VAR(1)-BEKK-GARCH(1,1) Model

[TomDoan]

Hi Tom,

I have several questions.

Is there are a way to obtain the three variance equations that I have posted earlier through RATS?

You can use SUMMARIZE to calculate the interaction terms and their standard errors:

viewtopic.php?p=12562#p12562

However, that's generally just a waste of space---there's nothing about that that's easier to interpret than the original structural estimates of the GARCH coefficients.

Also, I have seen people posting this command for multivariate diagnostics:

dec vect[series] zu(%nvar)
do time=%regstart(),%regend()
compute %pt(zu,time,%solve(%decomp(hh(time)),rd(time)))
end do time

What does this do?

That was how you computed jointly standardized residuals before the STDRESIDS option was added.

How are the hmatrices or the rvectors graphed?

For HMATRICES=HX option, something like

set h11 = hx(t)(1,1)
graph
# h11

[humyra]

That was how you computed jointly standardized residuals before the STDRESIDS option was added.

So no need for this now right?

Also how to export everything to Word or Excel?

Thanks.

[TomDoan]

Also how to export everything to Word or Excel?

For GARCH estimation output, re-open the output as a REPORT window (Window---Report Windows submenu) and export or copy/paste what you want. For the time-varying variances or correlations, pull out the series as you would to graph them and then either use a COPY instruction or open a Series List Window and export the ones you want.

[jack]

Hi Tom,

I am estimating a VAR-Bekk model.
But diagnostic tests show that there are serial correlation between residuals and squared residuals. I used many lags but the problem still is here.
I don't know what to do and what's the problem with my model. Here is the results and data.

MV-GARCH, BEKK - Estimation by BFGS
Convergence in    86 Iterations. Final criterion was  0.0000080 <=  0.0000100

Usable Observations                      1211
Log Likelihood                     -3719.0235

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
Mean Model(RTEPIX)
1.  RTEPIX{1}                     0.356840835  0.025307333     14.10029  0.00000000
2.  RTEPIX{2}                    -0.013958931  0.026616815     -0.52444  0.59997232
3.  RCOIN{1}                     -0.016331690  0.011754705     -1.38937  0.16471883
4.  RCOIN{2}                      0.018265675  0.012985721      1.40660  0.15954695
5.  REX{1}                        0.014792658  0.012804938      1.15523  0.24799595
6.  REX{2}                        0.023228690  0.012417720      1.87061  0.06139939
7.  Constant                     -0.001061212  0.010126336     -0.10480  0.91653675
Mean Model(RCOIN)
8.  RTEPIX{1}                     0.061440022  0.040571549      1.51436  0.12993399
9.  RTEPIX{2}                     0.017191004  0.041208453      0.41717  0.67655275
10. RCOIN{1}                      0.048180292  0.023319299      2.06611  0.03881787
11. RCOIN{2}                     -0.084908253  0.025030851     -3.39214  0.00069348
12. REX{1}                       -0.096842723  0.037185150     -2.60434  0.00920518
13. REX{2}                        0.069483173  0.033987836      2.04435  0.04091857
14. Constant                      0.033570688  0.024864922      1.35012  0.17697672
Mean Model(REX)
15. RTEPIX{1}                     0.000014895  0.013898893      0.00107  0.99914491
16. RTEPIX{2}                     0.016894471  0.012303875      1.37310  0.16972074
17. RCOIN{1}                      0.008707838  0.009760591      0.89214  0.37231656
18. RCOIN{2}                     -0.040621795  0.010642314     -3.81701  0.00013508
19. REX{1}                       -0.024894645  0.024594886     -1.01219  0.31144825
20. REX{2}                       -0.096227906  0.026927006     -3.57366  0.00035203
21. Constant                      0.007144155  0.007756727      0.92103  0.35703629

