The Mean Model of Triangular BEKK

[Zankawa]

Dear Tom,
I have estimated a triangular BEKK model and applied restrictions to the mean model following the RATS user's guide. In the output results, I obtained standard results for the variance equation but all coefficients, standard errors, and t-statistics in the mean model are zeros (except the constants). I don't understand why all of these values should be zeros given that in the unrestricted version (full BEKK model), I obtained non-zero coefficients and standard errors. I have provided the output results from RATS below, and will be glad if you could look at the results and advice me on what I am not doing right.
Thank you.

Code: Select all

dec symm[series] hhs(4,4)
clear(zeros) hhs
equation dlgsecieq dlgseci
#constant hhs(1,1)
equation dlexreq dlexr
#constant hhs(2,1) hhs(2,2)
equation dlsp500eq dlsp500
#constant hhs(3,1) hhs(3,2) hhs(3,3)
equation dlcopeq dlcop
#constant hhs(4,1) hhs(4,2) hhs(4,3) hhs(4,4)
group garchm dlgsecieq dlexreq dlsp500eq dlcopeq
GARCH(MODEL=GARCHM,P=1,Q=1,MV=TBEKK,ASYMMETRIC,ITERS=300,PMETHOD=BFGS,PITERS=15,hmatrices=hh,rvectors=rd) / DLGSECI DLEXR DLSP500 DLCOP

MV-GARCH, Triangular BEKK - Estimation by BFGS
Convergence in   185 Iterations. Final criterion was  0.0000019 <=  0.0000100
Monthly Data From 1991:02 To 2015:12
Usable Observations                       299
Log Likelihood                      2127.3012

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                      0.007287757  0.002922266      2.49387  0.01263581
2.  HHS(1,1)                      0.000000000  0.000000000      0.00000  0.00000000
3.  Constant                      0.000826671  0.000387885      2.13123  0.03307024
4.  HHS(2,1)                      0.000000000  0.000000000      0.00000  0.00000000
5.  HHS(2,2)                      0.000000000  0.000000000      0.00000  0.00000000
6.  Constant                      0.007470998  0.002435940      3.06699  0.00216228
7.  HHS(3,1)                      0.000000000  0.000000000      0.00000  0.00000000
8.  HHS(3,2)                      0.000000000  0.000000000      0.00000  0.00000000
9.  HHS(3,3)                      0.000000000  0.000000000      0.00000  0.00000000
10. Constant                      0.005727075  0.004805212      1.19185  0.23332146
11. HHS(4,1)                      0.000000000  0.000000000      0.00000  0.00000000
12. HHS(4,2)                      0.000000000  0.000000000      0.00000  0.00000000
13. HHS(4,3)                      0.000000000  0.000000000      0.00000  0.00000000
14. HHS(4,4)                      0.000000000  0.000000000      0.00000  0.00000000
15. C(1,1)                        0.034613887  0.002305262     15.01517  0.00000000
16. C(2,1)                        0.000874804  0.000468841      1.86589  0.06205718
17. C(2,2)                        0.000456205  0.000527188      0.86536  0.38684328
18. C(3,1)                        0.000901935  0.002253929      0.40016  0.68903778
19. C(3,2)                       -0.012983218  0.004191467     -3.09754  0.00195137
20. C(3,3)                        0.003382128  0.011754729      0.28772  0.77355736
21. C(4,1)                       -0.005618728  0.005869985     -0.95720  0.33846813
22. C(4,2)                        0.015542818  0.056620967      0.27451  0.78369548
23. C(4,3)                        0.066699124  0.013939963      4.78474  0.00000171
24. C(4,4)                        0.000025343  0.736505280 3.44099e-005  0.99997254
25. A(1,1)                        1.016137853  0.084164661     12.07321  0.00000000
26. A(2,1)                        0.010817680  0.007489986      1.44429  0.14865862
27. A(2,2)                        0.978339477  0.076422234     12.80177  0.00000000
28. A(3,1)                       -0.048196786  0.039851345     -1.20941  0.22650373
29. A(3,2)                        0.083924082  0.112085577      0.74875  0.45400787
30. A(3,3)                        0.017158161  0.081237668      0.21121  0.83272385
31. A(4,1)                        0.051823080  0.094125855      0.55057  0.58192696
32. A(4,2)                        0.151954774  0.280599206      0.54154  0.58813770
33. A(4,3)                       -0.028549195  0.350842007     -0.08137  0.93514503
34. A(4,4)                        0.117005112  0.092013090      1.27161  0.20351038
35. B(1,1)                        0.271326783  0.047252317      5.74208  0.00000001
36. B(2,1)                       -0.033803882  0.005915303     -5.71465  0.00000001
37. B(2,2)                        0.744532071  0.025640440     29.03741  0.00000000
38. B(3,1)                        0.015439658  0.038029000      0.40600  0.68474484
39. B(3,2)                       -0.037870259  0.049079703     -0.77161  0.44034702
40. B(3,3)                        0.836755182  0.053515031     15.63589  0.00000000
41. B(4,1)                        0.067987728  0.091446223      0.74347  0.45719582
42. B(4,2)                       -0.146075380  0.196044092     -0.74511  0.45620225
43. B(4,3)                       -0.200949576  0.372875362     -0.53892  0.58994282
44. B(4,4)                       -0.038945271  0.228487879     -0.17045  0.86465793
45. D(1,1)                        0.220802608  0.248872367      0.88721  0.37496465
46. D(2,1)                        0.003665049  0.015640566      0.23433  0.81472901
47. D(2,2)                        0.160011503  0.239120904      0.66917  0.50338979
48. D(3,1)                        0.144159672  0.059414949      2.42632  0.01525282
49. D(3,2)                       -0.176506194  0.227106980     -0.77719  0.43704429
50. D(3,3)                        0.530086765  0.086254712      6.14560  0.00000000
51. D(4,1)                       -0.740725784  0.243103775     -3.04695  0.00231174
52. D(4,2)                       -4.249638024  1.430087281     -2.97159  0.00296259
53. D(4,3)                        0.475161631  0.227899202      2.08496  0.03707257
54. D(4,4)                        0.499860582  0.100747177      4.96153  0.00000070

