The RATS Software Forum

Posted: **Thu Jun 30, 2016 5:10 pm**

Hello,
I am estimating a multivariate GARCH BEKK model with 4 variables. I am not sure about the interpretation of the coefficients of A and B.
My understanding is that B measures the ARCH effects and A measures the GARCH effects. My results have been pasted below. I am not sure if the coefficients for B and A are simply interpreted as the ARCH and GARCH effects or they need a different type of interpretation.
Thank you

Code: Select all

GARCH(P=1,Q=1,MV=BEKK,ASYMMETRIC,PMETHOD=BFGS,PITERS=10) / DLEXR DLGSMKT DLCOP DLUSSMKT

MV-GARCH, BEKK - Estimation by BFGS
NO CONVERGENCE IN 200 ITERATIONS
LAST CRITERION WAS  0.0080076
Monthly Data From 1991:01 To 2012:12
Usable Observations                       264
Log Likelihood                      1764.6500

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Mean(1)                       0.003619613  0.001013105      3.57279  0.00035320
2.  Mean(2)                       0.009916118  0.000675486     14.67999  0.00000000
3.  Mean(3)                       0.007004693  0.004918564      1.42413  0.15440778
4.  Mean(4)                       0.008834684  0.002354522      3.75222  0.00017528
5.  C(1,1)                        0.000425516  0.001193002      0.35668  0.72133363
6.  C(2,1)                        0.008049378  0.011706294      0.68761  0.49169772
7.  C(2,2)                       -0.007096474  0.009285438     -0.76426  0.44471325
8.  C(3,1)                        0.029214942  0.082241788      0.35523  0.72241555
9.  C(3,2)                        0.058563762  0.021291251      2.75060  0.00594858
10. C(3,3)                        0.031502808  0.097213796      0.32406  0.74589490
11. C(4,1)                        0.001121682  0.006904488      0.16246  0.87094604
12. C(4,2)                       -0.000103940  0.004289055     -0.02423  0.98066615
13. C(4,3)                        0.000811943  0.005037137      0.16119  0.87194265
14. C(4,4)                        0.000029647  0.006785850      0.00437  0.99651404
15. A(1,1)                        0.609558895  0.053391887     11.41670  0.00000000
16. A(1,2)                       -0.092649510  0.115815517     -0.79997  0.42372532
17. A(1,3)                       -0.000758974  0.205212391     -0.00370  0.99704904
18. A(1,4)                        0.058674487  0.101425220      0.57850  0.56292662
19. A(2,1)                       -0.067669340  0.019184193     -3.52735  0.00041974
20. A(2,2)                        2.378734897  0.138330922     17.19597  0.00000000
21. A(2,3)                       -0.336053191  0.114727166     -2.92915  0.00339889
22. A(2,4)                       -0.052787042  0.057039860     -0.92544  0.35473641
23. A(3,1)                       -0.012259161  0.009875003     -1.24143  0.21444560
24. A(3,2)                        0.496808077  0.072579742      6.84500  0.00000000
25. A(3,3)                        0.233999617  0.081467241      2.87232  0.00407476
26. A(3,4)                        0.003619490  0.034131115      0.10605  0.91554535
27. A(4,1)                       -0.160123337  0.017485697     -9.15739  0.00000000
28. A(4,2)                        0.843295784  0.128521433      6.56152  0.00000000
29. A(4,3)                       -0.010041657  0.147039187     -0.06829  0.94555288
30. A(4,4)                       -0.059729654  0.062741241     -0.95200  0.34109704
31. B(1,1)                        0.814226193  0.022542365     36.11982  0.00000000
32. B(1,2)                        0.237335542  0.155850854      1.52284  0.12779934
33. B(1,3)                       -0.343995260  0.255631362     -1.34567  0.17840921
34. B(1,4)                       -0.061063322  0.068203699     -0.89531  0.37062242
35. B(2,1)                        0.022203230  0.007494880      2.96245  0.00305198
36. B(2,2)                       -0.029637945  0.021009503     -1.41069  0.15833535
37. B(2,3)                       -0.061633213  0.051003537     -1.20841  0.22688937
38. B(2,4)                       -0.007553501  0.033012468     -0.22881  0.81901851
39. B(3,1)                       -0.002634439  0.019241424     -0.13691  0.89109799
40. B(3,2)                       -0.197213039  0.090975955     -2.16775  0.03017779
41. B(3,3)                       -0.147075655  0.164039982     -0.89658  0.36994084
42. B(3,4)                       -0.173214943  0.093957992     -1.84354  0.06525079
43. B(4,1)                       -0.033306773  0.012635098     -2.63605  0.00838769
44. B(4,2)                       -0.198598294  0.134682055     -1.47457  0.14032783
45. B(4,3)                       -0.247582976  0.288696223     -0.85759  0.39111893
46. B(4,4)                        0.802728262  0.095133396      8.43792  0.00000000
47. D(1,1)                        0.015264495  0.247511122      0.06167  0.95082407
48. D(1,2)                       -2.943273697  1.714395521     -1.71680  0.08601578
49. D(1,3)                       -1.167811259  2.388426074     -0.48895  0.62487996
50. D(1,4)                       -0.634074509  1.785905519     -0.35504  0.72255685
51. D(2,1)                       -0.027194687  0.010649774     -2.55355  0.01066322
52. D(2,2)                        0.980252297  0.229660034      4.26828  0.00001970
53. D(2,3)                       -0.165390509  0.072294039     -2.28775  0.02215223
54. D(2,4)                       -0.027201917  0.027967485     -0.97263  0.33073896
55. D(3,1)                       -0.010542013  0.015410382     -0.68409  0.49392143
56. D(3,2)                       -0.335269790  0.124591987     -2.69094  0.00712506
57. D(3,3)                        0.468809623  0.129585144      3.61777  0.00029715
58. D(3,4)                       -0.135936511  0.054315286     -2.50273  0.01232394
59. D(4,1)                        0.025176340  0.034755108      0.72439  0.46882479
60. D(4,2)                        0.507988761  0.209369589      2.42628  0.01525459
61. D(4,3)                        0.047292850  0.266421250      0.17751  0.85910659
62. D(4,4)                        0.644617646  0.109214075      5.90233  0.00000000