22. C(1,1)                        0.103914068  0.011591412      8.96475  0.00000000
23. C(2,1)                       -0.070948467  0.020223345     -3.50825  0.00045107
24. C(2,2)                        0.035304293  0.042335022      0.83393  0.40432245
25. C(3,1)                       -0.025324483  0.014714289     -1.72108  0.08523614
26. C(3,2)                        0.000132656  0.029473720      0.00450  0.99640889
27. C(3,3)                       -0.000053435  0.024701834     -0.00216  0.99827401
28. A(1,1)                        0.291953794  0.023175837     12.59734  0.00000000
29. A(1,2)                        0.028216551  0.027416775      1.02917  0.30339930
30. A(1,3)                       -0.050407145  0.014019247     -3.59557  0.00032369
31. A(2,1)                       -0.093510466  0.009153410    -10.21592  0.00000000
32. A(2,2)                        0.226824872  0.017036717     13.31388  0.00000000
33. A(2,3)                        0.068778361  0.008795413      7.81980  0.00000000
34. A(3,1)                        0.020153175  0.016652668      1.21021  0.22619948
35. A(3,2)                        0.149401167  0.032214834      4.63765  0.00000352
36. A(3,3)                        0.692727451  0.017306683     40.02659  0.00000000
37. B(1,1)                        0.930043085  0.008699801    106.90394  0.00000000
38. B(1,2)                       -0.005327872  0.012446308     -0.42807  0.66860127
39. B(1,3)                        0.046867599  0.008303195      5.64453  0.00000002
40. B(2,1)                        0.022896938  0.003121553      7.33511  0.00000000
41. B(2,2)                        0.976021276  0.004132489    236.18244  0.00000000
42. B(2,3)                       -0.003287474  0.003990009     -0.82393  0.40998129
43. B(3,1)                       -0.013286749  0.004941800     -2.68865  0.00717425
44. B(3,2)                       -0.061202492  0.008838985     -6.92415  0.00000000
45. B(3,3)                        0.841698669  0.005393331    156.06287  0.00000000


Multivariate Q Test
Test Run Over 3 to 1213
Lags Tested         10
Degrees of Freedom  78
D of F Correction   12
Q Statistic        232.8999
Signif Level         0.0000


Multivariate ARCH Test
Statistic Degrees Signif
   488.38     360 0.00001

[TomDoan]

Have you looked at your data? No (well-behaved) GARCH model is going to be able to explain the huge spike in volatility in the last hundred periods of your RCOIN and REX variables. Despite that, the residual ARCH test isn't all that bad---it's not even 1.5 x the degrees of freedom.

[jack]

Thank you very much for your reply.

Have you looked at your data? No (well-behaved) GARCH model is going to be able to explain the huge spike in volatility in the last hundred periods of your RCOIN and REX variables.

I deleted those data but the problem remains. Especially, @varlagselect suggests optimal lag is 15. I used those lags but there are correlations among residual and squared residuals.

Despite that, the residual ARCH test isn't all that bad---it's not even 1.5 x the degrees of freedom.

I'm sorry I didn't get the point (it's not even 1.5 x the degrees of freedom???). And, what about @MVQstat result? Is it OK to have correlation among residuals in a VAR-BEKK model because that is the variance equation that is important for me?

I also deleted the first 600 and the last hundred periods of my data (a period without spikes which markets are relatively calm) but I didn't get any reasonable result: correlation among residuals and squared residuals.

[jack]

Dear Tom,

I have some questions about the interpretation of coefficients of A and B matrices in a BEKK model. I have studied posts about this issue in this forum but I think we need more for calcifying in this regard (for students like me).

1. you said: "

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

". (https://estima.com/forum/viewtopic.php?f=11&t=2705#p12607).

I would be grateful if you could possibly give more information about this. Specifically, if I want to study the volatility or shock spillovers between markets can I just rely on the coefficients of A and ignore the coefficients of B?

2. You said:"

Negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction, which probably isn’t unreasonable in many situations

" (https://estima.com/ratshelp/index.html?garchmvrpf.html#GARCH_Output_BEKK).

What do you mean exactly when you say Negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction?
fore example suposse that A(1,2)= -0.22 and A(2,1)=0.07 and they are statically significant. How one can interpret them according to your comment?

[TomDoan]

Your questions are largely answered in the 2nd Edition of the ARCH/GARCH e-course. I would strongly recommend that you get it.

The problem with the diagnostic tests is covered also in the June 2018 newsletter.

The statistics in @VARLagSelect assume homoscedatic residuals, and so can provide no more than a rough guide if the residuals have GARCH properties. In general, it tends (even with SBC) to get too high a value.

