Bekk Interpretation issue

[Stanchev]

Hello I'm somewhat of a newbie dealing with the multivariate garch structures and wander how to interpret the BEKK coefficients signs. What bothers me in particularly is the minus signs infront of the A(11) and A(22) terms and how exactly are those parameters interpreted.
Thank you in advance!

Here is my output:

Code: Select all

GARCH(P=1,Q=1,MV=BEKK,ROBUST,ITERS=2000) / DCRUDE_OIL DWHEAT

MV-GARCH, BEKK - Estimation by BFGS
Convergence in    59 Iterations. Final criterion was  0.0000000 <=  0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Daily(5) Data From 2007:07:02 To 2013:06:13
Usable Observations                      1554
Log Likelihood                      4142.3217

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Mean(1)                       4.541567636  0.003804514   1193.73149  0.00000000
2.  Mean(2)                       6.564543294  0.005803557   1131.12412  0.00000000
3.  C(1,1)                        0.014945724  0.001230371     12.14733  0.00000000
4.  C(2,1)                        0.002034286  0.003008181      0.67625  0.49888117
5.  C(2,2)                       -0.021780629  0.001449500    -15.02631  0.00000000
6.  A(1,1)                       -0.974548136  0.021436936    -45.46117  0.00000000
7.  A(1,2)                        0.004802817  0.004723004      1.01690  0.30920161
8.  A(2,1)                       -0.008615723  0.003666927     -2.34958  0.01879480
9.  A(2,2)                       -0.977314701  0.019606939    -49.84535  0.00000000
10. B(1,1)                        0.200349088  0.099082415      2.02204  0.04317172
11. B(1,2)                        0.011969347  0.007646136      1.56541  0.11748653
12. B(2,1)                       -0.017865809  0.023281752     -0.76737  0.44285924
13. B(2,2)                        0.196727305  0.097557746      2.01652  0.04374546

[TomDoan]

You might want to look at http://www.estima.com/forum/viewtopic.php?f=11&t=970. The signs of the coefficients aren't determined---if you multiply an A or a B by -1, you get exactly the same model. Usually the signs resolve naturally as positives on the diagonals, because that's how the guess values are assigned. In your case, you have the somewhat non-standard situation where the A matrix dominates the B---generally it's the B matrix that has values closer to 1 on the diagonal and the A is much smaller. The standard guess values for GARCH lean more towards the latter, so it looks like the signs flipped in the course of correcting that.

Did you use PMETHOD=SIMPLEX? If you didn't, you might find that that helps get the more normal signs.

[Stanchev]

Thanks Tom, you were right that with the simplex method I got more normal signs. I'm also a bit surprised that A matrix diagonal elements are way larger than the B ones, since I noticed this is not the case in similar previous papers examining those effect between crude oil prices and agricultural commodities. What is your guess for the reason matrix A to dominate B ?

[TomDoan]

I'm not sure. You would have to post the data so I could look. However, the series seem to have very high means for something going into a GARCH. What are they?

[Stanchev]

Below I attached the 3 month futures crude oil and wheat prices. In particular I'm examining the volatility spillover effects between them. In the Garch-Bekk run I used the logs of this data. Let me know what you think Tom, and once again thanks for the support!

[TomDoan]

Wouldn't you need to either use log differences, or log forward - log spot? As just log(forward) prices, you have near unit-root processes when you need serially uncorrelated residuals.

