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I'm reading this paper by Abosedra et. al (2006), where they study the volatility of US natural gas prices.

They report the estimation from a AR(1)-GARCH(1,1) model and a AR(1)-GJR-GARCH(1,1) model as in the image below.

Why do they also report $ \alpha+ \beta + \frac12 \lambda $? And how does this turn out to be exactly the same as $ \alpha + \beta $ in the AR(1)-GARCH(1,1) model?

Note: $\alpha, \beta, \gamma$ here are the ARCH, GARCH and Asymmetry (GJR model) parameters respectively. The equation(s) estimated is given below the table, if necessary.

enter image description here

Estimated equation(s):

           enter image description here

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I am not sure if half the asymmetry parameter has any useful interpretation on its own, but it does in the context of $\alpha+\beta+\frac{1}{2}\lambda$. Judging from the table you have included, the authors probably want to measure persistence in volatility. As is clarified in the thread "Persistence in GJR-GARCH (asymmetric GARCH)", the persistence of GJR-GARCH(1,1) is $$\alpha+\beta+\frac{1}{2}\lambda$$ if the distribution of standardized innovations is symmetric. The persistence of vanilla GARCH(1,1) is $$\alpha+\beta.$$ You have both of these in the table.

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  • $\begingroup$ Is there any significance to the fact that the persistence measures for both the models are exactly the same? $\endgroup$
    – WorldGov
    Commented Mar 17, 2021 at 13:11
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    $\begingroup$ @WorldGov, this is an interesting finding. For those who care directly about persistence, this shows its measurement is robust to model choice between vanilla GARCH and GJR-GARCH. This holds for this specific time series but not necessarily in general. $\endgroup$
    – Richard Hardy
    Commented Mar 17, 2021 at 13:55

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