Element of covariance matrix in Multivariate GARCH-M model

[mengqi]

Hi,

I tried to estimate a VARX-MGARCH-MEAN model with covariance matrix considered in the mean equation using the following codes:

dec symm[series] hhs(2,2)
clear(zeros) hhs
System(model=varxmodel)
variables lrbtc lrltc
lags 1
deterministic constant tbtc tltc wbtc wltc hbtc hltc usd hhs(1,1) hhs(1,2) hhs(2,1) hhs(2,2)
end(system)
estimate(noprint)
garch(model=varxmodel,p=1,q=3,mvhseries=hhs,pmethod=simplex,piters=1000,iters=1000, mv=DCC,rseries=rs,stdresids=zu,derives=dd)

The results I obtained is the following:

Code: Select all

MV-DCC GARCH  - Estimation by BFGS
Convergence in     1 Iterations. Final criterion was  0.0000001 <=  0.0000100
LOW ITERATION COUNT ON BFGS MAY LEAD TO POOR ESTIMATES FOR STANDARD ERRORS

Daily(7) Data From 2013:07:19 To 2016:01:20
Usable Observations                       916
Log Likelihood                      4785.7106

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
Mean Model(LRBTC)
1.  LRBTC{1}                      0.211854063  0.018719216     11.31746  0.00000000
2.  LRLTC{1}                     -0.033708185  0.010467811     -3.22018  0.00128112
3.  Constant                     -0.000530048  0.000185371     -2.85939  0.00424453
4.  TBTC                         -0.000635944  0.000804251     -0.79073  0.42910301
5.  TLTC                         -0.000010182  0.000634547     -0.01605  0.98719797
6.  WBTC                         -0.003406835  0.001375931     -2.47602  0.01328557
7.  WLTC                         -0.006375154  0.002296862     -2.77559  0.00551011
8.  HBTC                         -0.017822534  0.004189831     -4.25376  0.00002102
9.  HLTC                         -0.024813830  0.007298461     -3.39987  0.00067418
10. USD                           0.750206166  0.170819593      4.39180  0.00001124
11. HHS(1,1)                      3.047917521  0.312133940      9.76477  0.00000000
12. HHS(2,1)                     -0.324163333  0.196634955     -1.64855  0.09923905
13. HHS(2,1)                      0.769942717  0.196634955      3.91559  0.00009018
14. HHS(2,2)                     -0.228350557  0.096502756     -2.36626  0.01796885
Mean Model(LRLTC)
15. LRBTC{1}                      0.659116430  0.014509460     45.42667  0.00000000
16. LRLTC{1}                     -0.199070977  0.018119532    -10.98654  0.00000000
17. Constant                     -0.003120156  0.000240310    -12.98390  0.00000000
18. TBTC                          0.005092588  0.001233223      4.12949  0.00003636
19. TLTC                         -0.000403394  0.000627429     -0.64293  0.52026825
20. WBTC                          0.002659721  0.001911144      1.39169  0.16401599
21. WLTC                         -0.007408928  0.003048082     -2.43069  0.01507031
22. HBTC                         -0.006490039  0.007233257     -0.89725  0.36958561
23. HLTC                          0.006547112  0.009378495      0.69810  0.48511565
24. USD                           0.084725482  0.292027523      0.29013  0.77171800
25. HHS(1,1)                      1.462281071  0.397413249      3.67950  0.00023369
26. HHS(2,1)                      2.462092086  0.311330206      7.90830  0.00000000
27. HHS(2,1)                     -0.062806097  0.311330140     -0.20173  0.84012412
28. HHS(2,2)                      0.347067040  0.174738823      1.98620  0.04701062

29. C(1)                          0.000027554  0.000000844     32.63330  0.00000000
30. C(2)                          0.000014304  0.000000582     24.59590  0.00000000
31. A{1}(1)                       0.315975394  0.007281194     43.39609  0.00000000
32. A{1}(2)                       0.442895966  0.002097217    211.18271  0.00000000
33. A{2}(1)                      -0.129825562  0.006379184    -20.35144  0.00000000
34. A{2}(2)                      -0.201961752  0.001864739   -108.30564  0.00000000
35. A{3}(1)                       0.020958048  0.005759580      3.63882  0.00027389
36. A{3}(2)                      -0.079581339  0.001702136    -46.75381  0.00000000
37. B(1)                          0.778773504  0.003469686    224.45073  0.00000000
38. B(2)                          0.868810083  0.001118097    777.04385  0.00000000
39. DCC(A)                        0.209348733  0.000864199    242.24588  0.00000000
40. DCC(B)                        0.789954569  0.000874832    902.97910  0.00000000

My question is, in both of the mean equations i get two HHS(2,1) rather than HHS(1,2) and HHS(2,1). Is there any explanation for this?

Kind regards
Menqi

[TomDoan]

First of all, you're way overdoing the simplex iterations---you're doing enough that it's converged already, so BFGS doesn't have enough chance to build up a good estimate of the covariance matrix. (That's why you get that message about the small number of iterations).

HHS is symmetric, so HHS(1,2) and HHS(2,1) are the same. As they are stored, they are both considered to be HHS(2,1). Leave the HHS(1,2) out of the list of regressors.

[mengqi]

Dear Tom,

In order to get bivariate garch in mean model. Do I include square root of both variance as well as the covariance in the conditional mean equation, ie: sqrt(HHS(1,1)), sqrt(HHS(2,2),HHS(2,1)? (Rather than using HHS(1,1), HHS(2,1), HHS(2,2)?)

However, there was an error when I tried to use square root of variance in conditional mean equation.

Do I need to use HADJUST in GARCH command?

Thanks

[TomDoan]

If you want the covariance, you almost certain would want the variances, not standard deviations.

[mengqi]

Thanks for your reply.

I mainly want to observe covariance.

However, I saw univariate GARCH in mean model includes standard deviation in conditional mean equation rather than variance.

So I am wondering in bivariate case, should I include two standard deviation of two asset returns as well as the covariance. Or should I just use two variances and the covariance.

Thanks very much.

[TomDoan]

Both variances and standard deviations get used, depending upon the situation. CAPM would argue for variances, but variances are on the order of units^2, while standard deviations are in units, so it appears than when risk vs return isn't considered to be the reason for the "M" effect, standard deviations get used. However, if you want to mix in covariances, you want to use variances since they have a scale compatible with the covariances.