The DCC framework for correlations is a useful modeling tool, however when the number of test assets becomes large the estimation can become unreliable and even breakdown completely. The DECO class of correlation models is designed to overcome some of these computational difficulties. As such, the documentation for DECO models will be formulated for many return series.

Consider a vector of $N$ different return series ${r}_{t}=\left[{r}_{1\mathrm{,t}},...,{r}_{N\mathrm{,t}}\right]$ where all series have been demeaned. Further, define the conditional covariance matrix of all return series as ${\mathbb{E}}_{t-1}\left[{r}_{t}{r}_{t}^{\prime}\right]={H}_{t}$. We can further decompose ${H}_{t}$ into the following:

$${H}_{t}={D}_{t}{R}_{t}{D}_{t}$$where ${D}_{t}=\mathrm{diag}\left({\sigma}_{i,t}\right)$ for $i=\left(1,2\right)$. Here, ${\sigma}_{i,t}$ is the conditional volatility of return series $i$ and is the $i$th diagonal entry of ${H}_{t}$. Finally, ${R}_{t}$ is the conditional correlation matrix for the two return series. The GARCH-DECO model puts specific parametric assumptions on the evolution of ${D}_{t}$ and ${R}_{t}$ separately.

Each individual return series’ conditional variance is modeled as a standard GARCH process. That is, we assume:

$${\mathbb{E}}_{t-1}\left[{r}_{\mathrm{i,t}}^{2}\right]={\sigma}_{\mathrm{i,t}}^{2}$$ $${\sigma}_{t}^{2}=\omega +\alpha {\epsilon}_{t-1}^{2}+\beta {\sigma}_{t-1}^{2}$$As with a standard GARCH, we can compute the residuals for each return series after we have fit the univariate GARCH model. Formally, these are defined as:

$${\epsilon}_{\mathrm{i,t}}=\frac{{r}_{\mathrm{i,t}}}{{\sigma}_{\mathrm{i,t}}}$$Importantly, the volatility residual vector ${\epsilon}_{t}={\left[{\epsilon}_{1\mathrm{,t}},...,{\epsilon}_{N\mathrm{,t}}\right]}^{\prime}$ will inherit the same correlation structure as the original two return series. We now turn to modeling this correlation structure.

The DECO model assumes the a specific parametric form for conditional correlation matrix ${R}_{t}$ . More specifically, on a given day the DECO model assumes that all pairwise correlations are identical. It turns out that despite this seemingly strong restriction, the DECO model can provide consistent estimates of DCC parameters in large systems. The correlation matrix ${R}_{t}$ is thus defined as an equicorrelation matrix and evolves as:

$${R}_{t}=\left(1-{\rho}_{t}\right){I}_{N}+{\rho}_{t}{J}_{N}$$ $${\rho}_{t}=\frac{2}{N\left(N-1\right)}\sum _{i>\mathrm{j}}\frac{{q}_{\mathrm{i,j,t}}}{\sqrt{{q}_{\mathrm{i,i,t}}{q}_{\mathrm{j,j,t}}}}$$ $${q}_{\mathrm{i,j,t}}={\overline{\rho}}_{\mathrm{i,j}}+{\alpha}_{\mathrm{DECO}}\left({\epsilon}_{\mathrm{i,t-1}}{\epsilon}_{\mathrm{j,t-1}}-{\overline{\rho}}_{\mathrm{i,j}}\right)+{\beta}_{\mathrm{DECO}}\left({q}_{\mathrm{i,j,t-1}}-{\overline{\rho}}_{\mathrm{i,j}}\right)$$where ${\overline{\rho}}_{\mathrm{i,j}}$ is the unconditional correlation between ${\epsilon}_{\mathrm{i,t}}$ and ${\epsilon}_{\mathrm{j,t-1}}$. By modeling the univariate return series as individual GARCH processes, and their standardized residual series as a DECO process, we form the complete GARCH-DECO specification.

VLAB estimates the parameters $\left({\omega}_{i=\mathrm{1,2}},{\alpha}_{i=\mathrm{1,2}},{\beta}_{i=\mathrm{1,2}},{\alpha}_{\mathrm{DECO}},{\beta}_{\mathrm{DECO}}\right)$ of the GARCH-DECO system via Quasi-Maximum likelihood. In order to implement maximum likelihood, we assume the stacked return series ${r}_{t}={\left[{r}_{1\mathrm{,t}},{r}_{2\mathrm{,t}}\right]}^{\prime}$ is multivariate normal with a conditional covariance ${H}_{t}$ as defined above. Specifically, ${r}_{t}\sim N\left(0,{H}_{t}\right)$ which leads to the natural definition of the likelihood function. It can be shown that the likelihood function can be decomposed into a volatility component and a correlation component, which naturally leads to a two step estimation procedure. First, we estimate univariate GARCH models to each return series. Next, using the stacked residuals ${\epsilon}_{t}={\left[{\epsilon}_{1\mathrm{,t}},{\epsilon}_{2\mathrm{,t}}\right]}^{\prime}={D}_{t}^{-1}{r}_{t}$, we estimate the correlation parameters ${\alpha}_{\mathrm{DECO}}$ and ${\beta}_{\mathrm{DECO}}$ by maximizing the following function:

$${L}_{c}\left({\alpha}_{\mathrm{DECO}},{\beta}_{\mathrm{DECO}}\right)=-\frac{1}{2}\sum _{t}\left(log\left|{R}_{t}\right|+{\epsilon}_{t}^{\prime}{R}_{t}^{-1}{\epsilon}_{t}-{\epsilon}_{t}^{\prime}{\epsilon}_{t}^{\prime}\right)$$Under reasonable regularity conditions, it can be shown that this two step procedure provides consistent estimates of both the volatility parameters and the DECO parameters. As mentioned above, the estimated parameters ${\alpha}_{\mathrm{DECO}}$, ${\beta}_{\mathrm{DECO}}$ provide consistent estimates in the event that the true correlations evolve as a DCC system, but with much less computational overhead.

As is similar with the GARCH, the single correlation ${\rho}_{t}$ will be stable and mean-reverting so long as ${\alpha}_{\mathrm{DECO}}>0$, ${\beta}_{\mathrm{DECO}}>0$, ${\alpha}_{\mathrm{DECO}}+{\beta}_{\mathrm{DECO}}<1$. The standard restrictions and properties of the univariate GARCH models that are used to model each individual return series’ volatilities also naturally still hold.