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egarch

Create EGARCH conditional variance model object

Create an egarch model object to represent an exponential generalized autoregressive conditional heteroscedastic (EGARCH) model. The EGARCH(P,Q) conditional variance model includes P past log conditional variances composing the GARCH polynomial, and Q past standardized innovations composing the ARCH and leverage polynomials.

Use egarch to create a model with known or unknown coefficients, and then estimate any unknown coefficients from data using estimate. You can also simulate or forecast conditional variances from fully specified models using simulate or forecast, respectively.

For more information about egarch model objects, see egarch.

Syntax

Mdl = egarch
Mdl = egarch(P,Q)
Mdl = egarch(Name,Value)

Description

example

Mdl = egarch creates a zero-degree conditional variance EGARCH model object.

example

Mdl = egarch(P,Q) creates an EGARCH model with GARCH polynomial degree P, and ARCH and leverage polynomials having degree Q.

example

Mdl = egarch(Name,Value) creates an EGARCH model with additional options specified by one or more Name,Value pair arguments. For example, you can specify a conditional variance model constant, the number of ARCH polynomial lags, and the innovation distribution.

Examples

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Create a default egarch model object and specify its parameter values using dot notation.

Create an EGARCH(0,0) model.

Mdl = egarch
Mdl = 
    EGARCH(0,0) Conditional Variance Model:
    -----------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 0
               Q: 0
        Constant: NaN
           GARCH: {}
            ARCH: {}
        Leverage: {}

Mdl is an egarch model. It contains an unknown constant, its offset is 0, and the innovation distribution is 'Gaussian'. The model does not have GARCH, ARCH, or leverage polynomials.

Specify two unknown ARCH and leverage coefficients for lags one and two using dot notation.

Mdl.ARCH = {NaN NaN};
Mdl.Leverage = {NaN NaN};
Mdl
Mdl = 
    EGARCH(0,2) Conditional Variance Model:
    -----------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 0
               Q: 2
        Constant: NaN
           GARCH: {}
            ARCH: {NaN NaN} at Lags [1 2]
        Leverage: {NaN NaN} at Lags [1 2]

The Q, ARCH, and Leverage properties update to 2, {NaN NaN}, {NaN NaN}, respectively. The two ARCH and leverage coefficients are associated with lags 1 and 2.

Create an egarch model object using the shorthand notation egarch(P,Q), where P is the degree of the GARCH polynomial and Q is the degree of the ARCH and leverage polynomial.

Create an EGARCH(3,2) model.

Mdl = egarch(3,2)
Mdl = 
    EGARCH(3,2) Conditional Variance Model:
    -----------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 3
               Q: 2
        Constant: NaN
           GARCH: {NaN NaN NaN} at Lags [1 2 3]
            ARCH: {NaN NaN} at Lags [1 2]
        Leverage: {NaN NaN} at Lags [1 2]

Mdl is an egarch model object. All properties of Mdl, except P, Q, and Distribution, are NaN values. By default, the software:

  • Includes a conditional variance model constant

  • Excludes a conditional mean model offset (i.e., the offset is 0)

  • Includes all lag terms in the GARCH polynomial up to lag P

  • Includes all lag terms in the ARCH and leverage polynomials up to lag Q

Mdl specifies only the functional form of an EGARCH model. Because it contains unknown parameter values, you can pass Mdl and time-series data to estimate to estimate the parameters.

Create an egarch model object using name-value pair arguments.

Specify an EGARCH(1,1) model. By default, the conditional mean model offset is zero. Specify that the offset is NaN. Include a leverage term.

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

Mdl is an egarch model object. The software sets all parameters to NaN, except P, Q, and Distribution.

Since Mdl contains NaN values, Mdl is appropriate for estimation only. Pass Mdl and time-series data to estimate. For a continuation of this example, see Estimate EGARCH Model.

Create an EGARCH(1,1) model with mean offset,

where

and is an independent and identically distributed standard Gaussian process.

Mdl = egarch('Constant',0.0001,'GARCH',0.75,...
    'ARCH',0.1,'Offset',0.5,'Leverage',{-0.3 0 0.01})
Mdl = 
    EGARCH(1,3) Conditional Variance Model with Offset:
    -----------------------------------------------------  
    Distribution: Name = 'Gaussian'
               P: 1
               Q: 3
        Constant: 0.0001
           GARCH: {0.75} at Lags [1]
            ARCH: {0.1} at Lags [1]
        Leverage: {-0.3 0.01} at Lags [1 3]
          Offset: 0.5

egarch assigns default values to any properties you do not specify with name-value pair arguments. An alternative way to specify the leverage component is 'Leverage',{-0.3 0.01},'LeverageLags',[1 3].