Posted: **Thu Jun 30, 2016 6:54 pm**

First of all, you're not converged. It's probably close, but you need to increase the number of iterations. There's some discussion about the coefficients in the BEKK at

https://estima.com/ratshelp/garchmvrpf. ... utput_BEKK

However, there is no simple explanation because of the interactions. However, your second variable has some coefficients that are hard to understand. Given that you are using only intercepts for the mean, I would first make sure that you are getting an adequate mean model.

Posted: **Fri Jul 01, 2016 3:14 pm**

Thank you for the reply. As you have suggested, I have increased the number of iterations and the model has now converged. I have pasted the new output below. I have also referred to the link you gave about the discussions of the coefficients of the BEKK model:
https://estima.com/ratshelp/garchmvrpf. ... utput_BEKK
However, the explanations offered in the link on how the coefficients of A and B are interpreted is not very clear to me.
My interpretation of the coefficients for example, are as follows;
A(3,1) is the GARCH effect which measures the effect of the residual of the third variable on the first variable, and the coefficient is -0.09 which is statistically significant.
B(3,1) is the ARCH effect which measures the spill over effect of the first variable on the third variable, and the coefficient is 0.06 which is statistically insignificant.
I would like to know if this interpretation accurate, or if the coefficients need any more transformations before they can be interpreted.
Thank you

Code: Select all

MV-GARCH, BEKK - Estimation by BFGS
Convergence in   136 Iterations. Final criterion was  0.0000081 <=  0.0000100
Monthly Data From 1991:01 To 2012:12
Usable Observations                       264
Log Likelihood                      1740.7289

