EGARCH conditional variance time series model

Use `egarch`

to specify a univariate EGARCH (exponential
generalized autoregressive conditional heteroscedastic) model. The
`egarch`

function returns an `egarch`

object
specifying the functional form of an EGARCH(*P*,*Q*) model, and stores its
parameter values.

The key components of an `egarch`

model include the:

GARCH polynomial, which is composed of lagged, logged conditional variances. The degree is denoted by

*P*.ARCH polynomial, which is composed of the magnitudes of lagged standardized innovations.

Leverage polynomial, which is composed of lagged standardized innovations.

Maximum of the ARCH and leverage polynomial degrees, denoted by

*Q*.

*P* is the maximum nonzero lag in the GARCH polynomial, and
*Q* is the maximum nonzero lag in the ARCH and leverage
polynomials. Other model components include an innovation mean model offset, a
conditional variance model constant, and the innovations distribution.

All coefficients are unknown (`NaN`

values) and estimable unless you specify their values using name-value pair argument syntax. To estimate models containing all or partially unknown parameter values given data, use `estimate`

. For completely specified models (models in which all parameter values are known), simulate or forecast responses using `simulate`

or `forecast`

, respectively.

`Mdl = egarch`

`Mdl = egarch(P,Q)`

`Mdl = egarch(Name,Value)`

creates a zero-degree
conditional variance `Mdl`

= egarch`egarch`

object.

creates an EGARCH conditional variance model object (`Mdl`

= egarch(`P`

,`Q`

)`Mdl`

)
with a GARCH polynomial with a degree of `P`

, and ARCH and
leverage polynomials each with a degree of `Q`

. All polynomials
contain all consecutive lags from 1 through their degrees, and all coefficients
are `NaN`

values.

This shorthand syntax enables you to create a template in which you specify the polynomial degrees explicitly. The model template is suited for unrestricted parameter estimation, that is, estimation without any parameter equality constraints. However, after you create a model, you can alter property values using dot notation.

sets properties or additional options using
name-value pair arguments. Enclose each name in quotes. For example,
`Mdl`

= egarch(`Name,Value`

)`'ARCHLags',[1 4],'ARCH',{0.2 0.3}`

specifies the two ARCH
coefficients in `ARCH`

at lags `1`

and
`4`

.

This longhand syntax enables you to create more flexible models.

The shorthand syntax provides an easy way for you to create model templates that are suitable for unrestricted parameter estimation. For example, to create an EGARCH(1,2) model containing unknown parameter values, enter:

Mdl = egarch(1,2);

`P`

— GARCH polynomial degreenonnegative integer

GARCH polynomial degree, specified as a nonnegative integer. In the
GARCH polynomial and at time *t*, MATLAB^{®} includes all consecutive logged conditional variance terms
from lag *t* – 1 through lag *t* –
`P`

.

You can specify this argument using the
`egarch`

`(P,Q)`

shorthand syntax only.

If `P`

> 0, then you must specify `Q`

as a positive integer.

**Example: **`egarch(1,1)`

**Data Types: **`double`

`Q`

— ARCH polynomial degreenonnegative integer

ARCH polynomial degree, specified as a nonnegative integer. In the
ARCH polynomial and at time *t*, MATLAB includes all consecutive magnitudes of standardized
innovation terms (for the ARCH polynomial) and all standardized
innovation terms (for the leverage polynomial) from lag
*t* – 1 through lag *t* –
`Q`

.

You can specify this argument using the
`egarch`

`(P,Q)`

shorthand syntax only.

If `P`

> 0, then you must specify `Q`

as a positive integer.

**Example: **`egarch(1,1)`

**Data Types: **`double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

The longhand syntax
enables you to create models in which some or all coefficients are known. During estimation,
`estimate`

imposes equality constraints on any known parameters.

