Likelihood Specification for an Asymmetric GARCH Model

Summary

The document compares a standard GARCH(1,1) conditional variance specification with a version that adds an asymmetric term. In the second model, a dummy variable activates an extra squared-residual contribution when the previous innovation is positive. The author asks whether this change requires a different log-likelihood function and, if so, how it should be written.

The post supplies model equations but no answer, estimation results, or distributional assumption for the innovations. Consequently, it identifies the key modeling issue without establishing a complete likelihood: the conditional variance recursion changes, while the likelihood’s form also depends on the assumed conditional distribution and mean specification. The asymmetry indicator’s sign convention matters as well, since the term as written activates for positive rather than negative innovations. Readers can learn what must be specified, but the document alone does not provide a derivation or empirical evidence about whether the asymmetric model fits better.

Key ideas

• The asymmetric specification adds a residual-dependent term to the conditional variance recursion.
• The indicator activates the added term when the lagged innovation is positive.
• A likelihood requires a conditional distribution assumption in addition to the variance recursion.
• The post asks how the likelihood changes but provides no derivation or estimation evidence.

Tags

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


# 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!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.