# Log likelihood function, GARCH(1,1) with asymmetric term

I am modelling a GARCH(1,1) and a GARCH(1,1) with an asymmetric term.

$$h(t)=\omega+\alpha\varepsilon(t-1)^2+\beta\sigma(t-1)^2$$

and

$$h(t)=\omega+\alpha u(t-1)^2+\beta\sigma(t-1)^2 + \gamma (u(t-1)^2D)$$

$$D$$ takes a value of 1 if u is positive.

Will my log likelihood function change between these two models?

If so, how will it change?

Thanks!

• The Gaussian quasi log-likelihood function for the the GJR-GARCH(1,1) will be similar, but somewhat different because the variance parameter $h(t)$ is different, provided you have some assumptions on the terms $u$ and $\gamma$. – FunnyBuzer Jan 24 at 11:06
• Great, how different, could you provide an example of a text where it will show me the difference. – user22485 Jan 25 at 7:54
• Try having a look at these books: book1 or book2 – FunnyBuzer Jan 25 at 12:24
• are you an expert with these books, could you direct me to specific pages? – user22485 Jan 25 at 14:11
• I used them a long time ago. There are sections on the GJR-GARCH model, and you should have the log-likelihood derivation there – FunnyBuzer Jan 25 at 14:30