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garch

Create GARCH conditional variance model object

Create a garch model object to represent a generalized autoregressive conditional heteroscedastic (GARCH) model. The GARCH(P,Q) conditional variance model includes P past conditional variances composing the GARCH polynomial, and Q past squared innovations composing the ARCH polynomial.

Use garch 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 garch model objects, see garch.

Syntax

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

Description

example

Mdl = garch creates a zero-degree conditional variance GARCH model object.

example

Mdl = garch(P,Q) creates a GARCH model with GARCH polynomial degree P and ARCH polynomial degree Q.

example

Mdl = garch(Name,Value) creates a GARCH 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 garch model object and specify its parameter values using dot notation.

Create a GARCH(0,0) model.

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

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

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

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

The Q and ARCH properties are updated to 2 and {NaN NaN}. The two ARCH coefficients are associated with lags 1 and 2.

Create a garch model using the shorthand notation garch(P,Q), where P is the degree of the GARCH polynomial and Q is the degree of the ARCH polynomial.

Create a GARCH(3,2) model.

Mdl = garch(3,2)
Mdl = 
    GARCH(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]

Mdl is a garch 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 ARCH and GARCH lag-operator polynomials up to lags Q and P, respectively

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

Create a garch model using name-value pair arguments.

Specify a GARCH(1,1) model. By default, the conditional mean model offset is zero. Specify that the offset is NaN.

Mdl = garch('GARCHLags',1,'ARCHLags',1,'Offset',NaN)
Mdl = 
    GARCH(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]
          Offset: NaN

Mdl is a garch model object. The software sets all parameters (the properties of the model object) to NaN, except P, Q, and Distribution.

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

Create a GARCH(1,1) model with mean offset,

where

and is an independent and identically distributed standard Gaussian process.

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

garch assigns default values to any properties you do not specify with name-value pair arguments.

Input Arguments

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Number of past consecutive 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. P also specifies the minimum number of presample conditional variances the software requires to initiate the model.

You can specify P using the garch(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: garch(3,2)

Data Types: double

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

You can specify this property using the garch(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: garch(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 squared innovation terms at lags 1 and 2 of the ARCH polynomial, 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 positive scalar.

Example: 'Constant',0.5

Data Types: double

Coefficients corresponding to the past conditional variance terms that compose the GARCH polynomial, specified as the comma-separated pair consisting of 'GARCH' and a cell vector of nonnegative 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 model. For details, see GARCH 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 past squared innovation terms that compose the ARCH polynomial, specified as the comma-separated pair consisting of 'ARCH' and a cell vector of nonnegative scalars.

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

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

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

Example: 'ARCH',{0.5 0 0.2}

Data Types: cell

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 Q, 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 Q.

Example: 'ARCHLags',[3 1 2]

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

Note

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

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

Output Arguments

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GARCH model, returned as a garch 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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GARCH Model

A GARCH 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, with coefficients for each term:

  • Past conditional variances (the GARCH component or polynomial)

  • Past squared innovations (the ARCH component or polynomial)

Consider the time series

yt=μ+εt,

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

σt2=κ+γ1σt12++γPσtP2+α1εt12++αQεtQ2.

In lag operator notation, the model is

(1γ1LγPLP)σt2=κ+(α1L++αQLQ)εt2.

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'
γi0GARCH component coefficients'GARCH'
αj0ARCH component coefficients'ARCH'
ztSeries of independent random variables with mean 0 and variance 1'Distribution'

For stationarity and positivity, GARCH models use these constraints:

  • κ>0

  • γi0,αj0

  • i=1Pγi+j=1Qαj<1

Engle’s original ARCH(Q) model is equivalent to a GARCH(0,Q) specification.

GARCH models are appropriate when positive and negative shocks of equal magnitude contribute equally to volatility [1].

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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