`'ARCHLags',[1 4],'ARCH',{NaN NaN}`

specifies an
EGARCH(0,4) model and unknown, but nonzero, ARCH coefficient matrices at lags
`1`

and `4`

.`'GARCHLags'`

— GARCH polynomial lags`1:P`

(default) | numeric vector of unique positive integersGARCH polynomial lags, specified as the comma-separated pair consisting of
`'GARCHLags'`

and a numeric vector of unique positive
integers.

`GARCHLags(`

is the lag corresponding to
the coefficient * j*)

`GARCH{``j`

}

. The lengths of
`GARCHLags`

and `GARCH`

must be equal.Assuming all GARCH coefficients (specified by the `GARCH`

property)
are positive or `NaN`

values, `max(GARCHLags)`

determines the value of the `P`

property.

**Example: **`'GARCHLags',[1 4]`

**Data Types: **`double`

`'ARCHLags'`

— ARCH polynomial lags `1:Q`

(default) | numeric vector of unique positive integersARCH polynomial lags, specified as the comma-separated pair consisting of
`'ARCHLags'`

and a numeric vector of unique positive
integers.

`ARCHLags(`

is the lag corresponding to
the coefficient * j*)

`ARCH{``j`

}

. The lengths of
`ARCHLags`

and `ARCH`

must be equal.Assuming all ARCH and leverage coefficients (specified by the
`ARCH`

and `Leverage`

properties) are positive
or `NaN`

values, `max([ARCHLags LeverageLags])`

determines the value of the `Q`

property.

**Example: **`'ARCHLags',[1 4]`

**Data Types: **`double`

`'LeverageLags'`

— Leverage polynomial lags`1:Q`

(default) | numeric vector of unique positive integersLeverage polynomial lags, specified as the comma-separated pair consisting of
`'LeverageLags'`

and a numeric vector of unique positive
integers.

`LeverageLags(`

is the lag corresponding
to the coefficient * j*)

`Leverage{``j`

}

. The
lengths of `LeverageLags`

and `Leverage`

must be
equal.Assuming all ARCH and leverage coefficients (specified by the
`ARCH`

and `Leverage`

properties) are positive
or `NaN`

values, `max([ARCHLags LeverageLags])`

determines the value of the `Q`

property.

**Example: **`'LeverageLags',1:4`

**Data Types: **`double`

`'Distribution'`

— Conditional probability distribution of innovation process`'Gaussian'`

(default) | `'t'`

| structure arrayConditional probability distribution of the innovation process, specified as the
comma-separated pair consisting of `'Distribution'`

and a string or
structure array.

Distribution | String | Structure Array |
---|---|---|

Gaussian | `"Gaussian"` | `struct('Name','Gaussian')` |

Student’s t | `"t"` | `struct('Name','t','DoF',DoF)` |

The `'DoF'`

field specifies the *t* distribution
degrees of freedom parameter.

`DoF`

> 2 or`DoF`

=`NaN`

.`DoF`

is estimable. If you want`estimate`

to estimate this parameter along with all other unknown parameters, then its value must be`NaN`

.If you specify

`"t"`

for`Distribution`

, then`DoF`

is`NaN`

. You can change its value by using dot notation after you create the model. For example,`Mdl.Distribution.DoF = 3`

.If you supply a structure array for

`Distribution`

to specify the Student's*t*distribution, then you must specify both the`'Name'`

and`'DoF'`

fields.

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

**Data Types: **`char`

| `string`

| `struct`

You can set writable property values when you create the model
object by using name-value pair argument syntax, or after you create model object by using dot
notation. For example, to create an EGARCH(1,1) model with unknown coefficients, and
then specify a *t* innovation distribution with unknown degrees of
freedom, enter:

Mdl = egarch('GARCHLags',1,'ARCHLags',1); Mdl.Distribution = "t";

`P`

— GARCH polynomial degreenonnegative integer

This property is read-only.

GARCH polynomial degree, specified as a nonnegative integer. `P`

is
the maximum lag in the GARCH polynomial with a coefficient that is positive or
`NaN`

. Lags that are less than `P`

can have
coefficients equal to 0.

