Persistence in GJR-GARCH (asymmetric GARCH)

Question

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 $\alpha+\beta$.

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

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

where $\mathbb{I}_{t-1}$ is the indicator function:

$\mathbb{I}_{t-1}(\varepsilon_{t-1})=\varepsilon_{t-1}$ for $\varepsilon_{t-1}>0$ and
$\mathbb{I}_{t-1}(\varepsilon_{t-1})=0$ otherwise.

Question: What is the persistence parameter in the GJR-GARCH model? Could someone provide some references where this is explained?

My guess is that the persistence parameter equals $\alpha+\gamma/2+\beta$, but I am not sure. The guess is based on the material in V-Lab and the similarities between the standard GARCH and the GJR-GARCH model.

References

• Glosten, L. R., R. Jagannathan, and D. E. Runkle, 1993. On The Relation between The Expected Value and The Volatility of Nominal Excess Return on stocks. Journal of Finance 48: 1779-1801.

• V-Lab

Answer 1

I found an answer in the "vignette" to the "rugarch" package in R. Here is a quote from pages 7-8 (emphasis is mine):

Because of the presence of the indicator function, the persistence of the model now crucially depends on the asymmetry of the conditional distribution used. The persistence of the model $\hat P$ is,

$$ \hat P = \sum_{j=1}^q \alpha_j + \sum_{j=1}^p \beta_j + \sum_{j=1}^q \gamma_j\kappa $$

where $\kappa$ is the expected value of the standardized residuals $z_t$ below zero (effectively the probability of being below zero),

$$ \kappa = \mathbb{E}(\mathbb{I}_{t-j} z_{t-j}^2) = \int_{-\infty}^0 f(z,1,0,\dotsc) dz $$

where $f$ is the standardized conditional density with any additional skew and shape parameters $(\dotsc)$. In the case of symmetric distributions the value of $\kappa$ is simply equal to 0.5.