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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.

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  • $\begingroup$ I made some formatting corrections. Please see if I might have missed a parenthesis or anything. $\endgroup$ Jan 13, 2022 at 11:01
  • $\begingroup$ @Azertux0, where do you got these equations from? $\endgroup$
    – Count
    Jan 13, 2022 at 11:16
  • $\begingroup$ 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. $\endgroup$
    – Count
    Jan 13, 2022 at 11:49
  • $\begingroup$ 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. $\endgroup$
    – Azertux0
    Jan 13, 2022 at 12:14
  • $\begingroup$ 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) $\endgroup$
    – Azertux0
    Jan 13, 2022 at 13:28

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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.

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