[rdnvm]

I estimated an trivariate asymmetric VAR(1)-GARCH-BEKK:

MV-GARCH, BEKK - Estimation by BFGS
Convergence in 294 Iterations. Final criterion was 0.0000000 <= 0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations 434
Log Likelihood -2750.2036

Variable Coeff Std Error T-Stat Signif
*************************************************************************************
Mean Model(TS)
1. TS{1} 0.090681066 0.061821904 1.46681 0.14242740
2. PS{1} -0.008695900 0.046630385 -0.18649 0.85206389
3. GI{1} 0.037201467 0.092418767 0.40253 0.68729289
4. Constant 0.029431847 0.108394450 0.27153 0.78598696
Mean Model(PS)
5. TS{1} 0.033340389 0.037668410 0.88510 0.37610157
6. PS{1} 0.227483652 0.037235466 6.10933 0.00000000
7. GI{1} 0.103786514 0.070441456 1.47337 0.14065059
8. Constant -0.164301046 0.063900484 -2.57120 0.01013462
Mean Model(GI)
9. TS{1} -0.047695869 0.032127383 -1.48459 0.13765348
10. PS{1} -0.020886450 0.028012944 -0.74560 0.45590910
11. GI{1} 0.118013966 0.052185492 2.26143 0.02373249
12. Constant -0.037304227 0.069332067 -0.53805 0.59054145

13. C(1,1) 0.250561715 0.176868249 1.41666 0.15658312
14. C(2,1) 0.872741576 0.189278203 4.61089 0.00000401
15. C(2,2) 0.000004912 0.452043680 1.08662e-005 0.99999133
16. C(3,1) 0.194196030 0.117710394 1.64978 0.09898833
17. C(3,2) -0.000444816 0.085388412 -0.00521 0.99584358
18. C(3,3) -0.000003632 0.017726310 -2.04913e-004 0.99983650
19. A(1,1) 0.230637272 0.076493962 3.01510 0.00256891
20. A(1,2) 0.105296592 0.071115190 1.48065 0.13870029
21. A(1,3) 0.065047149 0.028714904 2.26527 0.02349582
22. A(2,1) 0.071414376 0.103003488 0.69332 0.48810877
23. A(2,2) 0.615999252 0.143949837 4.27926 0.00001875
24. A(2,3) 0.080168648 0.040346843 1.98699 0.04692385
25. A(3,1) -0.239122527 0.093571316 -2.55551 0.01060320
26. A(3,2) -0.265516082 0.141745808 -1.87318 0.06104288
27. A(3,3) -0.080626604 0.032323648 -2.49435 0.01261868
28. B(1,1) -1.325594157 0.032582071 -40.68477 0.00000000
29. B(1,2) -0.393207076 0.101769550 -3.86370 0.00011168
30. B(1,3) -0.416711299 0.028227712 -14.76249 0.00000000
31. B(2,1) 0.480751213 0.266944353 1.80094 0.07171207
32. B(2,2) 0.760889965 0.127604757 5.96287 0.00000000
33. B(2,3) 0.067386590 0.064583105 1.04341 0.29675893
34. B(3,1) 1.461466232 0.141126434 10.35572 0.00000000
35. B(3,2) 0.276619940 0.086886909 3.18368 0.00145417
36. B(3,3) 1.256045943 0.020731050 60.58767 0.00000000
37. D(1,1) 0.226970238 0.167115478 1.35816 0.17441163
38. D(1,2) 0.162807749 0.132505419 1.22869 0.21918900
39. D(1,3) 0.024127007 0.065052271 0.37089 0.71072209
40. D(2,1) 0.110968571 0.161104111 0.68880 0.49094891
41. D(2,2) -0.245074190 0.163398869 -1.49985 0.13365264
42. D(2,3) -0.015991093 0.096490817 -0.16573 0.86837213
43. D(3,1) -0.324202053 0.249873011 -1.29747 0.19447046
44. D(3,2) -0.301904452 0.184396644 -1.63726 0.10157706
45. D(3,3) -0.100421674 0.102455649 -0.98015 0.32701319


But, I am puzzled with B(1,1) and B(3,3) coefficients. Usually, these coefficients are close to 1, indicating high persistence. However, here, they are more than 1.
What does this mean? That the variance is non-stationary? Is it a measurement problem of the model?

[TomDoan]

You can't really interpret individual coefficients. The BEKK recursion overall is stable. However, you seem to have a rather odd set of data. The first and third series appear to move almost opposite to each other while the second has a much weaker "GARCH" effect. Is that to be expected?