    Variable                             Coeff         Std Error          T-Stat       Signif
*************************************************************************************
1.  Mean(1)                       0.002225785  0.000557426       3.99297  0.00006525
2.  Mean(2)                       0.002033930  0.004922426       0.41320  0.67946255
3.  Mean(3)                       0.006154442  0.002823617       2.17963  0.02928485
4.  Mean(4)                       0.001315009  0.005235304       0.25118  0.80167416
5.  C(1,1)                       -0.001261747  0.000520058      -2.42616  0.01525934
6.  C(2,1)                       -0.042084068  0.004148740     -10.14382  0.00000000
7.  C(2,2)                        0.000000198  0.028187077  7.03634e-006  0.99999439
8.  C(3,1)                        0.006659217  0.004986039       1.33557  0.18168898
9.  C(3,2)                       -0.000000119  0.008641784 -1.37150e-005  0.99998906
10. C(3,3)                        0.000000117  0.008132830  1.44290e-005  0.99998849
11. C(4,1)                       -0.012990512  0.016159592      -0.80389  0.42146133
12. C(4,2)                       -0.000000398  0.027321059 -1.45701e-005  0.99998837
13. C(4,3)                        0.000000075  0.029136206  2.57692e-006  0.99999794
14. C(4,4)                       -0.000000120  0.025495456 -4.69823e-006  0.99999625
15. A(1,1)                        0.644396815  0.042857437      15.03582  0.00000000
16. A(1,2)                       -0.076105814  0.242316570      -0.31408  0.75346333
17. A(1,3)                        0.178173570  0.086487749       2.06010  0.03938881
18. A(1,4)                       -0.028394668  0.171408317      -0.16566  0.86842835
19. A(2,1)                       -0.083032662  0.016682262      -4.97730  0.00000064
20. A(2,2)                       -1.628267766  0.114583722     -14.21029  0.00000000
21. A(2,3)                       -0.037384170  0.033988907      -1.09989  0.27137863
22. A(2,4)                       -0.089857142  0.075560996      -1.18920  0.23436094
23. A(3,1)                       -0.090393854  0.019194583      -4.70934  0.00000249
24. A(3,2)                       -1.008933127  0.204008866      -4.94554  0.00000076
25. A(3,3)                       -0.059092410  0.045109685      -1.30997  0.19020541
26. A(3,4)                        0.465722115  0.088095533       5.28656  0.00000012
27. A(4,1)                       -0.039647484  0.007194235      -5.51101  0.00000004
28. A(4,2)                       -0.644424026  0.111809199      -5.76360  0.00000001
29. A(4,3)                        0.121259292  0.026106329       4.64482  0.00000340
30. A(4,4)                       -0.167573204  0.063029067      -2.65867  0.00784508
31. B(1,1)                        0.849048220  0.018902027      44.91837  0.00000000
32. B(1,2)                        0.152585468  0.192933497       0.79087  0.42901939
33. B(1,3)                        0.397226470  0.175841938       2.25900  0.02388355
34. B(1,4)                       -0.251268459  0.358768344      -0.70036  0.48369997
35. B(2,1)                       -0.032675546  0.008019792      -4.07436  0.00004614
36. B(2,2)                       -0.020209393  0.059407703      -0.34018  0.73371996
37. B(2,3)                       -0.005813228  0.021768229      -0.26705  0.78942988
38. B(2,4)                        0.076259106  0.043375904       1.75810  0.07873077
39. B(3,1)                        0.060862240  0.031990508       1.90251  0.05710460
40. B(3,2)                       -0.033392606  0.252389669      -0.13231  0.89474245
41. B(3,3)                       -0.727815268  0.041423767     -17.56999  0.00000000
42. B(3,4)                       -1.296677370  0.088734407     -14.61302  0.00000000
43. B(4,1)                       -0.023481617  0.010785338      -2.17718  0.02946718
44. B(4,2)                        0.152569518  0.069906442       2.18248  0.02907401
45. B(4,3)                        0.284621734  0.020489507      13.89110  0.00000000
46. B(4,4)                       -0.700412944  0.048437665     -14.46009  0.00000000

Posted: **Fri Jul 01, 2016 7:54 pm**

Zankawa wrote:Thank you for the reply. As you have suggested, I have increased the number of iterations and the model has now converged. I have pasted the new output below. I have also referred to the link you gave about the discussions of the coefficients of the BEKK model:
https://estima.com/ratshelp/garchmvrpf. ... utput_BEKK
However, the explanations offered in the link on how the coefficients of A and B are interpreted is not very clear to me.
My interpretation of the coefficients for example, are as follows;
A(3,1) is the GARCH effect which measures the effect of the residual of the third variable on the first variable, and the coefficient is -0.09 which is statistically significant.

On the variance for the first variable.

Zankawa wrote: B(3,1) is the ARCH effect which measures the spill over effect of the first variable on the third variable, and the coefficient is 0.06 which is statistically insignificant.

No. It's the other way (three on one) because of the fact that the convention is to premultiply by the transpose. Also, the effect of the B's is much harder to describe than the A's (actually pretty close to impossible to describe) since there is no simple decomposition into a rank one outer product.

