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 ($t=1,~2,\dots,T$)