While CC generally has a well-behaved likelihood function, and can handle a bigger set of variables than the more fully parameterized models, it does have the drawback of requiring the correlation to be constant. In some applications, time-varying correlations are essential. Engle(2002) proposed a method of handling this which he dubbed Dynamic Conditional Correlations. This adds two scalar parameters which govern a “GARCH(1,1)” model on the covariance matrix as a whole:

\begin{equation} {\bf{Q}}_t = (1 - a - b){\kern 1pt} {\bf{\bar Q}} + au_{t - 1} u'_{t - 1} + b{\kern 1pt} {\bf{Q}}_{t - 1} \label{eq:garch_dcc} \end{equation}

where \({\bf{\bar Q}}\) is the unconditional covariance matrix.

 

However, \(\bf{Q}\) isn’t the sequence of covariance matrices. Instead, it is used solely to provide the correlation matrix. The actual \(\bf{H}\) matrix is generated using univariate GARCH models for the variances, combined with the correlations produced by the \(\bf{Q}\).

\begin{equation} H_{ij,t} = Q_{ij,t} {{\sqrt {H_{ii,t} H_{jj,t} } } \mathord{\left/ {\vphantom {{\sqrt {H_{ii,t} H_{jj,t} } } {\sqrt {Q_{ii,t} Q_{jj,t} } }}} \right. } {\sqrt {Q_{ii,t} Q_{jj,t} } }} \end{equation}

In GARCHMV.RPF the example of DCC is:

 

garch(p=1,q=1,mv=dcc)  / xjpn xfra xsui

 

The output has the mean model coefficients first, then the coefficients from the variance models (this is the default VARIANCES=SIMPLE), then the DCC parameters, shown as DCC(A) and DCC(B), which are the \(a\) and \(b\) in the \(\bf{Q}\) recursion.

 

Note that while apparently a generalization of the CC model, the two don't formally nest as CC estimates freely the correlation matrix and (if \(n > 2\)) actually has more free parameters than the DCC model. While you can't do a standard likelihood ratio test to compare CC with DCC, in this case, DCC has a higher log likelihood by roughly 1000, so CC is clearly quite inadequate. The @TSECCTEST procedure offers a Lagrange Multiplier test for CC against a time-varying alternative, though that alternative is not DCC.

 

There’s a separate DCC option which chooses the form of the submodel \eqref{eq:garch_dcc}. The default is DCC=COVARIANCE, which is \eqref{eq:garch_dcc} as written, using the model residuals. DCC=CORRELATION is Engle’s original idea of

\begin{equation} {\bf{Q}}_t = (1 - a - b){\bf{\bar Q}} + a\varepsilon _{t - 1} \varepsilon '_{t - 1} + b{\bf{Q}}_{t - 1} \end{equation}

where \(\varepsilon _{t} \) is the vector of standardized residuals (residuals divided by the model estimate for its standard deviation) and \({\bf{\bar Q}}\) is the sample covariance matrix of those. Engle’s recursion is really designed for “two-step” estimation procedures which estimate univariate models first and take the standardized residuals as given in a second step of estimating the joint covariance matrix. However, it can't be applied easily with any of the VARIANCE models which have interactions among the variances from different equations (SPILLOVER, VARMA and KOUTMOS). It also, in practice, rarely fits better than DCC=COVARIANCE, and often fits quite a bit worse. (It tends to be more easily dominated by outliers.) DCC=COVARIANCE is the calculation that GARCH has been doing since DCC was added and remains the default.

 

DCC=CDCC (corrected DCC) is based upon the suggestion in Aielli(2013). Aielli shows the Engle’s DCC recursion isn't internally consistent, in the sense that \(E(\varepsilon _t \varepsilon '_t ) \ne {\bf{Q}}_t \) as it would be if the secondary model were actually “GARCH”, and he proposes an alternative which fixes that. Aielli’s model has the same problem as Engle’s that it doesn’t really apply except to models which can be done in two steps (mainly because the calculation of the implied \({\bf{\bar Q}}\) requires that the individual variances be known), so DCC=CDCC uses a “feasible” estimator of the sample correlation matrix.

MV=ADCC

This is similar to DCC, but adds to the \(\bf{Q}\) recursion an asymmetry term as described on page UG–308 with a scalar multiplier (labeled as DCC(G) in the output).

Output

   MV-DCC GARCH  - Estimation by BFGS

Convergence in    42 Iterations. Final criterion was  0.0000033 <=  0.0000100

Usable Observations                      6236

Log Likelihood                    -11814.4403

 

    Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Mean(XJPN)                    0.003987506  0.006060129      0.65799  0.51054435

2.  Mean(XFRA)                   -0.003133447  0.006152936     -0.50926  0.61056964

3.  Mean(XSUI)                   -0.003070966  0.007508317     -0.40901  0.68253341

 

4.  C(1)                          0.008499092  0.001136736      7.47675  0.00000000

5.  C(2)                          0.012485541  0.001314571      9.49780  0.00000000

6.  C(3)                          0.016566320  0.001742031      9.50977  0.00000000

7.  A(1)                          0.151662080  0.009226265     16.43808  0.00000000

8.  A(2)                          0.138382721  0.008036770     17.21870  0.00000000

9.  A(3)                          0.123692585  0.006958449     17.77588  0.00000000

10. B(1)                          0.852005668  0.008113585    105.00977  0.00000000

11. B(2)                          0.848527776  0.008240241    102.97366  0.00000000

12. B(3)                          0.858001790  0.007341456    116.87079  0.00000000

13. DCC(A)                        0.053230310  0.003340516     15.93476  0.00000000

14. DCC(B)                        0.939072327  0.003985276    235.63544  0.00000000

Copyright © 2025 Thomas A. Doan