In this chapter, we have already considered multiple univariate conditional volatility models. That is why in this recipe, we move to the multivariate setting. As a starting point, we consider Bollerslev's Constant Conditional Correlation GARCH (CCC-GARCH) model. The idea behind it is quite simple. The model consists of N univariate GARCH models, related to each other via a constant conditional correlation matrix R.
Like before, we start with the model's specification:
In the first equation, we represent the return series. The key difference between this representation and the one presented in previous recipes is the fact that, this time, we are considering multivariate returns, so rt is actually a vector of returns 
. The mean and error terms are represented analogically. To highlight this, we use the bold font when considering vectors or matrices.
The second equation shows that the error terms come from a Multivariate Normal distribution with zero means and a conditional covariance matrix: 
t (of size N x N).
The elements of the conditional covariance matrix can be defined as:
- diagonal:
for 
- off-diagonal:
for 
The third equation presents the decomposition of the conditional covariance matrix. Dt is a matrix containing the conditional standard deviations on the diagonal, and R is a correlation matrix.
Key ideas of the model:
- It avoids the problem of guaranteeing positive definiteness of
t by splitting it into variances and correlations. - The conditional correlations between error terms are constant over time.
- Individual conditional variances follow a univariate GARCH(1,1) model.
In this recipe, we estimate the CCC-GARCH model on a series of stock returns for three US tech companies. For more details about the estimation of the CCC-GARCH model, please refer to the How it works... section.