This is an example of a multivariate GARCH model with VAR and VARMA mean models. Don't confuse the "VARMA-GARCH" with the (Chan and McAleer) VARMA model for the variances.

While GARCH models with a VAR mean model are common, VARMA mean models are much less common and for good reason. In the time series literature, VAR's are overwhelmingly chosen for two reasons:

1. they're much simpler to estimate
2. they don't suffer from identification/cancellation problems. (See, for instance, Section 12.1.1 of Lutkepohl(2006)).

For a GARCH model, the first of these doesn't really apply since, either way, you have to use non-linear estimation. However, the second is still an issue, and this is particularly true with the typical application for GARCH models, which are return series with (at most) fairly weak serial correlation. There is very little harm in adding extra (unnecessary) VAR lags to a VAR-GARCH model unless the data set is quite short compared to the typical GARCH data set. However, adding an unnecessary MA to the mean can produce a model which is effectively unidentified.

While this problem with the mean model will generally have little effect on the estimates of the variance model, it can make it much more difficult to get the overall model to converge. This example has no problem estimating a DVECH and BEKK model to a VAR(1) mean, but uses the PMETHOD=GA (added with RATS 9.2) to one of the models with the VARMA(1,1) mean because of problems getting the model to behave, and needs almost 500 iterations to get the other to converge.

This uses @VARLAGSELECT to try various lag lengths for a VAR to explain the mean, the first using BIC (SBC) and the second AIC.

@varlagselect(lags=5,crit=bic)

# reuro rpound rsw

@varlagselect(lags=5,crit=aic)

# reuro rpound rsw

As one can see, the two have very different ideas. The calculations used in these don't allow for heteroscedasticity, so you really can't use them as another more than a guide. In general, it's best to start with a small model, and only go to a more complicated one if diagnostics show that the smaller model isn't adequate. In this case, we'll use a VAR(1) as the starting point. This sets up an estimates a DVECH model with a VAR(1) mean:

system(model=simplevar)

variables reuro rpound rsw

lags 1

det constant

end(system)

nlpar(derives=fourth,exactline)

garch(p=1,q=1,model=simplevar,distrib=t,$pmethod=simplex,piters=10,iters=500,stdresids=stdu) The NLPAR instruction to use slower but more accurate calculations isn't really necessary for this model, but is needed for later ones. This estimates a BEKK with the same mean model: garch(p=1,q=1,model=simplevar,distrib=t,mv=bekk,$

pmethod=simplex,piters=10,iters=500,stdresids=stdu)

The estimates of the parameters on the means are almost identical, and, while some of the lag coefficients are statistically significant, none of them are "economically" large—you would be hard-pressed to see any difference between these data and white noise in a graph. More to the point, the diagnostics (here for the VAR(1)-BEKK model) show that the VAR(1) model has done its job of producing serially uncorrelated standardized residuals (highly insignificant Q statistic). The test for residual ARCH might be disappointing, but that would require looking more carefully at the "GARCH" part of this model, not the mean (and is outside the scope of this example).

Even though there is no reason to continue to a more complicated VARMA(1,1) model here, we will do it for illustration. Again, you should never start with a VARMA model, even if you find a paper which has used a VARMA model on a similar data set. The mean model is specific to a data set, and you should choose the simplest one you can find that is able to produce serially uncorrelated residuals.

The setup of the mean model for a full VARMA (lagged residuals from each variable in each equation) is

dec vect[series] u(3)

system(model=varma)

variables reuro rpound rsw

lags 1

det constant u(1){1} u(2){1} u(3){1}

end(system)

*

clear(zeros) u

The U's are the residuals, with U(1) being the residual for the first equation (for REURO), U(2) for RPOUND, etc. The diagonal VECH with this mean model is done using:

