Regularity condition in GARCH models

[d952]

Hi, I want to test if regularity condition is satisfied in my GARCH models or not, as I know in a GARCH(1,1) model (Alpha+beta)<1 verifies the second moment condition and (3*Alpha^2+2(alpha*beta)+Alpha*beta)<1 vefiries the fourth moment condition. But the problem is that I am estimating GARCH(2,1) and GARCH(2,2) models- I know that the second moment condition is acheivable only with adding Alpha2 and beta2, but Im wondering how can I calculate the fourth moment condition?

I appreciate alot if you reply me
Thanks

[d952]

I read the paper by Trasvirta (1997) who explain how to calculate the fourth moment structure for GARCH(2,2) models but in the formula there is one thing that I dont undrestand, I attach the paper here, in that paper, equation number 36 is what I need, I undrestand it but my i dont know what is gama~12 in that equation, I mean the gama that has a tilde symbol.
can any body help me about it?
thanks

[trubador]

It corresponds to E(c1t,c2t), which is equal to gamma11*gamma21 under certain conditions. If you are interested, you can also refer to Ling, S. and McAleer, M. (2002). "Necessary and Sufficient Moment Conditions for the GARCH(r,s) and Asymmetric Power GARCH(r,s) Models. Econometric Theory, v. 18, pp. 722-729.

[d952]

It corresponds to E(c1t,c2t), which is equal to gamma11*gamma21 under certain conditions. If you are interested, you can also refer to Ling, S. and McAleer, M. (2002). "Necessary and Sufficient Moment Conditions for the GARCH(r,s) and Asymmetric Power GARCH(r,s) Models. Econometric Theory, v. 18, pp. 722-729.

Thank you very much Trubador.