`P`

specifies the minimum number of presample conditional variances
required to initialize the model.

If you use name-value pair arguments to create the model, then MATLAB implements one of these alternatives (assuming the coefficient of the
largest lag is positive or `NaN`

):

If you specify

`GARCHLags`

, then`P`

is the largest specified lag.If you specify

`GARCH`

, then`P`

is the number of elements of the specified value. If you also specify`GARCHLags`

, then`egarch`

uses`GARCHLags`

to determine`P`

instead.Otherwise,

`P`

is`0`

.

**Data Types: **`double`

`Q`

— Maximum degree of ARCH and leverage polynomialsnonnegative integer

This property is read-only.

Maximum degree of ARCH and leverage polynomials, specified as a nonnegative integer.
`Q`

is the maximum lag in the ARCH and leverage polynomials in the
model. In either type of polynomial, lags that are less than `Q`

can
have coefficients equal to 0.

`Q`

specifies the minimum number of presample innovations required to
initiate the model.

If you use name-value pair arguments to create the model, then MATLAB implements one of these alternatives (assuming the coefficients of the
largest lags in the ARCH and leverage polynomials are positive or `NaN`

):

If you specify

`ARCHLags`

or`LeverageLags`

, then`Q`

is the maximum between the two specifications.If you specify

`ARCH`

or`Leverage`

, then`Q`

is the maximum number of elements between the two specifications. If you also specify`ARCHLags`

or`LeverageLags`

, then`egarch`

uses their values to determine`Q`

instead.Otherwise,

`Q`

is`0`

.

**Data Types: **`double`

`Constant`

— Conditional variance model constant`NaN`

(default) | numeric scalarConditional variance model constant, specified as a numeric scalar or
`NaN`

value.

**Data Types: **`double`

`GARCH`

— GARCH polynomial coefficientscell vector of positive scalars or

`NaN`

valuesGARCH polynomial coefficients, specified as a cell vector of positive scalars or `NaN`

values.

If you specify

`GARCHLags`

, then the following conditions apply.The lengths of

`GARCH`

and`GARCHLags`

are equal.`GARCH{`

is the coefficient of lag}`j`

`GARCHLags(`

.)`j`

By default,

`GARCH`

is a`numel(GARCHLags)`

-by-1 cell vector of`NaN`

values.

Otherwise, the following conditions apply.

The length of

`GARCH`

is`P`

.`GARCH{`

is the coefficient of lag}`j`

.`j`

By default,

`GARCH`

is a`P`

-by-1 cell vector of`NaN`

values.

**Data Types: **`cell`

`ARCH`

— ARCH polynomial coefficientscell vector of positive scalars or

`NaN`

valuesARCH polynomial coefficients, specified as a cell vector of positive scalars or `NaN`

values.

If you specify

`ARCHLags`

, then the following conditions apply.The lengths of

`ARCH`

and`ARCHLags`

are equal.`ARCH{`

is the coefficient of lag}`j`

`ARCHLags(`

.)`j`

By default,

`ARCH`

is a`Q`

-by-1 cell vector of`NaN`

values. For more details, see the`Q`

property.

Otherwise, the following conditions apply.

The length of

`ARCH`

is`Q`

.`ARCH{`

is the coefficient of lag}`j`

.`j`

By default,

`ARCH`

is a`Q`

-by-1 cell vector of`NaN`

values.

**Data Types: **`cell`

`Leverage`

— Leverage polynomial coefficientscell vector of numeric scalars or

`NaN`

valuesLeverage polynomial coefficients, specified as a cell vector of numeric scalars or `NaN`

values.

If you specify

`LeverageLags`

, then the following conditions apply.The lengths of

`Leverage`

and`LeverageLags`

are equal.`Leverage{`

is the coefficient of lag}`j`

`LeverageLags(`

.)`j`

By default,

`Leverage`

is a`Q`

-by-1 cell vector of`NaN`

values. For more details, see the`Q`

property.

Otherwise, the following conditions apply.

