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

Cholesky

The Cholesky model (selected with MV=CHOLESKY) is a less common choice for a restricted covariance model. It has some similarities to CC and DCC, but one important difference—while the other models apply a univariate model to observed data, the Cholesky model uses ideas from structural VAR modeling to map the observable residuals (\(\bf{u}\)) to uncorrelated residuals (\(\bf{v}\)) using \({\bf{u}}_t = {\bf{Fv}}_t \), where \(\bf{F}\) is lower triangular. The difference with the VAR literature is that the components of \(\bf{v}\) are assumed to follow (univariate) GARCH processes rather than having a fixed (identity) covariance matrix.

As in the VAR literature, it’s necessary to come up with some normalization between the \(\bf{F}\) and the variances of \(\bf{v}\). For the Cholesky model in the VAR, the obvious choice is to fix the variances at 1. However, here the variances aren’t fixed, so it’s simpler to make the diagonals of \(\bf{F}\) equal to 1 and leave the component GARCH processes free, so the free parameters in \(\bf{F}\) will be the elements below the diagonal.

The Cholesky model is sensitive to the order of listing—by construction, the first \(\bf{v}\) is identical to the first \(\bf{u}\). Any of the choices for the VARIANCES option are permitted with MV=CHOLESKY; the example from GARCHMV.RPF uses the default VARIANCES=SIMPLE.

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

Output

The output has the mean model first, then the parameters for the variance model. The \(\bf{F}\) matrix is lower triangular with 1's on the diagonal, so the estimated coefficients are F(i,j) for \(i > j\).

MV-Cholesky GARCH - Estimation by BFGS

Convergence in 114 Iterations. Final criterion was 0.0000025 <= 0.0000100

Usable Observations 6236

Log Likelihood -12173.5176

Variable Coeff Std Error T-Stat Signif

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

1. Mean(XJPN) 0.000188257 0.006440919 0.02923 0.97668251

2. Mean(XFRA) -0.006990951 0.006479695 -1.07890 0.28063176

3. Mean(XSUI) -0.004037282 0.007499684 -0.53833 0.59035134

4. C(1) 0.007428328 0.001043410 7.11928 0.00000000

5. C(2) 0.007284154 0.000984301 7.40033 0.00000000

6. C(3) 0.005180700 0.000874994 5.92084 0.00000000

7. A(1) 0.176281319 0.009392566 18.76817 0.00000000

8. A(2) 0.146640647 0.011034329 13.28949 0.00000000

9. A(3) 0.183550187 0.014973932 12.25798 0.00000000

10. B(1) 0.837274108 0.007741453 108.15464 0.00000000

11. B(2) 0.841738507 0.010952723 76.85199 0.00000000

12. B(3) 0.807028041 0.015512532 52.02426 0.00000000

13. F(2,1) 0.561329167 0.010339981 54.28725 0.00000000

14. F(3,1) 0.675958192 0.010563103 63.99239 0.00000000

15. F(3,2) 0.919538839 0.007987286 115.12532 0.00000000