4.1 Dynamic conditional correlation

The most popular is Engle’s DCC(\(1,1\)) model which indirectly specifies conditional correlation matrix \(R_t\) by modelling the matrix of innovations (standardized error terms) \(Q_t\)

\[\begin{equation} \begin{aligned} \Sigma_t&=D_tR_tD_t \\ \\ D_t&=\begin{bmatrix} \sqrt{\sigma^2_{1,t}} & 0 \\ 0 & \sqrt{\sigma^2_{2,t}} \end{bmatrix} \\ \\ R_t&= \text{diag}(Q_t)^{-1/2} Q_t \, \text{diag}(Q_t)^{-1/2} \\ \\ Q_t&=(1-a_1- b_1) \bar{Q} + a_1 (\tilde{u}_{t-1} \tilde{u}_{t-1}^\top) + b_1 Q_{t-1} \\ \\ \text{diag}(Q_t)^{-1/2}&=\begin{bmatrix} \dfrac{1}{\sqrt{q_{11,t}}}& 0 \\ 0 & \dfrac{1}{\sqrt{q_{22,t}}} \end{bmatrix} \end{aligned} \tag{4.4} \end{equation}\]

  • Parameter \(a_1\) is assumed to be positive and \(b_1\) non–negative scalar with condition \(a_1+b_1<1\). If both parameters are zero DCC reduces to CCC model, therefore testing the joint null hypothesis \(H_0:~a_1=b_1=0\) is used to determine which model is more appropriate, the one with constant or with dynamic correlations

  • In more general case DCC(\(p,q\)) a matrix \(Q_t\) is 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 in two independent 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 log–likelihood function in the first stage. In the second stage, the parameters \(a_1\) and \(b_1\) of the matrix equation \(Q_t\) are estimated through an additional maximization of the log–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 \(a_1\) and \(b_1\) (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 (\(t=1,~2,\dots,T\))