E-GARCH Modeling

Model conditional variance with exponential general autoregressive conditional heteroskedastic (E-GARCH) model (Nelson 1991).

E-GARCH with NumXL

The exponential general autoregressive conditional heteroskedastic (E-GARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):

$$x_t = \mu + a_t$$ $$\ln\sigma_t^2 = \alpha_o + \sum_{i=1}^p {\alpha_i \left(\left|\epsilon_{t-i}\right|+\gamma_i\epsilon_{t-i}\right )}+\sum_{j=1}^q{\beta_j \ln\sigma_{t-j}^2}$$ $$a_t = \sigma_t \times \epsilon_t$$ $$\epsilon_t \sim P_{\nu}(0,1)$$ 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

  • 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

E-GARCH Modeling Examples

Tutorial Videos

E-GARCH Modeling Function

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