Time-Series Modeling and Decomposition

If an ARMA model is assumed for the error variance, the model is a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model introduced by Bollerslev in 1996. In that case, the GARCH ($p$, $q$) model (where $p$ is the order of the GARCH terms $\sigma_{t}^{2}$ and $q$ is the order of the ARCH terms $\varepsilon_{t}$ is given by

$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1} \varepsilon_{t}^{2}+\ldots+\alpha_{q} \varepsilon_{t-q}^{2}+\beta_{1} \sigma_{t-1}^{2}+\ldots+\beta_{p} \sigma_{t-p}^{2}=\alpha_{0}+\sum_{i=1}^{q} \alpha_{i} \varepsilon_{t-i}^{2}+\sum_{i=1}^{p} \beta_{i} \sigma_{t-i}^{2}$                      (28)

The Nonlinear GARCH (NGARCH) also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH) was introduced by Engle and Ng in 1993

$\sigma_{t}^{2}=\omega+\alpha\left(\varepsilon_{t-1}-\theta \sigma_{t-1}\right)^{2}+\beta \sigma_{t-1}^{2}$.                                                                                                                         (29)

$\alpha, \beta \geq 0 ; \omega>0$. For stock returns, parameter $\theta$ is usually estimated to be positive; in this case, it reflects the leverage effect, signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.