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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

where and

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

If the innovation distribution is Student's

*t*with*ν*> 2 degrees of freedom, then

The toolbox treats the EGARCH(*P*,*Q*)
model as an ARMA model for
Thus,
to ensure stationarity, all roots of the GARCH coefficient polynomial,
, 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,

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.

- Specify EGARCH Models Using egarch
- EGARCH Model Specifications
- Assess the EGARCH Forecast Bias Using Simulations

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