Documentation 
Create EGARCH time series model
egarch creates model objects for EGARCH models. The EGARCH model is a conditional variance model that models the logged conditional variance (as opposed to the conditional variance directly). The EGARCH(P,Q) model includes P lagged log conditional variances, Q lagged standardized innovations, and Q leverage terms.
Create model objects with known or unknown coefficients. Estimate unknown coefficients from data using estimate.
model = egarch creates a conditional variance EGARCH model of degrees zero.
model = egarch(P,Q) creates a conditional variance EGARCH model with GARCH degree P and ARCH degree Q.
model = egarch(Name,Value) creates an EGARCH model with additional options specified by one or more Name,Value pair arguments. Name can also be a property name and Value is the corresponding value. Name must appear inside single quotes (''). You can specify several namevalue pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Specify optional commaseparated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
'Constant' 
Scalar constant in the EGARCH model. Default: NaN 
'GARCH' 
Cell vector of coefficients associated with the lagged log conditional variances. When specified without GARCHLags, GARCH is a Pelement cell vector of coefficients at lags 1, 2,...,P. When specified with GARCHLags, GARCH is an equivalentlength cell vector of coefficients associated with the lags in GARCHLags. Default: Cell vector of NaNs with the same length as GARCHLags. 
'ARCH' 
Cell vector of coefficients associated with the magnitude of lagged standardized innovations. When specified without ARCHLags, ARCH is cell vector of coefficients at lags 1, 2,... to the number of ARCH coefficients in the model. When specified with ARCHLags, ARCH is an equivalentlength cell vector of coefficients associated with the lags in ARCHLags. Default: Cell vector of NaNs with the same length as ARCHLags. 
'Leverage' 
Cell vector of coefficients associated with lagged standardized innovations. When specified without LeverageLags, Leverage is a cell vector of coefficients at lags 1, 2,... to the number of leverage coefficients in the model. When specified with LeverageLags, Leverage is an equivalentlength cell vector of coefficients associated with the lags in LeverageLags. Default: Cell vector of NaNs with the same length as LeverageLags. 
'Offset' 
Scalar offset, or additive constant, associated with an innovation mean model. Default: 0 
'GARCHLags' 
Vector of positive integer lags associated with the GARCH coefficients. Default: Vector of integers 1, 2,...,P. 
'ARCHLags' 
Vector of positive integer lags associated with the ARCH coefficients. Default: Vector of integers 1, 2,... to the number of ARCH coefficients. 
'LeverageLags' 
Vector of positive integer lags associated with the leverage coefficients. Default: Vector of integers 1, 2,... to the number of leverage coefficients. 
'Distribution' 
Conditional probability distribution of the innovation process. Distribution is a string you specify as 'Gaussian' or 't'. Alternatively, specify it as a data structure with the field Name to store the distribution 'Gaussian' or 't'. If the distribution is 't', then the structure also needs the field DoF to store the degrees of freedom. Default: 'Gaussian' 
Notes:

estimate  Fit EGARCH conditional variance model to data 
filter  Filter disturbances with EGARCH model 
forecast  Forecast EGARCH process 
infer  Infer EGARCH model conditional variances 
Display parameter estimation results for EGARCH models  
simulate  Monte Carlo simulation of EGARCH models 
Consider a time series y_{t} with a constant mean offset,
$${y}_{t}=\mu +{\epsilon}_{t},$$
where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$ The EGARCH(P,Q) conditional variance process, $${\sigma}_{t}^{2}$$, is of the form
$$\mathrm{log}{\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}\mathrm{log}}{\sigma}_{ti}^{2}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}\left[\frac{\left{\epsilon}_{tj}\right}{{\sigma}_{tj}}E\left\{\frac{\left{\epsilon}_{tj}\right}{{\sigma}_{tj}}\right\}\right]+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}\left(\frac{{\epsilon}_{tj}}{{\sigma}_{tj}}\right),$$
where
$$E\left\{\frac{\left{\epsilon}_{tj}\right}{{\sigma}_{tj}}\right\}=E\left\{\left{z}_{tj}\right\right\}=\sqrt{\frac{2}{\pi}}$$
for a Gaussian innovation distribution, and
$$E\left\{\frac{\left{\epsilon}_{tj}\right}{{\sigma}_{tj}}\right\}=E\left\{\left{z}_{tj}\right\right\}=\sqrt{\frac{\nu 2}{\pi}}\frac{\Gamma \left(\frac{\nu 1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}$$
for a Student's t distribution with $$\nu \ge 2$$ degrees of freedom
The additive constant μ corresponds to the namevalue argument Offset.
The constant κ corresponds to the namevalue argument Constant.
The coefficients $${\gamma}_{i}$$ correspond to the namevalue argument GARCH.
The coefficients $${\alpha}_{j}$$ correspond to the namevalue argument ARCH.
The coefficients $${\xi}_{j}$$ correspond to the namevalue argument Leverage.
The distribution of z_{t} (the innovation distribution) corresponds to the namevalue argument Distribution, and can be Gaussian or Student's t.
egarch enforces stationarity by ensuring the roots of the Pdegree polynomial
$$(1{\gamma}_{1}L{\gamma}_{2}{L}^{2}\cdots {\gamma}_{P}{L}^{P})$$
lie outside the unit circle.
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.