[TomDoan]

You're missing the MVHSERIES option which is needed to redefine the HHS series.

Your first and fourth series have very odd behavior for a BEKK GARCH. #1 is "upside" down with a small B with a large A, which is very uncommon and #4 has no GARCH properties other than the asymmetry term, which is even more strange.

[TomDoan]

First of all, you're not converged, so I wouldn't bother interpreting until that's fixed:

MV-GARCH, Triangular BEKK - Estimation by BFGS
NO CONVERGENCE IN 200 ITERATIONS
LAST CRITERION WAS 0.0677245
Monthly Data From 1991:02 To 2015:12
Usable Observations 299
Log Likelihood 2175.2842

Those are variances (and are thus on the order of data^2) as regressors, which means that they're sensitive to scale of the data. If you multiply the data by x, those coefficients will all multiply by 1/x.

You have no lags in your mean model. Should you? Is there serial correlation in the mean in your data? If you're not dealing with that, that might explain any other odd results.

[Zankawa]

Hi Tom,
Sorry the message I sent you earlier contained a model that did not converge. I sent you another message with the model that actually achieved convergence. I am proving the results again below. My question was why I was obtaining extremely high coefficients in the mean equation. I have increased the lags up to lag 4 and still obtaining very high coefficients.
Thank you

Code: Select all

MV-GARCH, Triangular BEKK - Estimation by BFGS
Convergence in   112 Iterations. Final criterion was  0.0000067 <=  0.0000100
Monthly Data From 1991:05 To 2015:12
Usable Observations                       296
Log Likelihood                      2130.8178

