4 Multivariate GARCH models

  • In financial practice, it is often necessary to forecast both the mean and variance of returns for multiple assets simultaneously, as it has been observed that the volatility of one asset can influence the volatility of others. This influence may vary in intensity across different time periods.

  • As a result, the most relevant econometric models today are those in which the correlation matrix between the returns of different assets is not constant. In such cases, it is essential to analyze multivariate GARCH models (MGARCH), although this is just one approach to modeling a time-varying correlation matrix.

  • These models can provide answers to key questions, such as:

  1. Do shocks in one capital market increase or decrease volatility in another capital market, and to what extent?
  2. Does the volatility of one asset “spill over” on the volatility of other assets (the spillover effect)?
  3. Is the correlation between the returns of different assets higher during periods of increased volatility (stress periods)?
  4. Does the correlation between volatilities in different markets tend to increase in the long–run due to globalization and markets liberalization?
  5. During which periods is negative correlation observed — as evidence of a safe haven asset or a potential diversifier?

  • There are various MGARCH models, depending on how the covariance matrix is specified and modeled, but due to the issue of dimensionality, the most commonly used model is the Dynamic Conditional Correlation (DCC) model and its variants

  • DCC models belong to the class of nonlinear combinations of univariate GARCH models

  • The second group of models results from a direct generalization of univariate GARCH models (such as DVEC, BEKK, etc.), but they are less commonly used in empirical studies for two reasons: (a) the excessive number of parameters makes estimation computational demanding (despite many efforts to diagonalize and parametrize in achieving more parsimonious models), and (b) they lack flexibility in the sense that a different univariate GARCH specifications cannot be fitted for each asset individually.

  • If two assets are analyzed (\(k=2\)) then \(r_t\) is a two–dimensional vector, \(\mu\) is a vector of two constants \(\mu_1\) and \(\mu_2\), while \(u_t\) is a two–dimensional vector of innovations with zero means and time–varying covariance matrix \(\Sigma_t\) of dimensions \(2\times2\)

\[\begin{equation} r_t=\mu+u_t~~~~~u_t\sim(0,~\Sigma_t) \tag{4.1} \end{equation}\]

  • Equation (4.1) is the conditional mean equation, which in addition to constant terms may include AR terms (the same as Vector AutoRegression – VAR), but the issue is how to determine matrix \(\Sigma_t\)?

  • Nonlinear combination of univariate GARCH models can be described by following decomposition of the time–varying covariance matrix \(\Sigma_t\)

\[\begin{equation} \begin{aligned} \Sigma_t&=D_tR_tD_t \\ \\ D_t&=\begin{bmatrix} \sqrt{\sigma^2_{1,t}} & 0 \\ 0 & \sqrt{\sigma^2_{2,t}} \end{bmatrix}~~~~~~R_t=\begin{bmatrix} \rho_{11,t} & \rho_{12,t}\\ \rho_{21,t} & \rho_{22,t} \end{bmatrix} \end{aligned} \tag{4.2} \end{equation}\]

  • In decomposition (4.2) matrix \(D_t\) is diagonal matrix with conditional standard deviations \(\sigma_{i,t}\) following any univariate GARCH for assets \(i=1,~2,\dots,k\), while \(R_t\) is symmetric correlation matrix between \(k\) assets (\(\rho_{12,t}=\rho_{21,t}\)) that also varies over time

  • Due to above decomposition, a covariance is presented as a product of the conditional correlation coefficient and the conditional standard deviations of the two assets (nonlinear combination)

\[\begin{equation} \begin{aligned} \sigma_{i,t} \times \rho_{ij,t} \times \sigma_{j,t}&= \sigma_{ij,t}~~~off-diagonal~~elements~~of~~\Sigma_t \\ \sigma_{i,t} \times \underbrace{\rho_{ii,t}}_{=1} \times \sigma_{i,t}&=\sigma^2_{i,t}~~~~~~~~~~~~~~~diagonal~~elements~~of~~\Sigma_t \end{aligned} \tag{4.3} \end{equation}\]

  • Bollerslev was first who assumed a constant conditional correlation matrix \(R_t \equiv R\), where dynamics of the covariances are determined solely by the dynamics of the two conditional variances (standard deviations), but not by the dynamics of their correlation, and that’s why this model is usually referred to as the constant conditional correlation (CCC) model

