\[x_t = \mu + a_t\] \[\ln\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i \left(\left|\epsilon_{t-i}\right|+\gamma_i\epsilon_{t-i}\right )}+\sum_{j=1}^q{\beta_j \ln\sigma_{t-j}^2}\] \[a_t = \sigma_t \times \epsilon_t\] \[\epsilon_t \sim P_{\nu}(0,1)\] Where:
- \(x_t\) is the time series value at time t.
- \(\mu\) is the mean of GARCH model.
- \(a_t\) is the model's residual at time t.
- \(\sigma_t\) is the conditional standard deviation (i.e. volatility) at time t.
- \(p\) is the order of the ARCH component model.
- \(\alpha_o,\alpha_1,\alpha_2,...,\alpha_p\) are the parameters of the the ARCH component model.
- \(q\) is the order of the GARCH component model.
- \(\beta_1,\beta_2,...,\beta_q\) are the parameters of the the GARCH component model.
- \(\left[\epsilon_t\right]\) are the standardized residuals: \[\left[\epsilon_t \right] \sim i.i.d\] \[E\left[\epsilon_t\right]=0\] \[\mathit{VAR}\left[\epsilon_t\right]=1\]
-
\(P_{\nu}\) is the probability distribution function for \(\epsilon_t\). Currently, the following distributions are supported:
- Normal distribution \[P_{\nu} = N(0,1)\]
- Student's t-distribution \[P_{\nu} = t_{\nu}(0,1)\] \[\nu \succ 4\]
- Generalized error distribution (GED) \[P_{\nu} = \mathit{GED}_{\nu}(0,1)\] \[\nu \succ 1\]
- Remarks
-
- The E-GARCH model differs from GARCH in several ways. For instance, it used the logged conditional variances to relax the positiveness constraint of model coefficients
- EGARCH(p,q) model has 2p+q+2 estimated parameters