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                       0.01119649   0.00290556      3.85347  0.00011646
2.  HHS(1,1){4}                   -0.08536413   0.12527635     -0.68141  0.49561426
3.  Constant                       0.00084684   0.00046643      1.81558  0.06943458
4.  HHS(2,1){4}                    1.94888258   0.95352672      2.04387  0.04096661
5.  HHS(2,2){4}                    2.96400721   0.91164168      3.25129  0.00114884
6.  Constant                       0.00836886   0.00590864      1.41638  0.15666495
7.  HHS(3,1){4}                  -63.80052673  53.90882247     -1.18349  0.23661520
8.  HHS(3,2){4}                  -16.79605115  20.61840171     -0.81461  0.41529295
9.  HHS(3,3){4}                    1.16795806   1.63627448      0.71379  0.47535640
10. Constant                      -0.01047937   0.00877553     -1.19416  0.23241651
11. HHS(4,1){4}                    7.32237258   7.20456022      1.01635  0.30946157
12. HHS(4,2){4}                  -19.06214293   8.24773244     -2.31120  0.02082192
13. HHS(4,3){4}                   -4.22352411   3.12693274     -1.35069  0.17679401
14. HHS(4,4){4}                    2.74766332   1.25895734      2.18249  0.02907330
15. C(1,1)                         0.03454507   0.00260360     13.26820  0.00000000
16. C(2,1)                         0.00108157   0.00061617      1.75531  0.07920546
17. C(2,2)                         0.00090715   0.00059289      1.53005  0.12600333
18. C(3,1)                         0.00131248   0.00205622      0.63830  0.52328048
19. C(3,2)                        -0.01309572   0.01190372     -1.10014  0.27127271
20. C(3,3)                         0.01165778   0.01325264      0.87966  0.37904513
21. C(4,1)                        -0.00383023   0.00594231     -0.64457  0.51920699
22. C(4,2)                        -0.01315637   0.03872749     -0.33972  0.73407002
23. C(4,3)                         0.02294760   0.03945135      0.58167  0.56079017
24. C(4,4)                         0.05747416   0.01299520      4.42272  0.00000975
25. A(1,1)                         1.02126899   0.09042137     11.29455  0.00000000
26. A(2,1)                         0.00103628   0.00784959      0.13202  0.89497056
27. A(2,2)                         0.96412611   0.07208742     13.37440  0.00000000
28. A(3,1)                         0.00571280   0.00598680      0.95423  0.33996556
29. A(3,2)                         0.02781698   0.03543835      0.78494  0.43248878
30. A(3,3)                         0.03247660   0.03007723      1.07977  0.28024306
31. A(4,1)                        -0.01087936   0.03798988     -0.28638  0.77459080
32. A(4,2)                         0.45054528   0.18055732      2.49530  0.01258495
33. A(4,3)                        -0.01689652   0.16446536     -0.10274  0.91817246
34. A(4,4)                         0.10607649   0.08462180      1.25354  0.21001065
35. B(1,1)                         0.27442013   0.05242755      5.23427  0.00000017
36. B(2,1)                        -0.03132019   0.00768878     -4.07349  0.00004631
37. B(2,2)                         0.74392767   0.02709385     27.45744  0.00000000
38. B(3,1)                        -0.00882804   0.01247637     -0.70758  0.47920578
39. B(3,2)                        -0.00211997   0.03488024     -0.06078  0.95153563
40. B(3,3)                         0.77762962   0.07017975     11.08054  0.00000000
41. B(4,1)                         0.01143046   0.07699610      0.14845  0.88198370
42. B(4,2)                        -0.11388575   0.13105702     -0.86898  0.38485878
43. B(4,3)                        -0.66937890   0.26076315     -2.56700  0.01025827
44. B(4,4)                        -0.09590631   0.16055137     -0.59736  0.55026979
45. D(1,1)                        -0.16633782   0.12324469     -1.34966  0.17712664
46. D(2,1)                         0.00035480   0.01546861      0.02294  0.98170090
47. D(2,2)                         0.19986273   0.16859938      1.18543  0.23584755
48. D(3,1)                        -0.01324636   0.04578628     -0.28931  0.77234536
49. D(3,2)                        -0.14514097   0.09856296     -1.47257  0.14086672
50. D(3,3)                         0.61330252   0.10312294      5.94729  0.00000000
51. D(4,1)                        -0.63651813   0.20761864     -3.06580  0.00217085
52. D(4,2)                        -0.32856803   0.27541657     -1.19299  0.23287499
53. D(4,3)                         0.47486507   0.23929554      1.98443  0.04720801
54. D(4,4)                         0.47767276   0.10704341      4.46242  0.00000810

[TomDoan]

I have no idea what you're trying to do with your mean model. If you're taking this out of a paper, perhaps you could post the relevant page that describes the model.

The HHS are variances. They aren't going to help reduce serial correlation in the mean (if there is any), so I don't know why you think changing the lag number on them (you didn't increase the number of lags, just the timing) would be useful. But again, those coefficients will be sensitive to the scales of the data. Multiply the series by 100 and all of those will go down by a factor of 100. The A's, B's and D's aren't scale dependent, because those are variances explaining variances. In the mean model, you have variances explaining means, which is O(data^2) vs O(data).

