BEKK-GARCH constant

[fadimohamed]

Dear Tom,
i have a question regarding the the estimation of the BEKK-GARCH model, in the RATS Users' guide 6.2 page 418 the constant is defined like C'C where C is the lower triangle matrix while in the literature like Timo Terasvirta(2007) Multivariate GARCH models, it is written like CC' where C is the lower triangle matrix. which one i should follow? and why is this difference?

Many Thanks

Fadi

[TomDoan]

Dear Tom,
i have a question regarding the the estimation of the BEKK-GARCH model, in the RATS Users' guide 6.2 page 418 the constant is defined like C'C where C is the lower triangle matrix while in the literature like Timo Terasvirta(2007) Multivariate GARCH models, it is written like CC' where C is the lower triangle matrix. which one i should follow? and why is this difference?

Many Thanks

Fadi

There's no advantage to one over the other, and it's just a question of convenience for the way particular software handles triangular matrices. What matters is forcing the "constant" to be a p.s.d. matrix which wouldn't happen if it were simply parameterized as a SYMMETRIC.

[fadimohamed]

Thank you Tom your answer helped a lot,

I have another question, i am trying to estimate volatility spillover using the BEKK-GARCH among three markets, my question is that why the BEKK-GARCH results changes when i change the order of the three equations in the VECM model? the RATS version i am using is 6.3

Regards

Fadi

[TomDoan]

Thank you Tom your answer helped a lot,

I have another question, i am trying to estimate volatility spillover using the BEKK-GARCH among three markets, my question is that why the BEKK-GARCH results changes when i change the order of the three equations in the VECM model? the RATS version i am using is 6.3

Regards

Fadi

If the model is fully symmetric, the only difference should be the labeling of the variables. You'd have to post the program (and data) for us to check.

[fadimohamed]

Tom,
the only thing that i can post is the code and for the data i can't post it for confidentiality issues.

Code: Select all

equation eq1 dlbio
#constant dlbio{1} dlsunflower{1} dlcrude{1} residci{1} dummy3 dummy6

equation eq2 dlsunflower
#constant dlbio{1} dlsunflower{1} dlcrude{1} residci{1} dummy3 dummy6

equation eq3 dlcrude
#constant dlbio{1} dlsunflower{1} dlcrude{1} residci{1} dummy3 dummy6

group VECM eq1 eq2 eq3

garch(p=1,q=1,model=VECM, mv=bek,method=bfgs,iter=200,pmethod=simplex,piter=32,reg)

in the VECM, when i change the equation order i get different results for the BEKK-GARCH

[TomDoan]

Tom,
in the VECM, when i change the equation order i get different results for the BEKK-GARCH

Are you getting converged estimates with the different orders, or is it just that all the models are failing to converge in different ways?

[fadimohamed]

yes i do get convergence with different orders but the estimates chaanges and also the significance of the estimates changes.

[TomDoan]

yes i do get convergence with different orders but the estimates chaanges and also the significance of the estimates changes.

You may not be able to post it, but you'll have to send the program and data to support@estima.com so we can look at it.

[TomDoan]

You have only 98 usable data points (after allowing for lags) which isn't much to estimate a model that size. Also, the GARCH effects are fairly weak. If you do a two-step model (taking the residuals from the VAR and running them into GARCH), you get estimates of the GARCH process which are the same for any ordering:

Code: Select all

MV-GARCH, BEKK - Estimation by BFGS
Convergence in   109 Iterations. Final criterion was  0.0000000 <=  0.0000100
Weekly Data From 2007:11:06 To 2009:09:15
Usable Observations                        98
Log Likelihood                      -621.7316

    Variable                        Coeff      Std Error      T-Stat       Signif
*************************************************************************************
1.  Mean(1)                       0.835531639  0.403860566       2.06886  0.03855907
2.  Mean(2)                       0.059098946  0.073204691       0.80731  0.41948738
3.  Mean(3)                      -0.142325189  0.165264568      -0.86120  0.38913009
4.  C(1,1)                        0.766341717  0.682495052       1.12285  0.26149987
5.  C(2,1)                        0.344659200  0.169706041       2.03092  0.04226323
6.  C(2,2)                       -0.000030286  0.376768379 -8.03833e-005  0.99993586
7.  C(3,1)                        1.409773511  0.355715978       3.96320  0.00007395
8.  C(3,2)                       -0.000317812  3.958575791 -8.02844e-005  0.99993594
9.  C(3,3)                       -0.000016607  1.388019400 -1.19648e-005  0.99999045
10. A(1,1)                       -0.079537148  0.099145659      -0.80223  0.42242269
11. A(1,2)                       -0.020380544  0.017248781      -1.18156  0.23737863
12. A(1,3)                       -0.255017109  0.051558823      -4.94614  0.00000076
13. A(2,1)                        1.441254401  0.522705545       2.75730  0.00582814
14. A(2,2)                        0.340559037  0.100593507       3.38550  0.00071049
15. A(2,3)                        0.558354833  0.399864593       1.39636  0.16260618
16. A(3,1)                        0.543279489  0.272889715       1.99084  0.04649859
17. A(3,2)                       -0.009838266  0.064052362      -0.15360  0.87792731
18. A(3,3)                        0.982834823  0.163665017       6.00516  0.00000000
19. B(1,1)                        0.957194988  0.041770509      22.91557  0.00000000
20. B(1,2)                        0.025273016  0.013543218       1.86610  0.06202720
21. B(1,3)                        0.030861707  0.096013411       0.32143  0.74788363
22. B(2,1)                       -0.411697972  0.448809555      -0.91731  0.35897966
23. B(2,2)                        0.720456425  0.121676243       5.92109  0.00000000
24. B(2,3)                       -0.954708769  0.541083822      -1.76444  0.07765829
25. B(3,1)                       -0.064537177  0.307847881      -0.20964  0.83394879
26. B(3,2)                       -0.123061212  0.049982294      -2.46210  0.01381276
27. B(3,3)                       -0.050014359  0.154369909      -0.32399  0.74594535

However, this is on a boundary for the constant matrix; it's basically just a rank one matrix. Presumable, the global maximum would have a non-positive definite constant in the variance equation. With a small data set, it's quite possible that the requirement that all the data points can generate a p.d. covariance matrix without a p.s.d. constant. When you estimate the full model, with both the mean and the variance terms together, there are multiple modes, and the progression generated by the simplex iterations (which are sensitive to the order of parameters) manage to locate different modes for different orders.