The length of

`Leverage`

is`Q`

.`Leverage{`

is the coefficient of lag}`j`

.`j`

By default,

`Leverage`

is a`Q`

-by-1 cell vector of`NaN`

values.

**Data Types: **`cell`

`UnconditionalVariance`

— Model unconditional variancepositive scalar

This property is read-only.

The model unconditional variance, specified as a positive scalar.

The unconditional variance is

$${\sigma}_{\epsilon}^{2}=\mathrm{exp}\left\{\frac{\kappa}{(1-{\displaystyle {\sum}_{i=1}^{P}{\gamma}_{i}})}\right\}.$$

*κ* is the conditional variance model constant
(`Constant`

).

**Data Types: **`double`

`Offset`

— Innovation mean model offset`0`

(default) | numeric scalar | `NaN`

Innovation mean model offset, or additive constant, specified as a numeric scalar or `NaN`

value.

**Data Types: **`double`

`Distribution`

— Conditional probability distribution of innovation processstructure array

Conditional probability distribution of the innovations process, specified as a structure array.

The `Name`

field stores the name of the distribution, either `"Gaussian"`

for the Gaussian distribution or `"t"`

for the *t* distribution.

If `Name`

is `"t"`

, then `Distribution`

also contains the `DoF`

field, which stores the *t*-distribution degrees of freedom.

By default, `Distribution`

is `struct('Name',"Gaussian")`

.
When you create the object, if you specify that the underlying innovation process has a
*t* distribution by using the `Distribution`

name-value pair argument, then the `DoF`

field is
`NaN`

by default.

**Data Types: **`struct`

`Description`

— Model descriptionstring scalar | character vector

Model description, specified as a string scalar or character vector. By default, this property describes the parametric form of the model, for example,
```
"EGARCH(1,1) Conditional Variance Model (Gaussian
Distribution)"
```

.

**Data Types: **`string`

| `char`

All

`NaN`

-valued model parameters, which include coefficients and the*t*-innovation-distribution degrees of freedom (if present), are estimable. When you pass the resulting`egarch`

object and data to`estimate`

, MATLAB estimates all`NaN`

-valued parameters. During estimation,`estimate`

treats known parameters as equality constraints, that is,`estimate`

holds any known parameters fixed at their values.All

`GARCH`

,`ARCH`

, and`Leverage`

coefficients are subject to a near-zero tolerance exclusion test. That is, the software:Creates lag operator polynomials for each of the

`GARCH`

and`ARCH`

components.Compares each coefficient to the default lag operator zero tolerance,

`1e-12`

.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`

.Typically, the lags in the ARCH and leverage polynomials are the same, but their equality is not a requirement. Differing polynomials occur when:

Either

`ARCH{Q}`

or`Leverage{Q}`

meets the near-zero exclusion tolerance. In this case, MATLAB excludes the corresponding lag from the polynomial.You specify polynomials of differing lengths by specifying

`ARCHLags`

or`LeverageLags`

, or by setting the`ARCH`

or`Leverage`

property.

In either case,

`Q`

is the maximum lag between the two polynomials.

`estimate` | Fit conditional variance model to data |

`filter` | Filter disturbances through conditional variance model |

`forecast` | Forecast conditional variances from conditional variance models |

`infer` | Infer conditional variances of conditional variance models |

`simulate` | Monte Carlo simulation of conditional variance models |

`summarize` | Display estimation results of conditional variance model |

Create a default `egarch`

model object and specify its parameter values using dot notation.

Create an EGARCH(0,0) model.

Mdl = egarch

Mdl = egarch with properties: Description: "EGARCH(0,0) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 Q: 0 Constant: NaN GARCH: {} ARCH: {} Leverage: {} Offset: 0

`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 with properties: Description: "EGARCH(0,2) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 Q: 2 Constant: NaN GARCH: {} ARCH: {NaN NaN} at lags [1 2] Leverage: {NaN NaN} at lags [1 2] Offset: 0

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 with properties: Description: "EGARCH(3,2) Conditional Variance Model (Gaussian Distribution)" 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] Offset: 0

`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 with properties: Description: "EGARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: NaN GARCH: {NaN} at lag [1] ARCH: {NaN} at lag [1] Leverage: {NaN} at lag [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`

.

