BEKK stability check

[fatemaalaali]

Hi,

I am applying VAR(1)-GARCH(1,1) model using the following code:

Code: Select all

system(model=var1)
variables  RUSRet GOLDRET WTIRet
lags 1
det constant
end(system)
garch(p=1,q=1,model=var1,mv=BEKK,asymmetric,pmethod=simplex,piters=10,rseries=rs,mvhseries=hhs,$
stdresids=zu,derives=dd)
@mvqstat(lags=15)
# zu
@mvarchtest(lags=15)
# zu
@regcrits

I would like to check for the stationarity of the model using the roots of|I-AL-BL|=0. How can I get these roots?

Many thanks for your help.

[TomDoan]

I'm assuming you're talking about the roots of the GARCH recursion, and not the VAR. After the GARCH instruction:

@MVGARCHtoVECH(mv=BEKK)
eigen(cvalues=cv) %%vech_a+%%vech_b
disp "Eigenvalues from BEKK-t" *.### cv

(The I-A-B roots are 1- those).

[fatemaalaali]

Many thanks for your prompt reply.

I have used the posted code and got the following:

Eigenvalues from BEKK-t (0.965,0.000) (0.948,0.000) (0.941,0.000) (0.921,0.000) (0.905,0.000) (0.889,0.000)

Do these results mean that the model is not stationary?

[TomDoan]

Stationary is the wrong word. (Very strange-looking GARCH models can be stationary---see D. Nelson(1990)). And those are all stable roots (less than 1 in absolute value), so there is no obvious problem.

[fatemaalaali]

Thank you very much Tom.

I got confused because I was reading the following attached paper. On page 279, in the notes of the table. They got roots that are greater than 1.

Many thanks for your clarification. That was helpful.

[TomDoan]

Those are equivalent statements. What you did computes the eigenvalues of A+B. Those are the reciprocals of the roots of |I-(A+B)z|=0 so if the first are less than one, the latter are greater than one.

[fatemaalaali]

Got it now.

Thank you very much Tom.

[mengqi]

Hi Tom,
In the above example: (0.965,0.000) (0.948,0.000) (0.941,0.000) (0.921,0.000) (0.905,0.000) (0.889,0.000).
these eigenvalues are calculated for each conditional variance covariance equations? So each of these single equations have stable root?

I have the following results: Eigenvalues from BEKK-t (1.076,-0.000) (1.020,0.000) (0.966,0.000)
is it possible to calculate the dominant root for the whole model? or there is no point of doing so because my results are not good.

Thank you.

[TomDoan]

Hi Tom,
In the above example: (0.965,0.000) (0.948,0.000) (0.941,0.000) (0.921,0.000) (0.905,0.000) (0.889,0.000).
these eigenvalues are calculated for each conditional variance covariance equations? So each of these single equations have stable root?

No. Those are determined by the whole system. The entire system is stable.

I have the following results: Eigenvalues from BEKK-t (1.076,-0.000) (1.020,0.000) (0.966,0.000)
is it possible to calculate the dominant root for the whole model? or there is no point of doing so because my results are not good.

The dominant root is 1.076. If you forecast the covariance matrix from pretty much any set of initial conditions, eventually the variances will go up at an exponential rate of 1.076. If it were 1.02, I wouldn't be all that concerned (that's probably within sampling error), but 1.076 is pretty big.