To eliminate "causality" in variance, you need both the A and B coefficients to be zero. The A matrix coefficients control 1-step ahead effects, but non-zero B coefficients will permit multiple step effects.

What do you mean exactly when you say Negative coefficients in the off-diagonals of A mean that the variance is affected more when the shocks move in opposite directions than when they move in the same direction?

It means exactly what it says. If you are confused, just put some numbers in with different sign patterns and see what happens to the variance.

[jack]

Thank you very much for your reply.

This is the results that I got. Is there any problem with the results (especiall the low MARCHTEST' p-value):

MV-GARCH, BEKK - Estimation by BFGS
Convergence in   110 Iterations. Final criterion was  0.0000059 <=  0.0000100

Usable Observations                      1080
Log Likelihood                     -2751.5704

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
Mean Model(RTEPIX)
1.  RTEPIX{1}                     0.343003169  0.025891017     13.24796  0.00000000
2.  RTEPIX{2}                    -0.036916249  0.028387622     -1.30043  0.19345202
3.  RTEPIX{3}                     0.118814663  0.028526041      4.16513  0.00003112
4.  RTEPIX{4}                     0.064192838  0.027543490      2.33060  0.01977450
5.  RCOIN{1}                     -0.016038349  0.015466194     -1.03699  0.29973871
6.  RCOIN{2}                      0.032261995  0.013530195      2.38444  0.01710494
7.  RCOIN{3}                      0.016581483  0.013495586      1.22866  0.21919942
8.  RCOIN{4}                     -0.000545292  0.014338226     -0.03803  0.96966327
9.  REX{1}                        0.042152550  0.018897227      2.23062  0.02570625
10. REX{2}                        0.056525011  0.017068880      3.31158  0.00092770
11. REX{3}                        0.066663662  0.017823394      3.74023  0.00018385
12. REX{4}                        0.014835765  0.018558954      0.79939  0.42406671
13. Constant                     -0.028040784  0.010772655     -2.60296  0.00924229
Mean Model(RCOIN)
14. RTEPIX{1}                     0.015230944  0.038784506      0.39271  0.69453595
15. RTEPIX{2}                    -0.008182604  0.038016745     -0.21524  0.82958267
16. RTEPIX{3}                    -0.087619569  0.034338478     -2.55164  0.01072160
17. RTEPIX{4}                     0.062779864  0.032122074      1.95442  0.05065215
18. RCOIN{1}                     -0.038072374  0.025443194     -1.49637  0.13455785
19. RCOIN{2}                     -0.031702336  0.026136019     -1.21297  0.22513941
20. RCOIN{3}                      0.002056055  0.026849225      0.07658  0.93895940
21. RCOIN{4}                     -0.020624196  0.026468354     -0.77920  0.43586069
22. REX{1}                        0.060167592  0.034896067      1.72419  0.08467281
23. REX{2}                        0.123100150  0.036558714      3.36719  0.00075938
24. REX{3}                       -0.035047566  0.036828788     -0.95164  0.34128209
25. REX{4}                       -0.106830910  0.036266589     -2.94571  0.00322213
26. Constant                      0.032047276  0.023739842      1.34994  0.17703640
Mean Model(REX)
27. RTEPIX{1}                     0.005152688  0.019864336      0.25939  0.79533131
28. RTEPIX{2}                    -0.004941692  0.020610735     -0.23976  0.81051400
29. RTEPIX{3}                     0.008974481  0.021339945      0.42055  0.67408485
30. RTEPIX{4}                     0.020164755  0.019186873      1.05097  0.29327411
31. RCOIN{1}                      0.029296885  0.013454273      2.17752  0.02944215
32. RCOIN{2}                     -0.030636307  0.013177904     -2.32482  0.02008138
33. RCOIN{3}                      0.009736795  0.014085202      0.69128  0.48939066
34. RCOIN{4}                     -0.033521389  0.015835926     -2.11679  0.03427736
35. REX{1}                       -0.093832978  0.028107290     -3.33839  0.00084267
36. REX{2}                        0.020533402  0.026575800      0.77264  0.43973822
37. REX{3}                        0.043466847  0.029750016      1.46107  0.14399631
38. REX{4}                        0.050515566  0.031599073      1.59864  0.10990045
39. Constant                      0.024255527  0.009114889      2.66109  0.00778885