garch(p=1,q=1,model=varma,rvector=rv,uadjust=%pt(u,t,rv(t)),distrib=t,$pmethod=ga,piters=5,iters=500,stdresids=stdu) The key additions here are the RVECTOR=RV and UADJUST=%PT(U,T,RV(T)) options. UADJUST is an option which gets inserted into the GARCH calculation after the residuals have been computed for a given time period. The RVECTOR option supplies a SERIES[VECTOR], and it copies the residuals at time T into entry T of that (so RV(T) are the residuals at T). The %PT splits the VECTOR RV(T) into the separate elements of U so they can be used at the next period as the lag of U(1), ... Not surprisingly (given the fact that we saw no obvious serial correlation left behind by the VAR(1)), the log likelihoods for the VARMA models are only slightly better than the VAR and the more complicated models would be rejected by any of the standard information criteria. And the resulting (mean model) coefficient estimates, particularly in the BEKK model show all the signs of the VARMA identification problem with wild values with the AR and corresponding MA being nearly equal with opposite signs. Note that the GARCH coefficients are almost the same when you compare the two BEKK's with each other—the odd coefficients in the mean basically cancel each other out leaving roughly the same residuals. Full Program open data "exrates(daily).xls" calendar(d) 2000:1:3 data(format=xls,org=columns) 2000:01:03 2008:12:23 aust euro pound sw ca * * We generally recommend that the returns be multiplied up by 100, which * changes the scales on the variances by 10000. This puts the * coefficients in the GARCH within a few orders of magnitude of 1, which * makes them easier to read. It also makes it easier to estimate the * model. Note that this doesn't affect the lagged variance or lagged * squared residual terms in the GARCH specification; just the means * (which will be higher by x 100) and the constants in the GARCH, which * will be higher by x 10000. * set reuro = 100.0*log(euro/euro{1}) set rpound = 100.0*log(pound/pound{1}) set rsw = 100.0*log(sw/sw{1}) * spgraph(vfields=3,footer="Exchange rate returns") graph(header="Euro") # reuro graph(header="UK Pound") # rpound graph(header="Swiss Franc") # rsw spgraph(done) * @varlagselect(lags=5,crit=bic) # reuro rpound rsw @varlagselect(lags=5,crit=aic) # reuro rpound rsw * * VAR-GARCH * system(model=simplevar) variables reuro rpound rsw lags 1 det constant end(system) * * Diagonal VECH model for one lag VAR * nlpar(derives=fourth,exactline) garch(p=1,q=1,model=simplevar,distrib=t,$
pmethod=simplex,piters=10,iters=500,stdresids=stdu)
*
* BEKK on one lag VAR
*
garch(p=1,q=1,model=simplevar,distrib=t,mv=bekk,$pmethod=simplex,piters=10,iters=500,stdresids=stdu) * * Tests for residual ARCH, and for residual serial correlation * @mvarchtest(lags=5) # stdu @mvqstat(lags=5) # stdu * * VARMA(1,1)-GARCH * dec vect[series] u(3) system(model=varma) variables reuro rpound rsw lags 1 det constant u(1){1} u(2){1} u(3){1} end(system) * clear(zeros) u * * Diagonal VECH on VARMA(1,1) mean model * garch(p=1,q=1,model=varma,rvector=rv,uadjust=%pt(u,t,rv(t)),distrib=t,$
pmethod=ga,piters=5,iters=500,stdresids=stdu)
*
* BEKK on VARMA(1,1) mean model
*
pmethod=simplex,piters=20,iters=1000)
*
* Tests for residual ARCH, and for residual serial correlation
*
@mvarchtest(lags=5)
# stdu
@mvqstat(lags=5)
# stdu


Output

VAR Lag Selection

Lags SBC/BIC

0 3.10059950*

1 3.11338393

2 3.13335378

3 3.15647962

4 3.17731837

5 3.19583282

VAR Lag Selection

Lags AICC

0 3.09320859

1 3.08383353

2 3.08166378

3 3.08267000

4 3.08140919

5 3.07784420*

MV-GARCH - Estimation by BFGS

Convergence in   141 Iterations. Final criterion was  0.0000000 <=  0.0000100

Daily(5) Data From 2000:01:05 To 2008:12:23

Usable Observations                      2340

Log Likelihood                     -2567.2677

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

Mean Model(REURO)

1.  REURO{1}                      0.209378532  0.014739656     14.20512  0.00000000

2.  RPOUND{1}                    -0.054730387  0.025512607     -2.14523  0.03193453

3.  RSW{1}                       -0.164985452  0.015213740    -10.84450  0.00000000

4.  Constant                      0.020228580  0.009917240      2.03974  0.04137635

Mean Model(RPOUND)

5.  REURO{1}                      0.140880988  0.031692402      4.44526  0.00000878

6.  RPOUND{1}                     0.016754971  0.025728358      0.65123  0.51490073

7.  RSW{1}                       -0.133025025  0.027428857     -4.84982  0.00000124

8.  Constant                      0.017347793  0.009131388      1.89980  0.05745964

Mean Model(RSW)