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

$${y}_{t}=0.5+{\epsilon}_{t},$$

where $${\epsilon}_{t}={\sigma}_{t}{z}_{t},$$

$${\sigma}_{t}^{2}=0.0001+0.75\mathrm{log}{\sigma}_{t-1}^{2}+0.1(\frac{\left|{\epsilon}_{t-1}\right|}{{\sigma}_{t-1}}-\sqrt{\frac{2}{\pi}})-0.3\frac{{\epsilon}_{t-1}}{{\sigma}_{t-1}}+0.01\frac{{\epsilon}_{t-3}}{{\sigma}_{t-3}},$$

and $${z}_{t}$$ 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 with properties: Description: "EGARCH(1,3) Conditional Variance Model with Offset (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 3 Constant: 0.0001 GARCH: {0.75} at lag [1] ARCH: {0.1} at lag [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]`

.

Access the properties of a created `egarch`

model object using dot notation.

Create an `egarch`

model object.

Mdl = egarch(3,2)

Mdl = egarch with properties: Description: "EGARCH(3,2) Conditional Variance Model (Gaussian Distribution)" 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] Offset: 0

Remove the second GARCH term from the model. That is, specify that the GARCH coefficient of the second lagged conditional variance is `0`

.

Mdl.GARCH{2} = 0

Mdl = egarch with properties: Description: "EGARCH(3,2) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 3 Q: 2 Constant: NaN GARCH: {NaN NaN} at lags [1 3] ARCH: {NaN NaN} at lags [1 2] Leverage: {NaN NaN} at lags [1 2] Offset: 0

The GARCH polynomial has two unknown parameters corresponding to lags 1 and 3.

Display the distribution of the disturbances.

Mdl.Distribution

`ans = `*struct with fields:*
Name: "Gaussian"

The disturbances are Gaussian with mean 0 and variance 1.

Specify that the underlying disturbances have a *t* distribution with five degrees of freedom.

Mdl.Distribution = struct('Name','t','DoF',5)

Mdl = egarch with properties: Description: "EGARCH(3,2) Conditional Variance Model (t Distribution)" Distribution: Name = "t", DoF = 5 P: 3 Q: 2 Constant: NaN GARCH: {NaN NaN} at lags [1 3] ARCH: {NaN NaN} at lags [1 2] Leverage: {NaN NaN} at lags [1 2] Offset: 0

Specify that the ARCH coefficients are 0.2 for the first lag and 0.1 for the second lag.

Mdl.ARCH = {0.2 0.1}

Mdl = egarch with properties: Description: "EGARCH(3,2) Conditional Variance Model (t Distribution)" Distribution: Name = "t", DoF = 5 P: 3 Q: 2 Constant: NaN GARCH: {NaN NaN} at lags [1 3] ARCH: {0.2 0.1} at lags [1 2] Leverage: {NaN NaN} at lags [1 2] Offset: 0

To estimate the remaining parameters, you can pass `Mdl`

and your data to estimate and use the specified parameters as equality constraints. Or, you can specify the rest of the parameter values, and then simulate or forecast conditional variances from the GARCH model by passing the fully specified model to `simulate`

or `forecast`

, respectively.

Fit an EGARCH model to an annual time series of Danish nominal stock returns from 1922-1999.

Load the `Data_Danish`

data set. Plot the nominal returns (`RN`

).

load Data_Danish; nr = DataTable.RN; figure; plot(dates,nr); hold on; plot([dates(1) dates(end)],[0 0],'r:'); % Plot y = 0 hold off; title('Danish Nominal Stock Returns'); ylabel('Nominal return (%)'); xlabel('Year');

The nominal return series seems to have a nonzero conditional mean offset and seems to exhibit volatility clustering. That is, the variability is smaller for earlier years than it is for later years. For this example, assume that an EGARCH(1,1) model is appropriate for this series.

