VARMA GARCH Model

[TomDoan]

This is an example which estimates a multivariate GARCH model with VARMA mean models. It actually does both VAR-GARCH and VARMA-GARCH. This requires RATS version 8.1 or later for the simple coding using GARCH, as it makes use of the UADJUST option.

Note that this is only intended to show the process of estimating such a model. The VARMA(1,1) on this data set is almost certainly overparameterized, as the log likelihood gain (versus the simpler VAR-GARCH) is only about 11 for 9 extra parameters.

varmagarch.rpf

Program file

(1.75 KiB) Downloaded 329 times

exrates(daily).xls

Data file

(365.5 KiB) Downloaded 207 times

[IRJ]

Is there any reference to a paper/textbook that uses the VARMA GARCH model?

[TomDoan]

See Rahman and Serletis(2012), "Oil price uncertainty and the Canadian economy: Evidence from a VARMA, GARCH-in-Mean, asymmetric BEKK model", Energy Economics, Volume 34, Issue 2, March 2012, 603–610. Looks like it was done with RATS.

Note, BTW, that VARMA-GARCH is used for two completely different ideas. This is for a VARMA mean model. There's also the VARMA-GARCH variance process of Ling and McAleer, which is implemented in RATS using the options MV=CC and VARIANCES=VARMA. That allows use of the simpler CC model but with spillovers in the variances.

[lumengobobo46]

Dear Tom

I would assume that the option hmatrices=hh should be included in the GARCH model, especially in the case of a GARCH-M where the mean equation should include hh. therefore I have two questions: 1. what is the meaning of unadjust=%pt(u,t,rv(t)). 2. How to include the variance in the mean equation in the absence of hmatrices as option in the GARCH programme?

[TomDoan]

Dear Tom

I would assume that the option hmatrices=hh should be included in the GARCH model, especially in the case of a GARCH-M where the mean equation should include hh. therefore I have two questions: 1. what is the meaning of uadjust=%pt(u,t,rv(t)).

It takes the RV element just computed and splits it into the VEC[SERIES] u so lagged u's will be available to evaluate the mean at future values of t.

2. How to include the variance in the mean equation in the absence of hmatrices as option in the GARCH programme?

You use the HADJUST option, as in the following, which is a VARMA(2,1)-BEKK-GARCH-M (square root) model.

Code: Select all

dec vect[series] sqrth(2)
dec vect[series] eps(2)
*
system(model=varmah)
variables ipg infl
lags 1 2
det constant eps(1){1} eps(2){1} eps(1){2} eps(2){2} sqrth(1) sqrth(2)
end(system)
*
clear(zeros) sqrth eps
*
* Do three iterations on the VARMA part to get an initial value for the
* covariance matrix.
*
do iters=1,3
   estimate(noprint,resids=resids)
   set eps(1) %regstart() * = resids(1)
   set eps(2) %regstart() * = resids(2)
end do iters
*
compute cv0=%sigma
*
garch(model=varmah,mv=bekk,asymmetric,rvectors=rv,hmatrices=hh,presample=cv0,$
  uadjust=%pt(eps,t,rv(t)),hadjust=%pt(sqrth,t,%sqrt(%xdiag(hh(t)))),$
  pmethod=simplex,piters=20)

[lumengobobo46]

Thanks Tom; I am fully satisified with your response.

[Bella]

dear Tom,
I'd like to ask for a code for Wald test and likelihood ratio test within a VARMA(2,1)-asymmetric BEKK-GARCH-M (square root) model
for a12=b12=d12=a21=b21=d21=0, that is testing whether all coefficients including a12、b12、d12、a21、b21and d21 equal to 0?
the variance equation is showed in the following picture. I have tried to compile the code by myself but I don't know how to start it.
I would be very grateful for your instructions.

regards

Bella

[Bella]

dear Tom
I mean the null Hypothesis is: all the coefficients including a11、a12、b12、d12、a21、b21、d21are simultaneously zero
that is Null Hypothesis: a12=b12=d12=a21=b21=d21=0.

[TomDoan]

You can do a Wald test using

TEST(ZEROS)
# list of positions that you want to test

It would probably be easiest to just look at the output from the GARCH instruction to figure out what positions you want to test. For maximum likelihood, you would have to estimate the DBEKK model using MAXIMIZE to do the comparison. While I've seen DBEKK models used, I'm not sure I understand the point. The variance terms are allowed to vary, but the covariances aren't. A CC model will be similar, except it fixes correlations rather than covariances, which seems much more plausible.

[Bella]

Thank you for your answer. I'll tried to learn some about DBEKK.
Regards
Bella

[Bella]

I am sorry that i have another question. Are the coefficients from the variance model of a GARCH model stored in a vector like %BETA? How can i get the coefficients in A(1,2)、A(2,1)、B(1,2)、B(2,1)、D(1,2) and D(2,1).
regards
Bella

[Bella]

I am sorry that i have another question. Are the coefficients from the variance model of a GARCH model stored in a vector like %BETA? How can i get the coefficients in A(1,2)、A(2,1)、B(1,2)、B(2,1)、D(1,2) and D(2,1).

