# In a GJR model, is there any interpretation attributed to half of the asymmetry parameter?

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.

Estimated equation(s):

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.