## Multivariate GARCH Output

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### Multivariate GARCH Output

Hi guys,

I was given data to run through RATS using a Multivariate GARCH model and as part of the assignment we have to comment on the results. However we have recieved absolutely no guidelines or lessons on how to interpret the results. I ran the data for Gold spot and Gold futures prices and it gave me the following output:

GARCH Model - Estimation by BFGS
Convergence in 30 Iterations. Final criterion was 0.0000770 <= 0.0001000
Usable Observations 3913
Log Likelihood 25048.53595371
Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Mean(1) 1.66764e-04 1.56457e-04 1.06588 0.28648001
2. Mean(2) 2.00411e-04 1.69581e-04 1.18180 0.23728458
3. C(1,1) 2.04806e-06 3.78281e-07 5.41414 0.00000006
4. C(2,1) 1.56340e-05 1.68456e-06 9.28075 0.00000000
5. C(2,2) 2.30516e-06 5.48870e-07 4.19983 0.00002671
6. A(1,1) 0.05587 0.00714 7.83098 0.00000000
7. A(2,1) 0.00472 0.00789 0.59817 0.54972832
8. A(2,2) 0.04249 0.00711 5.97279 0.00000000
9. B(1,1) 0.92614 0.00907 102.15379 0.00000000
10. B(2,1) 0.37651 0.04039 9.32117 0.00000000
11. B(2,2) 0.93785 0.01063 88.26250 0.00000000

I'm wondering can anyone help me interpret this as I haven't really got a clue where to start and don't know what this output means. Any basics will be appreciated.
Jimbo

Posts: 4
Joined: Thu Mar 08, 2012 6:28 pm

### Re: Multivariate GARCH Output

The diagonal VECH model (which is the default with multivariate GARCH) has separate equations for each component of the covariance matrix. The covariance matrix H evolves as:

H(1,1)(t)=C(1,1)+A(1,1)e1(t-1)^2+B(1,1)*H(1,1)(t-1)
H(1,2)(t)=C(1,2)+A(1,2)e1(t-1)e2(t-1)+B(1,2)*H(1,2)(t-1)
H(2,2)(t)=C(2,2)+A(2,2)e2(t-1)^2+B(2,2)*H(2,2)(t-1)

where e1(t)=y1(t)-mean(1) and e2(t)=y2(t)-mean(2). The two "GARCH" equations (for the variances) are fairly normal looking, with B's around .9 and relatively small A's, with A+B summing to slightly less than 1. The cross term isn't as persistent as it would be if this were two or more exchange rates or the like.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Multivariate GARCH Output

Thanks Tom. Have you got any suggestions on what I could say about my ARCH output? Again anything basic would be greatly appreciated as this is the first time I've looked at this stuff.

Linear Regression - Estimation by Least Squares
Dependent Variable A
Usable Observations 189 Degrees of Freedom 183
Centered R**2 0.077630 R Bar **2 0.052429
Uncentered R**2 0.247085 T x R**2 46.699
Mean of Dependent Variable 0.0024900966
Std Error of Dependent Variable 0.0052627718
Standard Error of Estimate 0.0051229545
Sum of Squared Residuals 0.0048027732
Regression F(5,183) 3.0804
Significance Level of F 0.01076261
Log Likelihood 731.65980
Durbin-Watson Statistic 1.996481

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Constant 0.001589395 0.000485468 3.27394 0.00126820
2. A{1} 0.248472052 0.073906134 3.36199 0.00094217
3. A{2} 0.050957390 0.076055314 0.67000 0.50369958
4. A{3} -0.042352548 0.076109781 -0.55647 0.57857177
5. A{4} 0.055901508 0.076094909 0.73463 0.46350593
6. A{5} 0.050282793 0.075575888 0.66533 0.50667807

Chi-Squared(5)= 14.672073 with Significance Level 0.01185938

Linear Regression - Estimation by Least Squares
Dependent Variable A
Usable Observations 189 Degrees of Freedom 183
Centered R**2 0.081068 R Bar **2 0.055961
Uncentered R**2 0.270210 T x R**2 51.070
Mean of Dependent Variable 0.0024317892
Std Error of Dependent Variable 0.0047894149
Standard Error of Estimate 0.0046534765
Sum of Squared Residuals 0.0039628363
Regression F(5,183) 3.2288
Significance Level of F 0.00809425
Log Likelihood 749.82586
Durbin-Watson Statistic 1.996380

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Constant 0.001482390 0.000449998 3.29422 0.00118496
2. A{1} 0.235233190 0.073934584 3.18164 0.00172062
3. A{2} 0.088259374 0.075834416 1.16384 0.24600269
4. A{3} -0.032052478 0.076122076 -0.42107 0.67420042
5. A{4} 0.054569012 0.075839159 0.71954 0.47272857
6. A{5} 0.046046276 0.075913595 0.60656 0.54489358

Chi-Squared(5)= 15.321857 with Significance Level 0.00907208
Jimbo

Posts: 4
Joined: Thu Mar 08, 2012 6:28 pm

### Re: Multivariate GARCH Output

I assume those are tests for ARCH. (BTW, there's an ARCHTEST procedure (http://www.estima.com/forum/viewtopic.php?f=7&t=1554) which does this). The coefficients in that (or in almost any LM test like this) aren't really of interest---it's just the test statistic that matters, which here is giving clear signs of "ARCH-ness".
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Multivariate GARCH Output

Hi Tom,

Thanks for your help in the past. I'm back again with more results from another assignment and I need your help My RATS skills haven't improved too much. I ran a code from our Lecturer using WTI spot prices and WTI futures prices. I'm looking for an interpretation of the coefficents in the OLS and GARCH results in order to work an OHR , unfortunately I don't really know what I'm looking for, hence asking you for help as the RATS expert in my college is away on holidays.