Create an EGARCH(1,1) model. The conditional mean offset is zero by default. To estimate the offset, specify that it is `NaN`

. Include a leverage lag.

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

Fit the EGARCH(1,1) model to the data.

EstMdl = estimate(Mdl,nr);

EGARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ _________ Constant -0.62723 0.74401 -0.84304 0.39921 GARCH{1} 0.77419 0.23628 3.2766 0.0010507 ARCH{1} 0.38636 0.37361 1.0341 0.30107 Leverage{1} -0.0024984 0.19222 -0.012998 0.98963 Offset 0.10325 0.037727 2.7368 0.0062047

`EstMdl`

is a fully specified `egarch`

model object. That is, it does not contain `NaN`

values. You can assess the adequacy of the model by generating residuals using `infer`

, and then analyzing them.

To simulate conditional variances or responses, pass `EstMdl`

to `simulate`

.

To forecast innovations, pass `EstMdl`

to `forecast`

.

Simulate conditional variance or response paths from a fully specified `egarch`

model object. That is, simulate from an estimated `egarch`

model or a known `egarch`

model in which you specify all parameter values.

Load the `Data_Danish`

data set.

```
load Data_Danish;
rn = DataTable.RN;
```

Create an EGARCH(1,1) model with an unknown conditional mean offset. Fit the model to the annual, nominal return series. Include a leverage term.

Mdl = egarch('GARCHLags',1,'ARCHLags',1,'LeverageLags',1,'Offset',NaN); EstMdl = estimate(Mdl,rn);

EGARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ _________ Constant -0.62723 0.74401 -0.84304 0.3992 GARCH{1} 0.77419 0.23628 3.2766 0.0010508 ARCH{1} 0.38636 0.37361 1.0341 0.30107 Leverage{1} -0.0024992 0.19222 -0.013002 0.98963 Offset 0.10325 0.037727 2.7368 0.0062047

Simulate 100 paths of conditional variances and responses from the estimated EGARCH model.

numObs = numel(rn); % Sample size (T) numPaths = 100; % Number of paths to simulate rng(1); % For reproducibility [VSim,YSim] = simulate(EstMdl,numObs,'NumPaths',numPaths);

`VSim`

and `YSim`

are `T`

-by- `numPaths`

matrices. Rows correspond to a sample period, and columns correspond to a simulated path.

Plot the average and the 97.5% and 2.5% percentiles of the simulate paths. Compare the simulation statistics to the original data.

VSimBar = mean(VSim,2); VSimCI = quantile(VSim,[0.025 0.975],2); YSimBar = mean(YSim,2); YSimCI = quantile(YSim,[0.025 0.975],2); figure; subplot(2,1,1); h1 = plot(dates,VSim,'Color',0.8*ones(1,3)); hold on; h2 = plot(dates,VSimBar,'k--','LineWidth',2); h3 = plot(dates,VSimCI,'r--','LineWidth',2); hold off; title('Simulated Conditional Variances'); ylabel('Cond. var.'); xlabel('Year'); subplot(2,1,2); h1 = plot(dates,YSim,'Color',0.8*ones(1,3)); hold on; h2 = plot(dates,YSimBar,'k--','LineWidth',2); h3 = plot(dates,YSimCI,'r--','LineWidth',2); hold off; title('Simulated Nominal Returns'); ylabel('Nominal return (%)'); xlabel('Year'); legend([h1(1) h2 h3(1)],{'Simulated path' 'Mean' 'Confidence bounds'},... 'FontSize',7,'Location','NorthWest');

Forecast conditional variances from a fully specified `egarch`

model object. That is, forecast from an estimated `egarch`

model or a known `egarch`

model in which you specify all parameter values. The example follows from Estimate EGARCH Model.