The big problem with your estimates is the bizarre value for A(2,2). The difficulty isn't the - sign (since the signs generally aren't determined), but the large value. Diagonal A's in BEKK models (since they are own on own, and are "squared" in converting to variance) are typically in the .1 to .5 range.

Posted: **Sat Jul 02, 2016 12:14 pm**

Hi Tom,
Thank you so much for the reply. You stated that the coefficients of the Bs in the BEKK model are hard (almost impossible) to describe. I am actually a PhD student and I really do need to explain the output results I obtained in the BEKK in order to complete my PhD thesis. I really need some guidance as to how I can analyse the results in the BEKK, especially the coefficients of the Bs. I will appreciate it most if you could kindly offer me this assistance. You also mentioned that the value A(2,2) in my estimates is very large and does not fall within the acceptable range. Can you advise me on what to do in order to solve this problem. I intend checking the second variable for any possible outliers
Many thanks

Mutawakil Mumuni

Posted: **Sun Jul 03, 2016 3:04 pm**

Zankawa wrote:Hi Tom,
Thank you so much for the reply. You stated that the coefficients of the Bs in the BEKK model are hard (almost impossible) to describe. I am actually a PhD student and I really do need to explain the output results I obtained in the BEKK in order to complete my PhD thesis. I really need some guidance as to how I can analyse the results in the BEKK, especially the coefficients of the Bs.

As with many time series models, the BEKK coefficients in isolation are almost impossible to interpret. If you look at the vast majority of papers that do BEKK-GARCH models, very little attention is paid to them. It's more a question of what BEKK implies for the time series of variances and correlations.

Zankawa wrote: I will appreciate it most if you could kindly offer me this assistance. You also mentioned that the value A(2,2) in my estimates is very large and does not fall within the acceptable range. Can you advise me on what to do in order to solve this problem. I intend checking the second variable for any possible outliers

To be perfectly honest, I've never seen anything that extreme. It might be a pair of very large outliers in consecutive entries, or one fairly large residual followed by one that's off-the-scale.

Posted: **Sat Jul 23, 2016 12:00 pm**

Dear Tom,
Thank you so much for your replies to my questions about the BEKK model. In your last reply, you indicated that the coefficients of the BEKK model in isolation are almost impossible to interpret, and it is more a question of what the BEKK implies for the time series for variances and correlations. I am not too sure how the variances and correlations are obtained from the output results of the BEKK model. For example, the output results of the BEKK from the RATS software only produces the coefficients for A, G, and D, and these coefficients does not tell us anything about the variances and correlations. I will be very glad if you could explain to me exactly how to obtain the variance and correlations in the BEKK model.
Thank you

Posted: **Sat Jul 23, 2016 5:02 pm**

See Section 9.4.4. of the User's Guide. That has an excerpt from the GARCHMV.RPF example which shows most of what you can do with a multivariate GARCH model. The GARCH e-course would also be a good investment.

Posted: **Sat Aug 13, 2016 8:10 am**

Hi Tom,
In the BEKK model, the coefficients of a_ij, b_ij and d_ij are interpreted differently in different papers. some interpret the coefficients as the effect of variable j on variable i whilst others interpret them as the effect of variable i on variable j. I am not sure which one is correct, and I am seeking your advice on the most appropriate interpretation.
Thank you

Posted: **Sat Aug 13, 2016 9:06 am**

You would have to read the paper carefully with regards to how the matrices are used. From the RATS Help description:

Because of the standard use of the transpose of A as the pre-multiplying matrix, the coefficients (unfortunately) have the opposite interpretation as they do for almost all other forms of GARCH models: A(i,j) is the effect of residual i on variable j, rather than j on i. However, note that it is very difficult to interpret the individual coefficients anyway.

If a paper writes the model in the (more natural) form of Auu'A', then in A(i,j), i would be the target and j the source.

Posted: **Sat Aug 13, 2016 11:08 am**

Thank you so much for the reply. I am now clear about how to interpret A(i, j), but what about the B(i, j) and D(i,j). Can we interpret the coefficients of B and D as the effect of variable j on variable i?
Thank you

Posted: **Sat Aug 13, 2016 11:58 am**

Again, it's the other way around (i on j).

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