Input Arguments

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Number of past consecutive, logged conditional variance terms to include in the GARCH polynomial, specified as a nonnegative integer. That is, P is the degree of the GARCH polynomial, where the polynomial includes each lag term from t – 1 to tP.

You can specify P using the egarch(P,Q) shorthand syntax only. You cannot specify P in conjunction with Name,Value pair arguments.

If P > 0, then you must specify Q as a positive integer.

Example: egarch(3,2)

Data Types: double

Number of past consecutive standardized innovation terms to include in the ARCH and leverage polynomials, specified as a nonnegative integer. That is, Q is the degree of the ARCH and leverage polynomials, where each polynomial includes each lag term from t – 1 to tQ. Also, Q specifies the minimum number of presample innovations the software requires to initiate the model.

You can specify this property when using the egarch(P,Q) shorthand syntax only. You cannot specify Q in conjunction with Name,Value pair arguments.

If P > 0, then you must specify Q as a positive integer.

Example: egarch(3,2)

Data Types: double

Name-Value Pair Arguments

Specify optional comma-separated 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.

Example: 'Constant',0.5,'ARCHLags',2,'Distribution',struct('Name','t','DoF',5) specifies a conditional variance model constant of 0.5, two standardized innovation terms at lags 1 and 2 of the ARCH polynomial (but no leverage terms), and a t distribution with 5 degrees of freedom for the innovations.

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Conditional variance model constant, specified as the comma-separated pair consisting of 'Constant' and a scalar.

Example: 'Constant',-0.5

Data Types: double

Coefficients corresponding to the past logged conditional variance terms that compose the GARCH polynomial, specified as the comma-separated pair consisting of 'GARCH' and a cell vector of scalars.

If you specify GARCHLags, then GARCH is an equivalent-length cell vector of coefficients associated with the lags in GARCHLags. Otherwise, GARCH is a P-element cell vector of coefficients corresponding to lags 1, 2,..., P.

The coefficients must compose a stationary GARCH polynomial. For details, see EGARCH Model.

By default, GARCH is a cell vector of NaNs of length P (the degree of the GARCH polynomial) or numel(GARCHLags).

Example: 'GARCH',{0.1 0 0 0.02}

Data Types: cell

Coefficients corresponding to the magnitude of the past standardized innovation terms that compose the ARCH polynomial, specified as the comma-separated pair consisting of 'ARCH' and a cell vector of scalars.

If you do not specify ARCHLags, then ARCH is a cell vector of coefficients corresponding to lags 1 through the number of elements in ARCH.

If you specify ARCHLags, then ARCH is an equivalent-length cell vector of coefficients associated with the lags in ARCHLags.

By default, ARCH is a cell vector of NaNs with the same length as the ARCH polynomial degree or numel(ARCHLags).

Example: 'ARCH',{0.5 0 0.2}

Data Types: cell

Coefficients corresponding to the past standardized innovation terms that compose the leverage polynomial, specified as the comma-separated pair consisting of 'Leverage' and a cell vector of scalars.

If you specify LeverageLags, then Leverage is an equivalent-length cell vector of coefficients associated with the lags in LeverageLags. Otherwise, Leverage is a cell vector of coefficients corresponding to lags 1 through the number of elements in Leverage.

By default, Leverage is a cell vector of NaNs with the same length as the leverage polynomial degree or numel(LeverageLags).

Example: 'Leverage',{-0.1 0 0 0.03}

Innovation mean model offset or additive constant, specified as the comma-separated pair consisting of 'Offset' and a scalar.

Example: 'Offset',0.1

Data Types: double

Lags associated with the GARCH polynomial coefficients, specified as the comma-separated pair consisting of 'GARCHLags' and a vector of positive integers. The maximum value of GARCHLags determines P, the GARCH polynomial degree.

If you specify GARCH, then GARCHLags is an equivalent-length vector of positive integers specifying the lags of the corresponding coefficients in GARCH. Otherwise, GARCHLags indicates the lags of unknown coefficients in the GARCH polynomial.

By default, GARCHLags is a vector containing the integers 1 through P.

Example: 'GARCHLags',[1 2 4 3]

Data Types: double

Lags associated with the ARCH polynomial coefficients, specified as the comma-separated pair consisting of 'ARCHLags' and a vector of positive integers. The maximum value of ARCHLags determines the ARCH polynomial degree.

If you specify ARCH, then ARCHLags is an equivalent-length vector of positive integers specifying the lags of the corresponding coefficients in ARCH. Otherwise, ARCHLags indicates the lags of unknown coefficients in the ARCH polynomial.

