GJR conditional variance time series model

Use `gjr`

to specify a univariate GJR (Glosten, Jagannathan, and Runkle) model. The `gjr`

function returns a `gjr`

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

The key components of a `gjr`

model include the:

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

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

Leverage polynomial, which is composed of lagged squared, negative 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.

returns a zero-degree conditional variance `Mdl`

= gjr`gjr`

object.

creates a GJR conditional variance model object (`Mdl`

= gjr(`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 property name in quotes. For example, `Mdl`

= gjr(`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 a GJR(1,2) model containing unknown parameter values, enter:

Mdl = gjr(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 conditional variance terms from lag *t* – 1 through lag *t* – `P`

.

You can specify this argument using the
`gjr`

`(P,Q)`

shorthand syntax only.

If `P`

> 0, then you must specify `Q`

as a positive integer.

**Example: **`gjr(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 squared innovation terms (for the ARCH polynomial) and squared, negative innovation terms (for the leverage polynomial) from lag *t* – 1 through lag *t* – `Q`

.

You can specify this argument using the
`gjr`

`(P,Q)`

shorthand syntax only.

If `P`

> 0, then you must specify `Q`

as a positive integer.

**Example: **`gjr(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 a GJR(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`

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

Mdl = gjr('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`gjr`

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`gjr`

uses their values to determine`Q`

instead.Otherwise,

`Q`

is`0`

.

**Data Types: **`double`

`Constant`

— Conditional variance model constant`NaN`

(default) | positive scalarConditional variance model constant, specified as a positive 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.

The coefficients in `GARCH`

correspond to coefficients in an underlying `LagOp`

lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to `1e–12`

or below, `gjr`

excludes that coefficient and its corresponding lag in `GARCHLags`

from the model.

**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.

The coefficients in `ARCH`

correspond to coefficients in an underlying `LagOp`

lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to `1e–12`

or below, `gjr`

excludes that coefficient and its corresponding lag in `ARCHLags`

from the model.

**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.

The coefficients in `Leverage`

correspond to coefficients in an underlying `LagOp`

lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to `1e–12`

or below, `gjr`

excludes that coefficient and its corresponding lag in `LeverageLags`

from the model.

**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

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

*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 process`"Gaussian"`

(default) | `"t"`

| structure arrayConditional probability distribution of the innovation process, specified as a string or structure array. `gjr`

stores the value as a 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 specify

`"t"`

,`DoF`

is`NaN`

by default. 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 to specify the Student's

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

and`'DoF'`

fields.

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

`Description`

— Model descriptionstring scalar | character vector

Model description, specified as a string scalar or character vector. `gjr`

stores the value as a string scalar. The default value describes the parametric form of the model, for example
`"GJR(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`gjr`

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.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 `gjr`

model object and specify its parameter values using dot notation.

Create a GJR(0,0) model.

Mdl = gjr

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

`Mdl`

is a `gjr`

model object. 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 = gjr with properties: Description: "GJR(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}`

, and `{NaN NaN}`

, respectively. The two ARCH and leverage coefficients are associated with lags 1 and 2.

Create a `gjr`

model object using the shorthand notation `gjr(P,Q)`

, where `P`

is the degree of the GARCH polynomial and `Q`

is the degree of the ARCH and leverage polynomials.

Create an GJR(3,2) model.

Mdl = gjr(3,2)

Mdl = gjr with properties: Description: "GJR(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 a `gjr`

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 lags

`P`

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

`Q`

`Mdl`

specifies only the functional form of a GJR model. Because it contains unknown parameter values, you can pass `Mdl`

and time-series data to `estimate`

to estimate the parameters.

Create a `gjr`

model using name-value pair arguments.

Specify a GJR(1,1) model.

Mdl = gjr('GARCHLags',1,'ARCHLags',1,'LeverageLags',1)

Mdl = gjr with properties: Description: "GJR(1,1) Conditional Variance Model (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: 0

`Mdl`

is a `gjr`

model object. The software sets all parameters to `NaN`

, except `P`

, `Q`

, `Distribution`

, and `Offset`

(which is `0`

by default).

