EGARCH Model Specifications

Default EGARCH Model

This example shows how to use the shorthand egarch(P,Q) syntax to specify the default EGARCH(P, Q) model, $\varepsilon_t = \sigma_t z_t$ with a Gaussian innovation distribution and

$$\log \sigma _t^2 = \kappa  + \sum\limits_{i = 1}^P {{\gamma _i}\log } \sigma _{t - i}^2 + \sum\limits_{j = 1}^Q {{\alpha _j}\left[ {\frac{{\left| {{\varepsilon _{t - j}}} \right|}}{{{\sigma _{t - j}}}} - E\left\{ {\frac{{\left| {{\varepsilon _{t - j}}} \right|}}{{{\sigma _{t - j}}}}} \right\}} \right]}  + \sum\limits_{j = 1}^Q {{\xi _j}} \left( {\frac{{{\varepsilon _{t - j}}}}{{{\sigma _{t - j}}}}} \right).$$

By default, all parameters in the created model have unknown values.

Specify the default EGARCH(1,1) model:

Mdl = egarch(1,1)
Mdl = 

    EGARCH(1,1) Conditional Variance Model:
    -----------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 1
               Q: 1
        Constant: NaN
           GARCH: {NaN} at Lags [1]
            ARCH: {NaN} at Lags [1]
        Leverage: {NaN} at Lags [1]

The output shows that the created model, Mdl, has NaN values for all model parameters: the constant term, the GARCH coefficient, the ARCH coefficient, and the leverage coefficient. You can modify the created model using dot notation, or input it (along with data) to estimate.

EGARCH Model with a Mean Offset

This example shows how to specify an EGARCH(P, Q) model with a mean offset. Use name-value pair arguments to specify a model that differs from the default model.

Specify an EGARCH(1,1) model with a mean offset,

$$y_t = \mu + \varepsilon_t,$$

where $\varepsilon_t = \sigma_t z_t$ and

$$\log \sigma _t^2 = \kappa  + {\gamma _1}\log \sigma _{t - 1}^2 + {\alpha _1}\left[ {\frac{{\left| {{\varepsilon _{t - 1}}} \right|}}{{{\sigma _{t - 1}}}} - E\left\{ {\frac{{\left| {{\varepsilon _{t - 1}}} \right|}}{{{\sigma _{t - 1}}}}} \right\}} \right] + {\xi _1}\left( {\frac{{{\varepsilon _{t - 1}}}}{{{\sigma _{t - 1}}}}} \right).$$

Mdl = egarch('Offset',NaN,'GARCHLags',1,'ARCHLags',1,...
    'LeverageLags',1)
Mdl = 

    EGARCH(1,1) Conditional Variance Model with Offset:
    -----------------------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 1
               Q: 1
        Constant: NaN
           GARCH: {NaN} at Lags [1]
            ARCH: {NaN} at Lags [1]
        Leverage: {NaN} at Lags [1]
          Offset: NaN

The mean offset appears in the output as an additional parameter to be estimated or otherwise specified.

EGARCH Model with Nonconsecutive Lags

This example shows how to specify an EGARCH model with nonzero coefficients at nonconsecutive lags.

Specify an EGARCH(3,1) model with nonzero GARCH terms at lags 1 and 3. Include a mean offset.

Mdl = egarch('Offset',NaN,'GARCHLags',[1,3],'ARCHLags',1,...
    'LeverageLags',1)
Mdl = 

    EGARCH(3,1) Conditional Variance Model with Offset:
    -----------------------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 3
               Q: 1
        Constant: NaN
           GARCH: {NaN NaN} at Lags [1 3]
            ARCH: {NaN} at Lags [1]
        Leverage: {NaN} at Lags [1]
          Offset: NaN

The unknown nonzero GARCH coefficients correspond to lagged log variances at lags 1 and 3. The output shows only the nonzero coefficients.

Display the value of GARCH:

Mdl.GARCH
ans = 

    [NaN]    [0]    [NaN]

The GARCH cell array returns three elements. The first and third elements have value NaN, indicating these coefficients are nonzero and need to be estimated or otherwise specified. By default, egarch sets the interim coefficient at lag 2 equal to zero to maintain consistency with MATLAB® cell array indexing.

EGARCH Model with Known Parameter Values

This example shows how to specify an EGARCH model with known parameter values. You can use such a fully specified model as an input to simulate or forecast.

Specify the EGARCH(1,1) model

$$\log \sigma _t^2 = 0.1 + 0.6\log \sigma _{t - 1}^2 + 0.2\left[ {\frac{{\left| {{\varepsilon _{t - 1}}} \right|}}{{{\sigma _{t - 1}}}} - E\left\{ {\frac{{\left| {{\varepsilon _{t - 1}}} \right|}}{{{\sigma _{t - 1}}}}} \right\}} \right] - 0.1\left( {\frac{{{\varepsilon _{t - 1}}}}{{{\sigma _{t - 1}}}}} \right)$$

with a Gaussian innovation distribution.

Mdl = egarch('Constant',0.1,'GARCH',0.6,'ARCH',0.2,...
    'Leverage',-0.1)
Mdl = 

    EGARCH(1,1) Conditional Variance Model:
    -----------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 1
               Q: 1
        Constant: 0.1
           GARCH: {0.6} at Lags [1]
            ARCH: {0.2} at Lags [1]
        Leverage: {-0.1} at Lags [1]

Because all parameter values are specified, the created model has no NaN values. The functions simulate and forecast don't accept input models with NaN values.

EGARCH Model with a t Innovation Distribution

This example shows how to specify an EGARCH model with a Student's t innovation distribution.

Specify an EGARCH(1,1) model with a mean offset,

$$y_t = \mu + \varepsilon_t,$$

where $\varepsilon_t = \sigma_t z_t$ and

$$\log \sigma _t^2 = \kappa  + {\gamma _1}\log \sigma _{t - 1}^2 + {\alpha _1}\left[ {\frac{{\left| {{\varepsilon _{t - 1}}} \right|}}{{{\sigma _{t - 1}}}} - E\left\{ {\frac{{\left| {{\varepsilon _{t - 1}}} \right|}}{{{\sigma _{t - 1}}}}} \right\}} \right] + {\xi _1}\left( {\frac{{{\varepsilon _{t - 1}}}}{{{\sigma _{t - 1}}}}} \right).$$

Assume $z_{t}$ follows a Student's t innovation distribution with 10 degrees of freedom.

tDist = struct('Name','t','DoF',10);
Mdl = egarch('Offset',NaN,'GARCHLags',1,'ARCHLags',1,...
    'LeverageLags',1,'Distribution',tDist)
Mdl = 

    EGARCH(1,1) Conditional Variance Model with Offset:
    -----------------------------------------------------  
    Distribution: Name = 't', DoF = 10
               P: 1
               Q: 1
        Constant: NaN
           GARCH: {NaN} at Lags [1]
            ARCH: {NaN} at Lags [1]
        Leverage: {NaN} at Lags [1]
          Offset: NaN

The value of Distribution is a struct array with field Name equal to 't' and field DoF equal to 10. When you specify the degrees of freedom, they aren't estimated if you input the model to estimate.

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