The following is the output of a VARMA-GARCH-BEKK model.
regards
Bella

MV-GARCH, BEKK - Estimation by BFGS
NO CONVERGENCE IN 59 ITERATIONS
LAST CRITERION WAS 0.0000000
SUBITERATIONS LIMIT EXCEEDED.
ESTIMATION POSSIBLY HAS STALLED OR MACHINE ROUNDOFF IS MAKING FURTHER PROGRESS DIFFICULT
TRY HIGHER SUBITERATIONS LIMIT, TIGHTER CVCRIT, DIFFERENT SETTING FOR EXACTLINE OR ALPHA ON NLPAR
RESTARTING ESTIMATION FROM LAST ESTIMATES OR DIFFERENT INITIAL GUESSES MIGHT ALSO WORK
Monthly Data From 1995:03 To 2011:12
Usable Observations 202
Log Likelihood 971.6380

Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. SZ{1} -0.117936006 0.042492735 -2.77544 0.00551272
2. SZ{2} 0.141955468 0.011608641 12.22843 0.00000000
3. CPI{1} -0.589937420 0.051885908 -11.36990 0.00000000
4. CPI{2} -0.035847550 0.176604592 -0.20298 0.83914912
5. Constant -0.012676657 0.004790693 -2.64610 0.00814256
6. EPS(1){1} 0.167128638 0.028361500 5.89280 0.00000000
7. EPS(2){1} 1.161043728 0.075180334 15.44345 0.00000000
8. EPS(1){2} 0.006151929 0.021264000 0.28931 0.77234267
9. EPS(2){2} 0.958279096 0.270894842 3.53746 0.00040400
10. SQRTH(1) 0.118908118 0.047816686 2.48675 0.01289161
11. SQRTH(2) 1.575369594 0.633793325 2.48562 0.01293258
12. SZ{1} -0.014364113 0.001753380 -8.19224 0.00000000
13. SZ{2} -0.074823279 0.001158599 -64.58081 0.00000000
14. CPI{1} 0.086281249 0.003676484 23.46841 0.00000000
15. CPI{2} -0.878388590 0.012013064 -73.11945 0.00000000
16. Constant 0.007155170 0.000417686 17.13052 0.00000000
17. EPS(1){1} -0.007189545 0.001427283 -5.03722 0.00000047
18. EPS(2){1} 0.217551239 0.004243445 51.26760 0.00000000
19. EPS(1){2} 0.064254898 0.001542754 41.64949 0.00000000
20. EPS(2){2} 0.993663770 0.020914492 47.51078 0.00000000
21. SQRTH(1) 0.065441274 0.004677328 13.99117 0.00000000
22. SQRTH(2) -1.305522046 0.073720331 -17.70912 0.00000000
23. C(1,1) 0.006132869 0.003087594 1.98629 0.04700069
24. C(2,1) 0.002469577 0.000314131 7.86162 0.00000000
25. C(2,2) 0.003664780 0.000322173 11.37520 0.00000000
26. A(1,1) 0.332852470 0.051451335 6.46927 0.00000000
27. A(1,2) 0.008766603 0.003843537 2.28087 0.02255622
28. A(2,1) 2.929005849 0.698602279 4.19267 0.00002757
29. A(2,2) 0.421650503 0.033092739 12.74148 0.00000000
30. B(1,1) 0.909580947 0.017822438 51.03572 0.00000000
31. B(1,2) -0.002947026 0.003228065 -0.91294 0.36127467
32. B(2,1) -2.720891012 0.532671316 -5.10801 0.00000033
33. B(2,2) 0.277304168 0.028687648 9.66633 0.00000000
34. D(1,1) -0.177803372 0.110317189 -1.61175 0.10701706
35. D(1,2) -0.060930740 0.004176135 -14.59022 0.00000000
36. D(2,1) -0.004553960 1.497792448 -0.00304 0.99757408
37. D(2,2) 0.455855798 0.044716865 10.19427 0.00000000

[TomDoan]

Yes. %BETA. You can see the sequence numbers in the output. So %beta(31) is, for instance, B(1,2)

[Bella]

Dear Tom
I want to make a Wald test for null hypothesis: a12=b12=d12=0.

the code is
compute a12=%beta(27),b12=%beta(31),d12=%beta(35)
TEST(title="Wald Test",form=chisquared,ZEROS)
# a12 b12 d12

But I get the following error:
## SX22. Expected Type INTEGER, Got REAL Instead
>>>># a12 <<<<

I don't know how i am wrong?

[Bella]

I think I have known the error source that TEST(ZEROS)
# a12 b12 d12 should be replaced as TEST(ZEROS)
# 27 31 35.

Thanks Tom.

regards
Bella