By default, ARCHLags is a vector containing the integers 1 through the ARCH polynomial degree.

Example: 'ARCHLags',[3 1 2]

Data Types: double

Lags associated with the leverage polynomial coefficients, specified as the comma-separated pair consisting of 'LeverageLags' and a vector of positive integers. The maximum value of LeverageLags determines the leverage polynomial degree.

If you specify Leverage, then LeverageLags is an equivalent-length vector of positive integers specifying the lags of the corresponding coefficients in LeverageLags. Otherwise, LeverageLags indicates the lags of unknown coefficients in the leverage polynomial.

By default, LeverageLags is a vector containing the integers 1 through the leverage polynomial degree.

Example: 'LeverageLags',1:4

Data Types: double

Conditional probability distribution of the innovation process, specified as the comma-separated pair consisting of 'Distribution' and a value in this table.

DistributionValueStructure Array
Gaussian'Gaussian'struct('Name','Gaussian')
Student’s t
't'
By default, DoF is NaN.
struct('Name','t','DoF',DoF)
DoF > 2 or DoF = NaN

Example: 'Distribution',struct('Name','t','DoF',10)

Data Types: char | struct

Notes:

  • All GARCH, ARCH and Leverage coefficients are subject to a near-zero tolerance exclusion test. That is, the software:

    1. Creates lag operator polynomials for each of the GARCH, ARCH and Leverage components.

    2. Compares each coefficient to the default lag operator zero tolerance, 1e-12.

    3. Includes a coefficient in the model if its magnitude is greater than 1e-12, and excludes the coefficient otherwise. In other words, the software considers excluded coefficients to be sufficiently close to zero.

    For details, see LagOp.

  • The lengths of ARCH and Leverage might differ. The difference can occur because the software defines the property Q as the largest lag associated with nonzero ARCH and Leverage coefficients, or max(ARCHLags,LeverageLags). Typically, the number and corresponding lags of nonzero ARCH and Leverage coefficients are equivalent, but this is not a requirement.

Output Arguments

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EGARCH model, returned as an egarch model object.

For the property descriptions of Mdl, see Conditional Variance Model Properties.

If Mdl contains unknown parameters (indicated by NaNs), then you can specify them using dot notation. Alternatively, you can pass Mdl and time series data to estimate to obtain estimates.

If Mdl is fully specified, then you can simulate or forecast conditional variances using simulate or forecast, respectively.

More About

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EGARCH Model

An EGARCH model is an innovations process that addresses conditional heteroscedasticity. Specifically, the model posits that the current conditional variance is the sum of these linear processes:

  • Past logged conditional variances (the GARCH component or polynomial)

  • Magnitudes of past standardized innovations (the ARCH component or polynomial)

  • Past standardized innovations (the leverage component or polynomial)

Consider the time series

yt=μ+εt,

where εt=σtzt. The EGARCH(P,Q) conditional variance process, σt2, has the form

logσt2=κ+i=1Pγilogσti2+j=1Qαj[|εtj|σtjE{|εtj|σtj}]+j=1Qξj(εtjσtj).

The table shows how the variables correspond to the properties of the garch model object.

VariableDescriptionProperty
μInnovation mean model constant offset'Offset'
κ > 0Conditional variance model constant'Constant'
γjGARCH component coefficients'GARCH'
αjARCH component coefficients'ARCH'
ξjLeverage component coefficients'Leverage'
ztSeries of independent random variables with mean 0 and variance 1'Distribution'

If zt is Gaussian, then

E{|εtj|σtj}=E{|ztj|}=2π.

If zt is t distributed with ν > 2 degrees of freedom, then

E{|εtj|σtj}=E{|ztj|}=ν2πΓ(ν12)Γ(ν2).

To ensure a stationary EGARCH model, all roots of the GARCH lag operator polynomial, (1γ1LγPLP), must lie outside of 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(σt2)exp{E(logσt2)}.

EGARCH models are appropriate when positive and negative shocks of equal magnitude do not contribute equally to volatility [1].

Tips

  • An EGARCH(1,1) specification is complex enough for most applications. Typically in these models, the GARCH and ARCH coefficients are positive, and the leverage coefficients are negative. If you get these signs, then large unanticipated downward shocks increase the variance. If you get signs opposite to those expected, you might encounter difficulties inferring volatility sequences and forecasting. A negative ARCH coefficient is particularly problematic. In this case, an EGARCH model might not be the best choice for your application.

References

[1] Tsay, R. S. Analysis of Financial Time Series. 3rd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2010.

Introduced in R2012a

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