Load the `Data_Danish`

data set.

```
load Data_Danish;
nr = DataTable.RN;
```

Create an EGARCH(1,1) model with an unknown conditional mean offset and include a leverage term. Fit the model to the annual nominal return series.

Mdl = egarch('GARCHLags',1,'ARCHLags',1,'LeverageLags',1,'Offset',NaN); EstMdl = estimate(Mdl,nr);

EGARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ _________ Constant -0.62723 0.74401 -0.84304 0.3992 GARCH{1} 0.77419 0.23628 3.2766 0.0010508 ARCH{1} 0.38636 0.37361 1.0341 0.30107 Leverage{1} -0.0024992 0.19222 -0.013002 0.98963 Offset 0.10325 0.037727 2.7368 0.0062047

Forecast the conditional variance of the nominal return series 10 years into the future using the estimated EGARCH model. Specify the entire returns series as presample observations. The software infers presample conditional variances using the presample observations and the model.

numPeriods = 10; vF = forecast(EstMdl,numPeriods,nr);

Plot the forecasted conditional variances of the nominal returns. Compare the forecasts to the observed conditional variances.

v = infer(EstMdl,nr); figure; plot(dates,v,'k:','LineWidth',2); hold on; plot(dates(end):dates(end) + 10,[v(end);vF],'r','LineWidth',2); title('Forecasted Conditional Variances of Nominal Returns'); ylabel('Conditional variances'); xlabel('Year'); legend({'Estimation sample cond. var.','Forecasted cond. var.'},... 'Location','Best');

An *EGARCH model* is a dynamic model that
addresses conditional heteroscedasticity, or volatility clustering, in an
innovations process. Volatility clustering occurs when an innovations process does
not exhibit significant autocorrelation, but the variance of the process changes
with time.

An EGARCH 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

$${y}_{t}=\mu +{\epsilon}_{t},$$

where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$ The EGARCH(*P*,*Q*) conditional
variance process, $${\sigma}_{t}^{2}$$, has the form

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

The table shows how the variables correspond to the properties
of the `egarch`

object.

Variable | Description | Property |
---|---|---|

μ | Innovation mean model constant offset | `'Offset'` |

κ | Conditional variance model constant | `'Constant'` |

γ_{j} | GARCH component coefficients | `'GARCH'` |

α_{j} | ARCH component coefficients | `'ARCH'` |

ξ_{j} | Leverage component coefficients | `'Leverage'` |

z_{t} | Series of independent random variables with mean 0 and variance 1 | `'Distribution'` |

If *z _{t}* is Gaussian, then

$$E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}=E\left\{\left|{z}_{t-j}\right|\right\}=\sqrt{\frac{2}{\pi}}.$$

If *z _{t}* is

$$E\left\{\frac{\left|{\epsilon}_{t-j}\right|}{{\sigma}_{t-j}}\right\}=E\left\{\left|{z}_{t-j}\right|\right\}=\sqrt{\frac{\nu -2}{\pi}}\frac{\Gamma \left(\frac{\nu -1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)}.$$

To ensure a stationary EGARCH model, all roots of the GARCH lag operator polynomial, $$(1-{\gamma}_{1}L-\dots -{\gamma}_{P}{L}^{P})$$, 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,

$$E({\sigma}_{t}^{2})\ge \mathrm{exp}\{E(\mathrm{log}{\sigma}_{t}^{2})\}.$$

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

You can specify an

`egarch`

model as part of a composition of conditional mean and variance models. For details, see`arima`

.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 signs that are expected, you can encounter difficulties inferring volatility sequences and forecasting. A negative ARCH coefficient is problematic. In this case, an EGARCH model might not be the best choice for your application.

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

- Specify EGARCH Models
- Modify Properties of Conditional Variance Models
- Specify Conditional Mean and Variance Models
- Infer Conditional Variances and Residuals
- Compare Conditional Variance Models Using Information Criteria
- Assess EGARCH Forecast Bias Using Simulations
- Forecast a Conditional Variance Model
- Conditional Variance Models
- EGARCH Model

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