9.  REURO{1}                      0.161174596  0.014875119     10.83518  0.00000000

10. RPOUND{1}                    -0.057871752  0.028037476     -2.06409  0.03900960

11. RSW{1}                       -0.132581133  0.018848976     -7.03386  0.00000000

12. Constant                      0.013508000  0.010993365      1.22874  0.21916888

13. C(1,1)                        0.002634720  0.000973294      2.70701  0.00678913

14. C(2,1)                        0.002005687  0.000849551      2.36088  0.01823166

15. C(2,2)                        0.002493164  0.001211333      2.05820  0.03957100

16. C(3,1)                        0.002454412  0.000994691      2.46751  0.01360554

17. C(3,2)                        0.001731103  0.000789840      2.19171  0.02840028

18. C(3,3)                        0.002972899  0.001295928      2.29403  0.02178865

19. A(1,1)                        0.031943246  0.004991266      6.39983  0.00000000

20. A(2,1)                        0.027536881  0.004525767      6.08447  0.00000000

21. A(2,2)                        0.033246759  0.005332250      6.23503  0.00000000

22. A(3,1)                        0.029698073  0.004720785      6.29092  0.00000000

23. A(3,2)                        0.026493859  0.004139350      6.40049  0.00000000

24. A(3,3)                        0.029480817  0.004864696      6.06016  0.00000000

25. B(1,1)                        0.962696735  0.006296626    152.89089  0.00000000

26. B(2,1)                        0.965449092  0.006519007    148.09756  0.00000000

27. B(2,2)                        0.960353014  0.007564946    126.94778  0.00000000

28. B(3,1)                        0.965523559  0.006053045    159.51039  0.00000000

29. B(3,2)                        0.968442355  0.005470510    177.02964  0.00000000

30. B(3,3)                        0.965480199  0.006454291    149.58734  0.00000000

31. Shape                         7.094912590  0.506860722     13.99776  0.00000000

MV-GARCH, BEKK - Estimation by BFGS

Convergence in   185 Iterations. Final criterion was  0.0000013 <=  0.0000100

Daily(5) Data From 2000:01:05 To 2008:12:23

Usable Observations                      2340

Log Likelihood                     -2552.0530

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

Mean Model(REURO)

1.  REURO{1}                      0.202357368  0.049417457      4.09486  0.00004224

2.  RPOUND{1}                    -0.054588791  0.029286178     -1.86398  0.06232480

3.  RSW{1}                       -0.161712868  0.041338404     -3.91193  0.00009156

4.  Constant                      0.021629962  0.010249763      2.11029  0.03483348

Mean Model(RPOUND)

5.  REURO{1}                      0.136150241  0.044390267      3.06712  0.00216133

6.  RPOUND{1}                     0.016651036  0.027192528      0.61234  0.54031372

7.  RSW{1}                       -0.130990704  0.038082594     -3.43965  0.00058247

8.  Constant                      0.018778884  0.009416485      1.99426  0.04612405

Mean Model(RSW)

