x_1\right)\right)\cdot c_{43|2}\left(F\left(x_4|x_2\right),F\left(x_3|x_2\right)\right)\\ \cdot c_{25|3}\left(F\left(x_2|x_3\right),F\left(x_5|x_3\right)\right)\cdot c_{31|5}\left(F\left(x_3|x_5\right),F\left(x_1|x_5\right)\right)\\ \cdot c_{45|23}\left(F\left(x_4|x_2,x_3\right),F\left(x_5|x_2,x_3\right)\right)\cdot c_{21|35}\left(F\left(x_2|x_3,x_5\right),F\left(x_1|x_3,x_5\right)\right)\\ \cdot c_{41|235}\left(F\left(x_4|x_2,x_3,x_5\right),F\left(x_1|x_2,x_3,x_5\right)\right)\cdot \underset{i=1}{\overset{5}{{\displaystyle \prod }}}{f}_i\left(x_i\right)\end{array}$ (39)

The S-vine is a extension of the D-vine that assumes stationarity in the dependence structure. To simplify the expression of the S-vine, strict stationarity conditions are imposed on the D-vine [20] . These conditions ensure that dependency structure between variables does not change with time shifting. If $C\left(V\right)$ represents a vine copula, then it is considered to be translation invariant if the partial copulas associated with any edge $E_i$, $i=1,\cdots ,d$ are identical to those of the edge $E_i+s$, where $s$ represents a time shift by $s$ steps. According to the translation invariance condition, a D-vine copula in Figure 3 transitions into an S-vine if the copulas illustrating the relationships between the pairs (4,3), (2,3), (3,5), (5,1) and partial copulas $c_{43|2},c_{25|3},c_{31|5},c_{45|23},c_{21|35}$ are identical. Therefore, the S-vine has of four-pair copulas, offering a streamlined alternative to the D-vine configuration, which requires ten-pair copulas.

The parameters of the vine copulas are estimated using the Full Maximum Likelihood (FML) and sequential estimation method. Consider a three dimension vine copula with the density function given as:

$\begin{array}{c}{f}_{123}\left(x|\theta \right)=c_{31}\left(F_3\left(x_3\right),F_1\left(x_1\right)\right)\cdot c_{12}\left(F_1\left(x_1\right),F_2\left(x_2\right)\right)\\ \cdot c_{32|1}\left(F\left(x_3|x_1\right),F\left(x_2|x_1\right)\right)\cdot \underset{i=1}{\overset{3}{{\displaystyle \prod }}}{f}_i\left(x_i\right)\end{array}$ (40)

The FML method estimates all vine structure parameters simultaneously. The joint log-likelihood for the three-dimensional vine copula is given as:

$\begin{array}{c}l\left(\theta \right)=\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log c_{31}\left(F_3\left(x_3\right),F_1\left(x_1\right);{\theta }_{31}\right)+\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log c_{12}\left(F_1\left(x_1\right),F_2\left(x_2\right);{\theta }_{12}\right)\\ +\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log c_{32|1}\left(F\left(x_3|x_1\right),F\left(x_2|x_1\right);{\theta }_{32|1}\right)\end{array}$ (41)

where $\theta =\left({\theta }_{31},{\theta }_{12},{\theta }_{32|1}\right)$ represents all the parameters that need to be estimated. By maximising $l\left(\theta \right)$ and solving

$\frac{\partial l\left(\theta \right)}{\partial \theta }=0$ (42)

$\hat{\theta }$ is obtained. However, in practice for some high dimensional cases, it is computationally expensive.

The sequential estimation method is a stepwise approach that estimates parameters for all pair copulas in the vine structure sequentially. This makes it computationally efficient and suitable for high-dimensional datasets. In step one, ${\theta }_{31}$ and ${\theta }_{12}$ are separately estimated, by maximizing $l\left(\theta \right)$ in Equation (41) by using the bivariate sub-sample $\left(F_1\left(x_{1t}\right),F_2\left(x_{2t}\right)\right)$ and $\left(F_2\left(x_{2t}\right),F_3\left(x_{3t}\right)\right)$ for $t=1,\cdots ,T$, respectively. The corresponding estimates are given by:

$\begin{array}{l}{\hat{\theta }}_{31}=\arg \underset{{\theta }_{31}}{\max }\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log c_{31}\left(F_3\left(x_3\right),F_1\left(x_1\right);{\theta }_{31}\right)\\ {\hat{\theta }}_{12}=\arg \underset{{\theta }_{12}}{\max }\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log c_{12}\left(F_1\left(x_1\right),F_2\left(x_2\right);{\theta }_{12}\right)\end{array}$ (43)

In the second step, the pseudo conditional copula observations are formed as $F\left(x_{3,t|1,{\hat{\theta }}_{31}}\right)=C_{3|1}\left(F\left(x_{3,t}|x_{1,t},{\hat{\theta }}_{31}\right)\right)$ and $F\left(x_{1,t|2,{\hat{\theta }}_{12}}\right)=C_{1|2}\left(F\left(x_{1,t}|x_{2,t},{\hat{\theta }}_{12}\right)\right)$ for

$t=1,\cdots ,T$ to be used to estimate ${\theta }_{32|1}$ by maximizing

$l\left({\theta }_{13|2}\right)=\underset{t=1}{\overset{T}{{\displaystyle \sum }}}\log c_{32|1}\left(F\left(x_3|x_1\right),F\left(x_2|x_1\right);{\theta }_{32|1}\right)$ (44)

over ${\theta }_{32|1}$. Thus, ${\hat{\theta }}_{32|1}$ is given as

${\hat{\theta }}_{32|1}=\text{arg}\underset{{\theta }_{32|1}}{\text{max}}l\left({\theta }_{32|1}\right)$ (45)

This section introduces hybrid models that combine MGARCH models with vine copulas. The MGARCH will model the time-varying volatilities and conditional correlation and vine copula will model the dependency structure separately but concurrently for non-normal distributions. The VC-MGARCH model eliminates linear correlation from the variables and forms dependent but uncorrelated innovations controlled by a vine copula, while MGARCH controls the correlation. The VC-MGARCH models are defined in a identical way to convectional MGARCH models, but the multivariate distribution function of the residuals or innovations are modelled using a vine copula function.

For $d$ assets, let $r_t={\left(r_{1,t},\cdots ,r_{d,t}\right)}^{\prime }$ be the log-returns vector, ${\eta }_t={\left({\eta }_{d,t},\cdots ,{\eta }_{d,t}\right)}^{\prime }$ be the uncorrelated dependent errors and ${\xi }_t={\left({\xi }_{1,t},\cdots ,{\xi }_{d,t}\right)}^{\prime }$ be random variables from the copula distribution, such that:

${\xi }_t|{ℰ}_{t-1}~F_{1,\cdots ,d}\left({\xi }_t\right)=C_{1,\cdots ,d}\left(F_1\left({\xi }_{1,t}\right),\cdots ,F_d\left({\xi }_{d,t}\right);{\theta }_t\right)$

where ${\theta }_t$ is the parameter associates with the copula $C\left(.,.\right)$. Instead of assuming a simple multivariate normal distribution for ${\eta }_t$ in Equation (3), [22] , proposed the transformation of these residuals using a copula-based structure:

${\eta }_t={\Sigma }_t^{-1/2}{\xi }_t$ (46)