[Zankawa]

Dear Tom,
Using 4 lags in the mean equation of the TBEKK model produced the results below. As you can see, some of the coefficient estimates of the mean model as high as -63.8 which is very unusual. The autocorrelation test suggests that there is serial correlation in the mean model. Do you think the serial correlation might be the cause of the odd coefficients in the mean model? If so, how can I deal with the autocorrelation because it doesn't look like the addition of more lags will solve the problem.
Thank you

Code: Select all

MV-GARCH, Triangular BEKK - Estimation by BFGS
Convergence in   112 Iterations. Final criterion was  0.0000067 <=  0.0000100
Monthly Data From 1991:05 To 2015:12
Usable Observations                       296
Log Likelihood                      2130.8178

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                       0.01119649   0.00290556      3.85347  0.00011646
2.  HHS(1,1){4}                   -0.08536413   0.12527635     -0.68141  0.49561426
3.  Constant                       0.00084684   0.00046643      1.81558  0.06943458
4.  HHS(2,1){4}                    1.94888258   0.95352672      2.04387  0.04096661
5.  HHS(2,2){4}                    2.96400721   0.91164168      3.25129  0.00114884
6.  Constant                       0.00836886   0.00590864      1.41638  0.15666495
7.  HHS(3,1){4}                  -63.80052673  53.90882247     -1.18349  0.23661520
8.  HHS(3,2){4}                  -16.79605115  20.61840171     -0.81461  0.41529295
9.  HHS(3,3){4}                    1.16795806   1.63627448      0.71379  0.47535640
10. Constant                      -0.01047937   0.00877553     -1.19416  0.23241651
11. HHS(4,1){4}                    7.32237258   7.20456022      1.01635  0.30946157
12. HHS(4,2){4}                  -19.06214293   8.24773244     -2.31120  0.02082192
13. HHS(4,3){4}                   -4.22352411   3.12693274     -1.35069  0.17679401
14. HHS(4,4){4}                    2.74766332   1.25895734      2.18249  0.02907330
15. C(1,1)                         0.03454507   0.00260360     13.26820  0.00000000
16. C(2,1)                         0.00108157   0.00061617      1.75531  0.07920546
17. C(2,2)                         0.00090715   0.00059289      1.53005  0.12600333
18. C(3,1)                         0.00131248   0.00205622      0.63830  0.52328048
19. C(3,2)                        -0.01309572   0.01190372     -1.10014  0.27127271
20. C(3,3)                         0.01165778   0.01325264      0.87966  0.37904513
21. C(4,1)                        -0.00383023   0.00594231     -0.64457  0.51920699
22. C(4,2)                        -0.01315637   0.03872749     -0.33972  0.73407002
23. C(4,3)                         0.02294760   0.03945135      0.58167  0.56079017
24. C(4,4)                         0.05747416   0.01299520      4.42272  0.00000975
25. A(1,1)                         1.02126899   0.09042137     11.29455  0.00000000
26. A(2,1)                         0.00103628   0.00784959      0.13202  0.89497056
27. A(2,2)                         0.96412611   0.07208742     13.37440  0.00000000
28. A(3,1)                         0.00571280   0.00598680      0.95423  0.33996556
29. A(3,2)                         0.02781698   0.03543835      0.78494  0.43248878
30. A(3,3)                         0.03247660   0.03007723      1.07977  0.28024306
31. A(4,1)                        -0.01087936   0.03798988     -0.28638  0.77459080
32. A(4,2)                         0.45054528   0.18055732      2.49530  0.01258495
33. A(4,3)                        -0.01689652   0.16446536     -0.10274  0.91817246
34. A(4,4)                         0.10607649   0.08462180      1.25354  0.21001065
35. B(1,1)                         0.27442013   0.05242755      5.23427  0.00000017
36. B(2,1)                        -0.03132019   0.00768878     -4.07349  0.00004631
37. B(2,2)                         0.74392767   0.02709385     27.45744  0.00000000
38. B(3,1)                        -0.00882804   0.01247637     -0.70758  0.47920578
39. B(3,2)                        -0.00211997   0.03488024     -0.06078  0.95153563
40. B(3,3)                         0.77762962   0.07017975     11.08054  0.00000000
41. B(4,1)                         0.01143046   0.07699610      0.14845  0.88198370
42. B(4,2)                        -0.11388575   0.13105702     -0.86898  0.38485878
43. B(4,3)                        -0.66937890   0.26076315     -2.56700  0.01025827
44. B(4,4)                        -0.09590631   0.16055137     -0.59736  0.55026979
45. D(1,1)                        -0.16633782   0.12324469     -1.34966  0.17712664
46. D(2,1)                         0.00035480   0.01546861      0.02294  0.98170090
47. D(2,2)                         0.19986273   0.16859938      1.18543  0.23584755
48. D(3,1)                        -0.01324636   0.04578628     -0.28931  0.77234536
49. D(3,2)                        -0.14514097   0.09856296     -1.47257  0.14086672
50. D(3,3)                         0.61330252   0.10312294      5.94729  0.00000000
51. D(4,1)                        -0.63651813   0.20761864     -3.06580  0.00217085
52. D(4,2)                        -0.32856803   0.27541657     -1.19299  0.23287499
53. D(4,3)                         0.47486507   0.23929554      1.98443  0.04720801
54. D(4,4)                         0.47767276   0.10704341      4.46242  0.00000810