9.  REURO{1}                      0.151980119  0.052812017      2.87776  0.00400514

10. RPOUND{1}                    -0.057125209  0.031981121     -1.78622  0.07406420

11. RSW{1}                       -0.126641992  0.045417971     -2.78837  0.00529744

12. Constant                      0.014522802  0.011348255      1.27974  0.20063705

13. C(1,1)                        0.033314660  0.010536408      3.16186  0.00156764

14. C(2,1)                        0.048656603  0.011643402      4.17890  0.00002929

15. C(2,2)                       -0.001807824  0.008481230     -0.21316  0.83120537

16. C(3,1)                        0.014180181  0.013571313      1.04486  0.29608565

17. C(3,2)                       -0.023470938  0.008102760     -2.89666  0.00377159

18. C(3,3)                        0.000000403  0.180368205  2.23542e-06  0.99999822

19. A(1,1)                        0.228246320  0.040490239      5.63707  0.00000002

20. A(1,2)                        0.124476004  0.042162724      2.95228  0.00315441

21. A(1,3)                       -0.035090265  0.042202712     -0.83147  0.40570849

22. A(2,1)                        0.027015850  0.025432890      1.06224  0.28812643

23. A(2,2)                        0.174821831  0.025708908      6.80005  0.00000000

24. A(2,3)                        0.000099984  0.024813469      0.00403  0.99678501

25. A(3,1)                       -0.105428817  0.032071604     -3.28729  0.00101155

26. A(3,2)                       -0.118505953  0.033529656     -3.53436  0.00040876

27. A(3,3)                        0.157890704  0.036983481      4.26922  0.00001962

28. B(1,1)                        0.963427203  0.009213238    104.56989  0.00000000

29. B(1,2)                       -0.026675663  0.010298271     -2.59030  0.00958910

30. B(1,3)                        0.005216031  0.009343084      0.55828  0.57665510

31. B(2,1)                       -0.016279312  0.005018437     -3.24390  0.00117905

32. B(2,2)                        0.973336158  0.005690512    171.04545  0.00000000

33. B(2,3)                       -0.004625606  0.004879547     -0.94796  0.34315080

34. B(3,1)                        0.034784451  0.008986944      3.87055  0.00010859

35. B(3,2)                        0.032941371  0.010418409      3.16184  0.00156774

36. B(3,3)                        0.989573991  0.009842454    100.54139  0.00000000

37. Shape                         7.253427847  0.587890218     12.33807  0.00000000

Test for Multivariate ARCH

Statistic Degrees Signif

346.97     180 0.00000

Multivariate Q(5)=      33.52557

Significance Level as Chi-Squared(45)=       0.89599

MV-GARCH - Estimation by BFGS

Convergence in   178 Iterations. Final criterion was  0.0000000 <=  0.0000100

Daily(5) Data From 2000:01:05 To 2008:12:23

Usable Observations                      2340

Log Likelihood                     -2561.8457

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

Mean Model(REURO)

1.  REURO{1}                     -0.014068067  0.061817711     -0.22757  0.81997790

2.  RPOUND{1}                    -0.071901125  0.086224632     -0.83388  0.40434776

3.  RSW{1}                        0.296274478  0.132965742      2.22820  0.02586708

4.  Constant                      0.018708139  0.009637512      1.94118  0.05223652

5.  U(1){1}                       0.213008646  0.052628964      4.04737  0.00005180

6.  U(2){1}                       0.015875540  0.080567403      0.19705  0.84379100

7.  U(3){1}                      -0.450225763  0.125213645     -3.59566  0.00032357

Mean Model(RPOUND)

8.  REURO{1}                      1.029854862  0.360632235      2.85569  0.00429430

9.  RPOUND{1}                    -0.018288971  0.253789264     -0.07206  0.94255128

10. RSW{1}                       -0.711495672  0.271801522     -2.61770  0.00885238

11. Constant                      0.007024168  0.009524326      0.73750  0.46081980

12. U(1){1}                      -0.902671268  0.375303541     -2.40518  0.01616466

13. U(2){1}                       0.032383172  0.261206380      0.12398  0.90133473

14. U(3){1}                       0.590538189  0.279278450      2.11451  0.03447139

Mean Model(RSW)

15. REURO{1}                      0.243841113  0.119037874      2.04843  0.04051759

16. RPOUND{1}                     0.050986040  0.097698763      0.52187  0.60176094

17. RSW{1}                        0.022280890  0.078779112      0.28283  0.77730916

18. Constant                      0.007428452  0.009648878      0.76988  0.44137266

19. U(1){1}                      -0.096995349  0.130298561     -0.74441  0.45662941

20. U(2){1}                      -0.104644228  0.093434722     -1.11997  0.26272590

21. U(3){1}                      -0.146766672  0.082000314     -1.78983  0.07348115

22. C(1,1)                        0.002519787  0.000827763      3.04409  0.00233384

23. C(2,1)                        0.001920914  0.000764742      2.51185  0.01201015

24. C(2,2)                        0.002493618  0.001048227      2.37889  0.01736479

25. C(3,1)                        0.002327588  0.000824076      2.82448  0.00473573

26. C(3,2)                        0.001660800  0.000658103      2.52362  0.01161546

27. C(3,3)                        0.002815421  0.001010467      2.78626  0.00533206

28. A(1,1)                        0.030797429  0.004863732      6.33206  0.00000000

29. A(2,1)                        0.026822131  0.004319575      6.20944  0.00000000

30. A(2,2)                        0.033002494  0.005172154      6.38080  0.00000000

31. A(3,1)                        0.028573522  0.004543561      6.28879  0.00000000

32. A(3,2)                        0.025880691  0.003667200      7.05734  0.00000000

33. A(3,3)                        0.028392681  0.004530741      6.26667  0.00000000

34. B(1,1)                        0.964114424  0.006212524    155.18884  0.00000000

35. B(2,1)                        0.966574557  0.006798803    142.16835  0.00000000

36. B(2,2)                        0.960648145  0.007529196    127.58974  0.00000000

37. B(3,1)                        0.966947751  0.005870076    164.72492  0.00000000

38. B(3,2)                        0.969368920  0.005351879    181.12683  0.00000000

39. B(3,3)                        0.966892257  0.005908982    163.63095  0.00000000

40. Shape                         7.096699684  0.586627405     12.09746  0.00000000

MV-GARCH, BEKK - Estimation by BFGS

Convergence in   473 Iterations. Final criterion was  0.0000028 <=  0.0000100

Daily(5) Data From 2000:01:05 To 2008:12:23

Usable Observations                      2340

Log Likelihood                     -2545.7617

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

Mean Model(REURO)

