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

Discussions of ARCH, GARCH, and related models

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

humyra wrote: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.

humyra wrote: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.

humyra wrote:How are the hmatrices or the rvectors graphed?

For HMATRICES=HX option, something like

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

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

TomDoan wrote: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.
humyra

Posts: 30
Joined: Fri Jun 02, 2017 4:26 am

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

humyra wrote: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.
TomDoan

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

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 BFGSConvergence in    86 Iterations. Final criterion was  0.0000080 <=  0.0000100Usable Observations                      1211Log Likelihood                     -3719.0235    Variable                        Coeff      Std Error      T-Stat      Signif************************************************************************************Mean Model(RTEPIX)1.  RTEPIX{1}                     0.356840835  0.025307333     14.10029  0.000000002.  RTEPIX{2}                    -0.013958931  0.026616815     -0.52444  0.599972323.  RCOIN{1}                     -0.016331690  0.011754705     -1.38937  0.164718834.  RCOIN{2}                      0.018265675  0.012985721      1.40660  0.159546955.  REX{1}                        0.014792658  0.012804938      1.15523  0.247995956.  REX{2}                        0.023228690  0.012417720      1.87061  0.061399397.  Constant                     -0.001061212  0.010126336     -0.10480  0.91653675Mean Model(RCOIN)8.  RTEPIX{1}                     0.061440022  0.040571549      1.51436  0.129933999.  RTEPIX{2}                     0.017191004  0.041208453      0.41717  0.6765527510. RCOIN{1}                      0.048180292  0.023319299      2.06611  0.0388178711. RCOIN{2}                     -0.084908253  0.025030851     -3.39214  0.0006934812. REX{1}                       -0.096842723  0.037185150     -2.60434  0.0092051813. REX{2}                        0.069483173  0.033987836      2.04435  0.0409185714. Constant                      0.033570688  0.024864922      1.35012  0.17697672Mean Model(REX)15. RTEPIX{1}                     0.000014895  0.013898893      0.00107  0.9991449116. RTEPIX{2}                     0.016894471  0.012303875      1.37310  0.1697207417. RCOIN{1}                      0.008707838  0.009760591      0.89214  0.3723165618. RCOIN{2}                     -0.040621795  0.010642314     -3.81701  0.0001350819. REX{1}                       -0.024894645  0.024594886     -1.01219  0.3114482520. REX{2}                       -0.096227906  0.026927006     -3.57366  0.0003520321. Constant                      0.007144155  0.007756727      0.92103  0.3570362922. C(1,1)                        0.103914068  0.011591412      8.96475  0.0000000023. C(2,1)                       -0.070948467  0.020223345     -3.50825  0.0004510724. C(2,2)                        0.035304293  0.042335022      0.83393  0.4043224525. C(3,1)                       -0.025324483  0.014714289     -1.72108  0.0852361426. C(3,2)                        0.000132656  0.029473720      0.00450  0.9964088927. C(3,3)                       -0.000053435  0.024701834     -0.00216  0.9982740128. A(1,1)                        0.291953794  0.023175837     12.59734  0.0000000029. A(1,2)                        0.028216551  0.027416775      1.02917  0.3033993030. A(1,3)                       -0.050407145  0.014019247     -3.59557  0.0003236931. A(2,1)                       -0.093510466  0.009153410    -10.21592  0.0000000032. A(2,2)                        0.226824872  0.017036717     13.31388  0.0000000033. A(2,3)                        0.068778361  0.008795413      7.81980  0.0000000034. A(3,1)                        0.020153175  0.016652668      1.21021  0.2261994835. A(3,2)                        0.149401167  0.032214834      4.63765  0.0000035236. A(3,3)                        0.692727451  0.017306683     40.02659  0.0000000037. B(1,1)                        0.930043085  0.008699801    106.90394  0.0000000038. B(1,2)                       -0.005327872  0.012446308     -0.42807  0.6686012739. B(1,3)                        0.046867599  0.008303195      5.64453  0.0000000240. B(2,1)                        0.022896938  0.003121553      7.33511  0.0000000041. B(2,2)                        0.976021276  0.004132489    236.18244  0.0000000042. B(2,3)                       -0.003287474  0.003990009     -0.82393  0.4099812943. B(3,1)                       -0.013286749  0.004941800     -2.68865  0.0071742544. B(3,2)                       -0.061202492  0.008838985     -6.92415  0.0000000045. B(3,3)                        0.841698669  0.005393331    156.06287  0.00000000Multivariate Q TestTest Run Over 3 to 1213Lags Tested         10Degrees of Freedom  78D of F Correction   12Q Statistic        232.8999Signif Level         0.0000Multivariate ARCH TestStatistic Degrees Signif   488.38     360 0.00001`
Attachments
data.xlsx
jack

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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

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.
TomDoan

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

TomDoan wrote: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.

