# Long-Run Variance LRV for TGARCH and GJR-GARCH

As LRV calculation from GARCH parameters is on annual basis:

$$LRV = \frac{\omega}{1 - \alpha - \beta} \cdot 252$$

I wonder if it's not a composition of unconditional variance divided by the model persistence?

As the spec of TGARCH is based simply on $$\sigma$$ and not on $$\sigma^2$$, what could be the LRV knowing that its persistence?

My potential answer: $$LRV = \bigg(\frac{\omega}{1- \beta - \alpha/\sqrt{2 \pi} - \theta/\sqrt{2 \pi}}\bigg)^2 \cdot 252$$

And for GJR-GARCH what would it be?

My potential answer: $$LRV = \frac{\omega}{1 - ( \alpha + \theta/2 + \beta)} \cdot 252$$

Thank you for your help.

• I made some formatting corrections. Please see if I might have missed a parenthesis or anything. Jan 13, 2022 at 11:01
• @Azertux0, where do you got these equations from? Jan 13, 2022 at 11:16
• In my opinion, it is strange to annualize volatility using the square root of time rule when dealing with a model that explicitly assumes that returns are not iid. Just from a theoretical point of view, this seems to be a bit odd but I am pretty sure that this is general practice. Jan 13, 2022 at 11:49
• RichardHardy, thank you @Lars, the GARCH LRV equation comes from my former courses on the subject. Unfortunately, I don't have any detail of how it's obtained. The TGARCH and GJR LRV are just hypothesis. Jan 13, 2022 at 12:14
• The sqrt of time comes from my comprehension of the TGARCH model as follows: We do actually try to estimate the volatility through σ(t) and not variance σ^2(t) and that the parameters are to the power of 1 by opposition to GARCH : σ(t)= Ω + αϵ(t-1) * (1 if (ϵ(t-1) > 0) else 0) + Θϵ(t-1) * (1 if (ϵ(t-1) < 0) else 0) + βσ(t−1) Jan 13, 2022 at 13:28

## 1 Answer

In your case, the (annualized) long run variance can be calculated as $$LRV=E(\sigma_t^2) \cdot 252$$ or in terms of standard deviations as $$LRV=\sqrt{E(\sigma_t^2)}\sqrt{252}$$ assuming that there are $$252$$ trading days within a year and that you estimate your model on daily returns. Now, depending on your model, the expression for $$E(\sigma_t^2)$$ differs.

For instance, for the GARCH(1,1) you have: $$E(\sigma_t)^2=\frac{\omega}{1-\alpha-\beta}$$ For the GJR-GARCH(1,1) you get $$E(\sigma_t^2)=\frac{\omega}{1-\alpha-\frac{\theta}2{-\beta}}$$ if you assume a symmetric distribution for the error term.