Exponential General Autoregressive Conditional Heteroskedastic (EGARCH) Model

The exponential general autoregressive conditional heteroskedastic (EGARCH) is another form of the GARCH model. E-GARCH model was proposed by Nelson (1991) to overcome the weakness in GARCH handling of financial time series. In particular, to allow for asymmetric effects between positive and negative asset returns.

Formally, an E-GARCH(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 the 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 ARCH component model.
  • $q$ is the order of the GARCH component model.
  • $\beta_1,\beta_2,...,\beta_q$ are the parameters of 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
      $$P_{\nu} = t_{\nu}(0,1) $$
      $$\nu \gt 4 $$
    3. Generalized Error Distribution (GED)
      $$P_{\nu} = \mathit{GED}_{\nu}(0,1) $$ 
      $$\nu \gt 1 $$

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. E-GARCH(p,q) model with normal-distributed innovation has 2p+q+2 estimated parameters
  3. E-GARCH(p,q) model with GED or student's t-distributed innovation has 2p+q+3 estimated parameters

References

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