The exponential GARCH (EGARCH) model is a GARCH variant that models the logarithm of the conditional variance process. In addition to modeling the logarithm, the EGARCH model has additional leverage terms to capture asymmetry in volatility clustering.
The EGARCH(P,Q) model has P GARCH coefficients associated with lagged log variance terms, Q ARCH coefficients associated with the magnitude of lagged standardized innovations, and Q leverage coefficients associated with signed, lagged standardized innovations. The form of the EGARCH(P,Q) model in Econometrics Toolbox™ is
$${y}_{t}=\mu +{\epsilon}_{t},$$
where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$ and
$$\mathrm{log}{\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}\mathrm{log}}{\sigma}_{t-i}^{2}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}\left[\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}-E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}\right]}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}\left(\frac{{\epsilon}_{t-j}}{{\sigma}_{t-j}}\right).$$
Note:
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The form of the expected value terms associated with ARCH coefficients in the EGARCH equation depends on the distribution of z_{t}:
If the innovation distribution is Gaussian, then
$$E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}=E\left\{\left|{z}_{t-j}\right|\right\}=\sqrt{\frac{2}{\pi}}.$$
If the innovation distribution is Student's t with ν > 2 degrees of freedom, then
$$E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}=E\left\{\left|{z}_{t-j}\right|\right\}=\sqrt{\frac{\nu -2}{\pi}}\frac{\Gamma \left(\frac{\nu -1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}.$$
The toolbox treats the EGARCH(P,Q) model as an ARMA model for$$\mathrm{log}{\sigma}_{t}^{2}.$$ Thus, to ensure stationarity, all roots of the GARCH coefficient polynomial,$$(1-{\gamma}_{1}L-\dots -{\gamma}_{P}{L}^{P})$$, must lie outside the unit circle.
The EGARCH model is unique from the GARCH and GJR models because it models the logarithm of the variance. By modeling the logarithm, positivity constraints on the model parameters are relaxed. However, forecasts of conditional variances from an EGARCH model are biased, because by Jensen's inequality,
$$E({\sigma}_{t}^{2})\ge \mathrm{exp}\{E(\mathrm{log}{\sigma}_{t}^{2})\}.$$
An EGARCH(1,1) specification will be complex enough for most applications. For an EGARCH(1,1) model, the GARCH and ARCH coefficients are expected to be positive, and the leverage coefficient is expected to be negative; large unanticipated downward shocks should increase the variance. If you get signs opposite to those expected, you might encounter difficulties inferring volatility sequences and forecasting (a negative ARCH coefficient can be particularly problematic). In this case, an EGARCH model might not be the best choice for your application.