"TomDoan wrote: 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

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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

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?
jack

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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

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.
TomDoan

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

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 BFGSConvergence in   110 Iterations. Final criterion was  0.0000059 <=  0.0000100Usable Observations                      1080Log Likelihood                     -2751.5704    Variable                        Coeff      Std Error      T-Stat      Signif************************************************************************************Mean Model(RTEPIX)1.  RTEPIX{1}                     0.343003169  0.025891017     13.24796  0.000000002.  RTEPIX{2}                    -0.036916249  0.028387622     -1.30043  0.193452023.  RTEPIX{3}                     0.118814663  0.028526041      4.16513  0.000031124.  RTEPIX{4}                     0.064192838  0.027543490      2.33060  0.019774505.  RCOIN{1}                     -0.016038349  0.015466194     -1.03699  0.299738716.  RCOIN{2}                      0.032261995  0.013530195      2.38444  0.017104947.  RCOIN{3}                      0.016581483  0.013495586      1.22866  0.219199428.  RCOIN{4}                     -0.000545292  0.014338226     -0.03803  0.969663279.  REX{1}                        0.042152550  0.018897227      2.23062  0.0257062510. REX{2}                        0.056525011  0.017068880      3.31158  0.0009277011. REX{3}                        0.066663662  0.017823394      3.74023  0.0001838512. REX{4}                        0.014835765  0.018558954      0.79939  0.4240667113. Constant                     -0.028040784  0.010772655     -2.60296  0.00924229Mean Model(RCOIN)14. RTEPIX{1}                     0.015230944  0.038784506      0.39271  0.6945359515. RTEPIX{2}                    -0.008182604  0.038016745     -0.21524  0.8295826716. RTEPIX{3}                    -0.087619569  0.034338478     -2.55164  0.0107216017. RTEPIX{4}                     0.062779864  0.032122074      1.95442  0.0506521518. RCOIN{1}                     -0.038072374  0.025443194     -1.49637  0.1345578519. RCOIN{2}                     -0.031702336  0.026136019     -1.21297  0.2251394120. RCOIN{3}                      0.002056055  0.026849225      0.07658  0.9389594021. RCOIN{4}                     -0.020624196  0.026468354     -0.77920  0.4358606922. REX{1}                        0.060167592  0.034896067      1.72419  0.0846728123. REX{2}                        0.123100150  0.036558714      3.36719  0.0007593824. REX{3}                       -0.035047566  0.036828788     -0.95164  0.3412820925. REX{4}                       -0.106830910  0.036266589     -2.94571  0.0032221326. Constant                      0.032047276  0.023739842      1.34994  0.17703640Mean Model(REX)27. RTEPIX{1}                     0.005152688  0.019864336      0.25939  0.7953313128. RTEPIX{2}                    -0.004941692  0.020610735     -0.23976  0.8105140029. RTEPIX{3}                     0.008974481  0.021339945      0.42055  0.6740848530. RTEPIX{4}                     0.020164755  0.019186873      1.05097  0.2932741131. RCOIN{1}                      0.029296885  0.013454273      2.17752  0.0294421532. RCOIN{2}                     -0.030636307  0.013177904     -2.32482  0.0200813833. RCOIN{3}                      0.009736795  0.014085202      0.69128  0.4893906634. RCOIN{4}                     -0.033521389  0.015835926     -2.11679  0.0342773635. REX{1}                       -0.093832978  0.028107290     -3.33839  0.0008426736. REX{2}                        0.020533402  0.026575800      0.77264  0.4397382237. REX{3}                        0.043466847  0.029750016      1.46107  0.1439963138. REX{4}                        0.050515566  0.031599073      1.59864  0.1099004539. Constant                      0.024255527  0.009114889      2.66109  0.0077888540. C(1,1)                        0.140945005  0.027682657      5.09146  0.0000003641. C(2,1)                        0.106277455  0.121742549      0.87297  0.3826800542. C(2,2)                        0.470825814  0.076125137      6.18489  0.0000000043. C(3,1)                       -0.059915638  0.017939794     -3.33982  0.0008383344. C(3,2)                        0.065232880  0.021819747      2.98963  0.0027931945. C(3,3)                        0.000002096  0.035404120  5.91896e-05  0.9999527746. A(1,1)                        0.522976780  0.040593179     12.88337  0.0000000047. A(1,2)                       -0.206172422  0.075411006     -2.73398  0.0062573248. A(1,3)                       -0.063560630  0.036528796     -1.74001  0.0818565049. A(2,1)                        0.067963724  0.020114448      3.37885  0.0007278950. A(2,2)                        0.301266938  0.040777197      7.38812  0.0000000051. A(2,3)                        0.102880995  0.025223281      4.07881  0.0000452752. A(3,1)                        0.019772645  0.018175817      1.08785  0.2766593453. A(3,2)                        0.150691818  0.037966418      3.96908  0.0000721554. A(3,3)                        0.436998981  0.025645081     17.04027  0.0000000055. B(1,1)                        0.814646709  0.029108352     27.98670  0.0000000056. B(1,2)                        0.148581929  0.063777417      2.32970  0.0198222857. B(1,3)                        0.067329676  0.032521694      2.07030  0.0384242258. B(2,1)                       -0.105321273  0.029512903     -3.56865  0.0003588259. B(2,2)                        0.725324532  0.078647034      9.22253  0.0000000060. B(2,3)                       -0.045118619  0.028141333     -1.60329  0.1088714261. B(3,1)                       -0.006338606  0.012246844     -0.51757  0.6047579262. B(3,2)                       -0.010685700  0.031937534     -0.33458  0.7379410063. B(3,3)                        0.897541169  0.010556796     85.02022  0.00000000Multivariate Q TestTest Run Over 4 to 1091Lags Tested        10Degrees of Freedom 90Q Statistic        99.80531Signif Level        0.22508Multivariate ARCH TestStatistic Degrees Signif   419.90     360 0.01603`
jack

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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

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.
TomDoan

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

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.
jack

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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

That's correct.
TomDoan

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

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.
jack

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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

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.
TomDoan

Posts: 7236
Joined: Wed Nov 01, 2006 5:36 pm

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

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).
jack

Posts: 118
Joined: Tue Sep 27, 2016 11:44 am

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