Statistics and Algorithms / GARCH Models / GARCH Models (Multivariate) / MV GARCH Restricted Covariance Models /

CC (Constant Correlation)

MV=CC gives you the Constant Correlation specification. The covariances are given by

\begin{equation} H_{ij,t} = R_{ij} \sqrt {H_{ii,t} {\kern 1pt} {\kern 1pt} H_{jj,t} } \end{equation}

where the off-diagonal (lower triangular) elements of \(\bf{R}\) are estimated parameters. The default variance model (governed by the VARIANCES option) is a simple univariate GARCH model which takes the form

\begin{equation} {H_{ii,t}} = {c_i} + {a_i}u_{i,t - 1}^2 + {b_i}{H_{ii,t - 1}} \end{equation}

In GARCHMV.RPF the example of CC is:

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

The output from CC has the mean model first, then the coefficients for the variance models, then the coefficients from the correlation matrix. Only the off-diagonal of the constant correlation \({\bf{R}}\) matrix needs to be estimated, since it's symmetric with ones on the diagonal. In the output, R(i,j) is the correlation between the residuals for variables i and j.

Output Example

MV-CC GARCH - Estimation by BFGS

Convergence in 47 Iterations. Final criterion was 0.0000099 <= 0.0000100

Usable Observations 6236

Log Likelihood -12817.3747

Variable Coeff Std Error T-Stat Signif

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

1. Mean(XJPN) -0.000775585 0.007081544 -0.10952 0.91278842

2. Mean(XFRA) -0.004266997 0.007222367 -0.59080 0.55465240

3. Mean(XSUI) 0.003648561 0.008713667 0.41872 0.67542294

4. C(1) 0.016827996 0.002079419 8.09264 0.00000000

5. C(2) 0.028385668 0.002678169 10.59891 0.00000000

6. C(3) 0.032306057 0.003053511 10.57997 0.00000000

7. A(1) 0.164130916 0.011353043 14.45700 0.00000000

8. A(2) 0.133203602 0.008892333 14.97960 0.00000000

9. A(3) 0.112696962 0.007653219 14.72543 0.00000000

10. B(1) 0.812634601 0.012102769 67.14452 0.00000000

11. B(2) 0.804346693 0.012209911 65.87654 0.00000000

12. B(3) 0.831322084 0.010479082 79.33158 0.00000000

13. R(2,1) 0.564320088 0.008522088 66.21852 0.00000000

14. R(3,1) 0.579295504 0.008318390 69.64034 0.00000000

15. R(3,2) 0.828697594 0.003977519 208.34536 0.00000000