40. C(1,1)                        0.140945005  0.027682657      5.09146  0.00000036
41. C(2,1)                        0.106277455  0.121742549      0.87297  0.38268005
42. C(2,2)                        0.470825814  0.076125137      6.18489  0.00000000
43. C(3,1)                       -0.059915638  0.017939794     -3.33982  0.00083833
44. C(3,2)                        0.065232880  0.021819747      2.98963  0.00279319
45. C(3,3)                        0.000002096  0.035404120  5.91896e-05  0.99995277
46. A(1,1)                        0.522976780  0.040593179     12.88337  0.00000000
47. A(1,2)                       -0.206172422  0.075411006     -2.73398  0.00625732
48. A(1,3)                       -0.063560630  0.036528796     -1.74001  0.08185650
49. A(2,1)                        0.067963724  0.020114448      3.37885  0.00072789
50. A(2,2)                        0.301266938  0.040777197      7.38812  0.00000000
51. A(2,3)                        0.102880995  0.025223281      4.07881  0.00004527
52. A(3,1)                        0.019772645  0.018175817      1.08785  0.27665934
53. A(3,2)                        0.150691818  0.037966418      3.96908  0.00007215
54. A(3,3)                        0.436998981  0.025645081     17.04027  0.00000000
55. B(1,1)                        0.814646709  0.029108352     27.98670  0.00000000
56. B(1,2)                        0.148581929  0.063777417      2.32970  0.01982228
57. B(1,3)                        0.067329676  0.032521694      2.07030  0.03842422
58. B(2,1)                       -0.105321273  0.029512903     -3.56865  0.00035882
59. B(2,2)                        0.725324532  0.078647034      9.22253  0.00000000
60. B(2,3)                       -0.045118619  0.028141333     -1.60329  0.10887142
61. B(3,1)                       -0.006338606  0.012246844     -0.51757  0.60475792
62. B(3,2)                       -0.010685700  0.031937534     -0.33458  0.73794100
63. B(3,3)                        0.897541169  0.010556796     85.02022  0.00000000


Multivariate Q Test
Test Run Over 4 to 1091
Lags Tested        10
Degrees of Freedom 90
Q Statistic        99.80531
Signif Level        0.22508


Multivariate ARCH Test
Statistic Degrees Signif
   419.90     360 0.01603

[TomDoan]

Did you read the article in the June newsletter? No, it would be hard to expect you to get a smaller test statistic than that.

[jack]

I read the newsletter. Based on the newsletter, I think the variance part of model is Ok because those autocorrelations are statistically significant, but practically insignificant. Please correct me if I am wrong.

[TomDoan]

That's correct.

[jack]

Dear Tom,

I have three variables: 1) an index of stock market, 2) exchange rate, and 3) price of a special gold coin. I want to study the volatility spillovers between those markets using GARCH-Bekk model. The price of gold coin equals to:(0.82)*(global price of gold(ounce))*(exchange rate). It is somehow the local price of gold in local currency but not exactly. In our country, both the global price of gold and exchange rate affect the price of the gold coin but it is the exchange rate that has the dominant effect on it. Can I use a VAR-BEKK model for this purpose (using returns series)? I mean how I can manage the relationship between the gold coin and exchange rate. It seems that there is a finite relationship between these variable.

[TomDoan]

If (X,Y) has a BEKK representation then so does (X+Y,Y) or any other linear combinations of the original variables. You can't fit a model if two or three variables in the model are connected by an identity (you couldn't do world gold, local gold and exchange rate), but you can include any two of the three.

[jack]

Dear Tom,

1. Suppose I have daily data for two markets: exchange rate and stock market.
Suppose that a shock occurs in one of these markets (exchange rate, fore example) and the other market (stock market) reacts but after a month (or even later) to that news or shock (for any reason). Can a GARCH-BEKK model with daily data capture this reaction of the second market to the first market's shock? (a12 or b12 ≠0 )

2. When I want to look at autocorrelations of residuals, which residuals I should use: stdseries=zu option OR rseries=rs option (and its standardized form)?

3. How I can determine DFC for @regcorrs after a VAR-BEKK model? (univariate standardized residuals and univariate squared standardized residuals) (in the arch-garch course (1ed) you used dfc=2 for squared standardized residuals of a MV-GARCH Diagonal Model with three variable).