This is the output I got:

GARCH Model - Estimation by BFGS
Convergence in 31 Iterations. Final criterion was 0.0000016 <= 0.0000100
Usable Observations 496
Log Likelihood 1289.00698439

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Mean 0.0010132696 0.0007843899 1.29179 0.19642873
2. C 0.0000302154 0.0000164380 1.83815 0.06604058
3. A 0.0793684789 0.0310971319 2.55228 0.01070215
4. B 0.8295110293 0.0650354910 12.75474 0.00000000

Linear Regression - Estimation by Least Squares
Dependent Variable A
Usable Observations 491 Degrees of Freedom 485
Centered R**2 0.027800 R Bar **2 0.017777
Uncentered R**2 0.327998 T x R**2 161.047
Mean of Dependent Variable 0.0003333894
Std Error of Dependent Variable 0.0004993166
Standard Error of Estimate 0.0004948586
Sum of Squared Residuals 0.0001187692
Regression F(5,485) 2.7737
Significance Level of F 0.01751800
Log Likelihood 3043.43780
Durbin-Watson Statistic 2.008190

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Constant 0.000243326 0.000037210 6.53928 0.00000000
2. A{1} 0.095470486 0.045164493 2.11384 0.03503887
3. A{2} 0.009482925 0.045355037 0.20908 0.83447196
4. A{3} 0.081384483 0.045206470 1.80028 0.07243704
5. A{4} -0.021235085 0.045217504 -0.46962 0.63883695
6. A{5} 0.104456010 0.045032079 2.31959 0.02077739

Chi-Squared(5)= 13.649659 with Significance Level 0.01799486

Linear Regression - Estimation by Least Squares
Dependent Variable A
Usable Observations 491 Degrees of Freedom 485
Centered R**2 0.026655 R Bar **2 0.016621
Uncentered R**2 0.351464 T x R**2 172.569
Mean of Dependent Variable 0.0003200257
Std Error of Dependent Variable 0.0004526690
Standard Error of Estimate 0.0004488914
Sum of Squared Residuals 0.0000977292
Regression F(5,485) 2.6564
Significance Level of F 0.02206881
Log Likelihood 3091.30591
Durbin-Watson Statistic 2.010548

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Constant 0.000244748 0.000036834 6.64460 0.00000000
2. A{1} 0.034134768 0.045055037 0.75762 0.44904417
3. A{2} -0.047345329 0.045038058 -1.05123 0.29367675
4. A{3} 0.078097763 0.044958738 1.73710 0.08300462
5. A{4} 0.042736616 0.044918052 0.95144 0.34185739
6. A{5} 0.126551047 0.044938565 2.81609 0.00505926

Chi-Squared(5)= 13.087813 with Significance Level 0.02256962

GARCH Model - Estimation by BFGS
Convergence in 1 Iterations. Final criterion was 0.0000720 <= 0.0001000
Usable Observations 496
Log Likelihood 3072.34273694

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. Mean(1) 0.00142 2.69735e-04 5.25780 0.00000015
2. Mean(2) 0.00163 2.23508e-04 7.27171 0.00000000
3. C(1,1) 1.48498e-04 3.33028e-07 445.90226 0.00000000
4. C(2,1) 1.52868e-04 1.44658e-07 1056.75765 0.00000000
5. C(2,2) 1.71258e-04 3.55069e-07 482.32221 0.00000000
6. A(1,1) 0.20414 0.00177 115.55775 0.00000000
7. A(2,1) 0.18285 8.81192e-04 207.49829 0.00000000
8. A(2,2) 0.18834 0.00167 112.84038 0.00000000
9. B(1,1) 0.39774 9.89860e-04 401.81733 0.00000000
10. B(2,1) 0.37546 5.36088e-04 700.36653 0.00000000
11. B(2,2) 0.35432 0.00104 340.16245 0.00000000
Jimbo

Posts: 4
Joined: Thu Mar 08, 2012 6:28 pm

### Re: Multivariate GARCH Output

Don't you have a book that handles GARCH models? RATS output isn't that much different from what you will see in a book.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Multivariate GARCH Output

No Tom unfortunately I don't have any book. Our lecturer who deals with RATS and GARCH models is away until September hence why I came on here for help as my first port of call having helped me in the past.
Jimbo

Posts: 4
Joined: Thu Mar 08, 2012 6:28 pm

### Re: Multivariate GARCH Output

You probably need to purchase a book then, or at least look up some basic ARCH/GARCH articles on the web. A detailed description of the standard notation for such models (beyond what we already provide in the User's Guide) is really beyond the scope of what we can provide for you here on the forum.

Regards,
Tom Maycock
Estima
moderator