dec vect[series] zu(%nvar)
do time=%regstart(),%regend()
compute %pt(zu,time,%solve(%decomp(hh(time)),rd(time)))
end do time
@mvqstat(lags=12)
# zu
Multivariate Q(12)=     239.61648
Significance Level as Chi-Squared(192)=       0.01109

[TomDoan]

Please read what I wrote. I don't think what you're actually doing is what you intend to do.

[Zankawa]

Thank you for the posts. I am just estimating returns spillover and volatility spillover among variables and I am using the BEKK model to estimate these relationships. My understanding is that the mean equation of the BEKK model measures the return spillovers while the variance equation measures the shock and volatility spillovers. Hence, I intend using the mean equation to estimate the linkages in terms of returns between the variables I am studying. If the HHS are variances, is there a way I can model a mean equation of the BEKK that estimates the returns among variables?
Thank you

[TomDoan]

If that's what you intend, then you're doing it wrong. You want lagged returns (not current variances) in the mean equations.

[Zankawa]

What is the option for lagged returns in the mean equation then? I am currently using the following procedure for the lags in the mean equation:
dec symm[series] hhs(4,4)
clear(zeros) hhs
equation dlgsecieq dlgseci
#constant hhs(1,1){04}
equation dlexreq dlexr
#constant hhs(2,1){04} hhs(2,2){04}
equation dlsp500eq dlsp500
#constant hhs(3,1){04} hhs(3,2){04} hhs(3,3){04}
equation dlcopeq dlcop
#constant hhs(4,1){04} hhs(4,2){04} hhs(4,3){04} hhs(4,4){04}
group garchm dlgsecieq dlexreq dlsp500eq dlcopeq

Thank you

[TomDoan]

Your returns series are dlgseci, dlexr, dlsp500, dlcop. You want lags of those. Those are what will give you return "spillover". What you really need to do is to write out (mathematically) the model that you want to estimate.

[Zankawa]

I am looking to estimate a mean returns model which will generate output in the following format;
constant
dlgseci
constant
dlgseci
dlexr
constant
dlgseci
dlexr
dlsp500
constant
dlgseci
dlexr
dlsp500
dlcop

It is like a restricted VAR model that restricts some returns from affecting others (e.g. dlgseci does not affect the other return series). Estimating the unrestricted return spillovers (unrestricted VAR) in the BEKK model is straight forward. However, applying the above restrictions in the model is the challenge I am faced with. I would be glad if you could give me the clue as to how to estimate this model. An alternative will be to estimate the mean equation for return spillovers separately from the variance equation. But this is not ideal because as in standard practice, the mean and variance equations in GARCH models are modelled together.
Thank you

[TomDoan]

First of all, you do not want those to be current values---they need to be lags. Use four EQUATIONS that you group into a model.

equation secieq dlgseci
# constant dlgseci{1}
equation exreq dlgexr
# constant dlgseci{1} dlgexr{1}
...
group rvar secieq exreq ...
garch(model=rvar,...)

[Zankawa]

Dear Tom,
I am forever grateful to you for all your replies to my questions. I am now able to estimate my models the way I want them to be. But one thing that I want to know is; If a model does not pass the @mvarchtest test at several lags, what can I do to the model to make sure it passes this test? And if happens that the model cannot not pass this test all, can the model still be used to make inferences or analysis? One of my models did not pass this test and I tried increasing the ARCH terms in the model but that did not solve this problem.
Thank you

[TomDoan]

Have you tried doing a regular BEKK instead of triangular? Usually, there's not much you can do about a significant @MVARCHTEST, but the TBEKK isn't including some of the interactions that the test is checking.