  • When fitting a CCC model, there are several alternatives for the estimation of the constant conditional correlation matrix (the sample correlation matrix is used, and no further MLE estimation of \(R\) is carried out, or the sample correlation matrix is used as the initial estimate, and the final estimate of \(R\) is obtained as part of the MLE method)

  • Since the assumption of constant conditional correlations is often unrealistic, various models with dynamic conditional correlations (DCC) have been developed

  • The most popular is Engle’s DCC model which indirectly specifies conditional correlation matrix \(R_t\) by modelling the matrix of standardized innovations (standardized residuals) \(Q_t\)

\[\begin{equation} \begin{aligned} \Sigma_t&=D_tR_tD_t \\ \\ D_t&=\begin{bmatrix} \sqrt{\sigma^2_{1,t}} & 0 \\ 0 & \sqrt{\sigma^2_{2,t}} \end{bmatrix} \\ \\ R_t&= \text{diag}(Q_t)^{-1/2} Q_t \, \text{diag}(Q_t)^{-1/2} \\ \\ Q_t&=(1-a- b) \bar{Q} + a (\tilde{u}_{t-1} \tilde{u}_{t-1}^\top) + b Q_{t-1} \\ \\ \text{diag}(Q_t)^{-1/2}&=\begin{bmatrix} \dfrac{1}{\sqrt{q_{11,t}}}& 0 \\ 0 & \dfrac{1}{\sqrt{q_{22,t}}} \end{bmatrix} \end{aligned} \tag{4.4} \end{equation}\]

  • Parameter \(a\) is assumed to be positive and \(b\) non–negative scalar with condition \(a+b<1\). If both parameters are zero DCC reduces to CCC model, therefore testing the null hypothesis \(H_0:~a=b=0\) can be used to determine which model is more appropriate – one with constant or with dynamic correlations

  • In more general case DCC(\(p,q\)) a matrix \(Q_t\) can be defined as

\[\begin{equation} Q_t=\bar{Q} + \sum_{i=1}^p a_i (\tilde{u}_{t-1} \tilde{u}_{t-1}^\top - \bar{Q}) + \sum_{j=1}^q b_j (Q_{t-j}- \bar{Q}) \\ \tag{4.5} \end{equation}\]

  • The standard procedure for parameter estimation in the DCC(\(1,1\)) model involves the application of the MLE method, executed in two stages. The parameters of the univariate GARCH(\(1,1\)) models (\(\mu\), \(\omega\), \(\alpha_1\) and \(\beta_1\) for every asset individually) are estimated by maximizing the likelihood function in the first stage. In the second stage, the parameters \(a\) and \(b\) of the matrix equation \(Q_t\) are estimated through an additional maximization of the likelihood function.

  • Estimation of Engle’s DCC(\(1,1\)) model requires following steps:

Step 1 – specify univariate GARCH(\(1,1\)) model for each asset and estimate the parameters of both equations (conditional mean and conditional variance)

Step 2 – compute the residuals from the mean equation and standardize them by conditional standard deviation from the variance equation of each asset (in matrix notation \(\tilde{u}=D^{-1}_t u_t\); keep in mind that \(D_t\) is diagonal matrix of conditional standard deviations from univariate GARCH models)

Step 3 – specify dynamic conditional correlation of standardized residuals by defining equation of \(Q_t\) and estimate the parameters \(a\) and \(b\) (uncoditional correlation matrix of standardized residulas \(\bar{Q}\) is typically estimated as a sample average of the outer product of the standardized two–dimensional residual vector, i.e. \(\bar{Q}=\dfrac{1}{T}\displaystyle\sum_{t=1}^{T}\tilde{u}_t\tilde{u}^\top_t\))

Step 4 – after the elements of matrix \(Q_t\) have been generated from previous step, they are normalized backward to obtain the dynamic correlation matrix \(R_t\)

Step 5 – matrices \(R_t\) (from step \(4\)) and \(D_t\) (from step \(2\)) are combined to obtain the covariance matrix \(\Sigma_t\) for each trading day (\(t=1,~2,\dots,T\))