Since `Mdl`

contains `NaN`

values, `Mdl`

is only appropriate for estimation only. Pass `Mdl`

and time-series data to `estimate`

.

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

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

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

$${\sigma}_{t}^{2}=0.0001+0.35{\sigma}_{t-1}^{2}+0.1{\epsilon}_{t-1}^{2}+0.03{\epsilon}_{t-1}^{2}I({\epsilon}_{t-1}<0)+0.01{\epsilon}_{t-3}^{2}I({\epsilon}_{t-3}<0),$$

and $${z}_{t}$$ is an independent and identically distributed standard Gaussian process.

Mdl = gjr('Constant',0.0001,'GARCH',0.35,... 'ARCH',0.1,'Offset',0.5,'Leverage',{0.03 0 0.01})

Mdl = gjr with properties: Description: "GJR(1,3) Conditional Variance Model with Offset (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 3 Constant: 0.0001 GARCH: {0.35} at lag [1] ARCH: {0.1} at lag [1] Leverage: {0.03 0.01} at lags [1 3] Offset: 0.5

`gjr`

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',{0.03 0.01},'LeverageLags',[1 3]`

.

Access the properties of a `gjr`

model object using dot notation.

Create a `gjr`

model object.

Mdl = gjr(3,2)

Mdl = gjr with properties: Description: "GJR(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 = gjr with properties: Description: "GJR(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 = gjr with properties: Description: "GJR(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 = gjr with properties: Description: "GJR(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 a GJR model to an annual time series of stock price index returns from 1861-1970.

Load the Nelson-Plosser data set. Convert the yearly stock price indices (`SP`

) to returns. Plot the returns.

load Data_NelsonPlosser; sp = price2ret(DataTable.SP); figure; plot(dates(2:end),sp); hold on; plot([dates(2) dates(end)],[0 0],'r:'); % Plot y = 0 hold off; title('Returns'); ylabel('Return (%)'); xlabel('Year'); axis tight;

The return series does not seem to have a 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 GJR(1,1) model is appropriate for this series.

Create a GJR(1,1) model. The conditional mean offset is zero by default. The software includes a conditional variance model constant by default.

Mdl = gjr('GARCHLags',1,'ARCHLags',1,'LeverageLags',1);

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

EstMdl = estimate(Mdl,sp);

GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ ________ Constant 0.0045728 0.0044199 1.0346 0.30086 GARCH{1} 0.55808 0.24 2.3253 0.020057 ARCH{1} 0.20461 0.17886 1.144 0.25263 Leverage{1} 0.18066 0.26802 0.67406 0.50027

`EstMdl`

is a fully specified `gjr`

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 `gjr`

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

model or a known `gjr`

model in which you specify all parameter values.

Load the Nelson-Plosser data set. Convert the yearly stock price indices to returns.

```
load Data_NelsonPlosser;
sp = price2ret(DataTable.SP);
```

Create a GJR(1,1) model. Fit the model to the return series.

Mdl = gjr(1,1); EstMdl = estimate(Mdl,sp);

GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ ________ Constant 0.0045728 0.0044199 1.0346 0.30086 GARCH{1} 0.55808 0.24 2.3253 0.020057 ARCH{1} 0.20461 0.17886 1.144 0.25263 Leverage{1} 0.18066 0.26802 0.67406 0.50027

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

numObs = numel(sp); % 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 simulated paths. Compare the simulation statistics to the original data.

dates = dates(2:end); 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'); axis tight; 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'); axis tight; legend([h1(1) h2 h3(1)],{'Simulated path' 'Mean' 'Confidence bounds'},... 'FontSize',7,'Location','NorthWest');

Forecast conditional variances from a fully specified `gjr`

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

model or a known `gjr`

model in which you specify all parameter values.

Load the Nelson-Plosser data set. Convert the yearly stock price indices (`SP`

) to returns.

```
load Data_NelsonPlosser;
sp = price2ret(DataTable.SP);
```

Create a GJR(1,1) model and fit it to the return series.

Mdl = gjr('GARCHLags',1,'ARCHLags',1,'LeverageLags',1); EstMdl = estimate(Mdl,sp);

GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ ________