# Persistence in TGARCH

Consider the standard GARCH model:

$$\sigma^2_t = \omega + \alpha\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}.$$

The so-called persistence parameter is defined as the sum $$\alpha+\beta$$.

And consider the GJR-GARCH model by Glosten et al. (1993):

$$\sigma^2_t = \omega + (\alpha+\gamma \mathbb{I}_{\varepsilon_{t-1} < 0})\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}$$

The persistence parameter equals $$\alpha+\gamma/2+\beta$$.

Now consider the TGARCH model by Zakoian (1994):

$$\sigma_t = \omega + (\alpha\mathbb{I}_{\varepsilon_{t-1} \geq 0}-\gamma \mathbb{I}_{\varepsilon_{t-1} < 0})\varepsilon_{t-1} + \beta\sigma_{t-1}.$$

Question: What is the persistence parameter in the TGARCH model?

The first important point is to understand where the "persistence parameter" comes from in the GJR-GARCH model. The specification depends on a normality assumption for the innovations $$\varepsilon_t$$:

$$E[\gamma \mathbb{I}_{\varepsilon < 0} \varepsilon^2] = \intop_{-\infty}^\infty \gamma \mathbb{I}_{\varepsilon < 0} \frac{\varepsilon^2 }{ \sqrt{2 \sigma^2 \pi}} \exp\left(-\frac{\varepsilon^2 }{ 2\sigma^2}\right) d\varepsilon = \intop_{-\infty}^0 \gamma \frac{\varepsilon^2 }{ \sqrt{2 \sigma^2 \pi}} \exp\left(-\frac{\varepsilon^2 }{ 2\sigma^2}\right) d\varepsilon = \frac{\gamma}{2}\sigma^2$$

The TGARCH model, as you present it, is specified in terms of $$\sigma_t$$ and not $$\sigma_t^2$$. We can perform similar calculations to obtain the persistance of $$\sigma_t$$

$$E[-\gamma \mathbb{I}_{\varepsilon < 0} \varepsilon] = \intop_{-\infty}^\infty -\gamma \mathbb{I}_{\varepsilon < 0} \frac{\varepsilon }{ \sqrt{2 \sigma^2 \pi}} \exp\left(-\frac{\varepsilon^2 }{ 2\sigma^2}\right) d\varepsilon = \intop_{-\infty}^0 -\gamma \frac{\varepsilon }{ \sqrt{2 \sigma^2 \pi}} \exp\left(-\frac{\varepsilon^2 }{ 2\sigma^2}\right) d\varepsilon = \frac{\gamma}{\sqrt{2 \pi}}\sigma$$

and

$$E[\alpha \mathbb{I}_{\varepsilon \geq 0} \varepsilon] = \intop_{-\infty}^\infty \alpha \mathbb{I}_{\varepsilon \geq 0} \frac{\varepsilon }{ \sqrt{2 \sigma^2 \pi}} \exp\left(-\frac{\varepsilon^2 }{ 2\sigma^2}\right) d\varepsilon = \intop_{0}^\infty \alpha \frac{\varepsilon }{ \sqrt{2 \sigma^2 \pi}} \exp\left(-\frac{\varepsilon^2 }{ 2\sigma^2}\right) d\varepsilon = \frac{\alpha}{\sqrt{2 \pi}}\sigma$$

Thus, we can say that the persistance of $$\sigma_t$$ is given by

$$\frac{\alpha}{\sqrt{2 \pi}} + \frac{\gamma}{\sqrt{2 \pi}} + \beta$$

• Is $\sigma$ the unconditional variance of $\varepsilon$? Jul 20, 2021 at 6:13
• Good answer ! But do we need the normal assumption for the GJR-GARCH ? In my opinion, if we assume that the density of $\epsilon_t$ is symmetric around zero, we should get the same result. The same should be valid for TGARCH as well.
– Lars
Jul 20, 2021 at 7:08
• Also what do we exactly mean if we talk about persistence in GARCH models ? I usually calculate the ACF of $\epsilon_t^2$ and then look at which rate the ACF converges to zero and call this the persistence of volatility. However, depending on the model this can be quite difficult. I know that your results for T-GARCH are correct, but how can we infer that the ACF of $\epsilon_t^2$ converges to zero at this rate without calculating the ACF ? Did I misunderstood something ?
– Lars
Jul 20, 2021 at 7:24