# EGARCH Model

$x_t = \mu + a_t$ $\ln\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i \left(\left|\epsilon_{t-i}\right|+\gamma_i\epsilon_{t-i}\right )}+\sum_{j=1}^q{\beta_j \ln\sigma_{t-j}^2}$ $a_t = \sigma_t \times \epsilon_t$ $\epsilon_t \sim P_{\nu}(0,1)$ 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:
1. Normal distribution $P_{\nu} = N(0,1)$
2. Student's t-distribution $P_{\nu} = t_{\nu}(0,1)$ $\nu \succ 4$
3. Generalized error distribution (GED) $P_{\nu} = \mathit{GED}_{\nu}(0,1)$ $\nu \succ 1$
Remarks
1. 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
2. EGARCH(p,q) model has 2p+q+2 estimated parameters