1.  REURO{1}                      1.197973087  1.088459189      1.10061  0.27106482

2.  RPOUND{1}                    -0.375758570  0.429955364     -0.87395  0.38214665

3.  RSW{1}                       -0.978191605  1.073632934     -0.91110  0.36224041

4.  Constant                      0.017536012  0.011823086      1.48320  0.13802105

5.  U(1){1}                      -1.000174100  1.090184332     -0.91744  0.35891434

6.  U(2){1}                       0.322980232  0.431329341      0.74880  0.45397661

7.  U(3){1}                       0.819366218  1.075548905      0.76181  0.44617213

Mean Model(RPOUND)

8.  REURO{1}                      2.629415188  1.209259048      2.17440  0.02967496

9.  RPOUND{1}                    -0.441259163  0.464093225     -0.95080  0.34170665

10. RSW{1}                       -2.333492926  1.159377359     -2.01271  0.04414492

11. Constant                      0.003944563  0.014978619      0.26335  0.79228372

12. U(1){1}                      -2.500322236  1.206782390     -2.07189  0.03827556

13. U(2){1}                       0.457294637  0.458770350      0.99678  0.31886969

14. U(3){1}                       2.206587053  1.159738083      1.90266  0.05708495

Mean Model(RSW)

15. REURO{1}                      1.519626499  1.132095336      1.34231  0.17949447

16. RPOUND{1}                    -0.293750512  0.411214445     -0.71435  0.47501157

17. RSW{1}                       -1.382372944  1.136911016     -1.21590  0.22402207

18. Constant                      0.007000370  0.013015217      0.53786  0.59067343

19. U(1){1}                      -1.363200982  1.122993183     -1.21390  0.22478606

20. U(2){1}                       0.239052070  0.408971080      0.58452  0.55887003

21. U(3){1}                       1.249578535  1.133583297      1.10233  0.27031990

22. C(1,1)                        0.034213022  0.010113951      3.38276  0.00071763

23. C(2,1)                        0.050192336  0.011277299      4.45074  0.00000856

24. C(2,2)                       -0.001915260  0.008698759     -0.22018  0.82573390

25. C(3,1)                        0.014928358  0.013094839      1.14002  0.25427860

26. C(3,2)                       -0.022782208  0.008005922     -2.84567  0.00443182

27. C(3,3)                       -0.000002833  0.164597998 -1.72109e-05  0.99998627

28. A(1,1)                        0.232921972  0.040242930      5.78790  0.00000001

29. A(1,2)                        0.134609352  0.041568816      3.23823  0.00120274

30. A(1,3)                       -0.029775690  0.041756608     -0.71308  0.47579792

31. A(2,1)                        0.024957849  0.025218424      0.98967  0.32233678

32. A(2,2)                        0.173336601  0.025413602      6.82062  0.00000000

33. A(2,3)                       -0.000333230  0.024379477     -0.01367  0.98909447

34. A(3,1)                       -0.108166526  0.031655198     -3.41702  0.00063310

35. A(3,2)                       -0.125795359  0.033249927     -3.78333  0.00015475

36. A(3,3)                        0.153512259  0.036365828      4.22133  0.00002429

37. B(1,1)                        0.962886503  0.009097083    105.84564  0.00000000

38. B(1,2)                       -0.028205676  0.010179868     -2.77073  0.00559306

39. B(1,3)                        0.004078178  0.009093497      0.44847  0.65381265

40. B(2,1)                       -0.016557174  0.004987052     -3.32003  0.00090007

41. B(2,2)                        0.972707541  0.005647346    172.24153  0.00000000

42. B(2,3)                       -0.004833512  0.004825034     -1.00176  0.31646093

43. B(3,1)                        0.035292419  0.008823122      3.99999  0.00006334

44. B(3,2)                        0.034450463  0.010279357      3.35142  0.00080398

45. B(3,3)                        0.990702221  0.009526221    103.99740  0.00000000

46. Shape                         7.307117875  0.584838219     12.49426  0.00000000

Test for Multivariate ARCH

Statistic Degrees Signif

353.50     180 0.00000

Multivariate Q(5)=      29.89063

Significance Level as Chi-Squared(45)=       0.95945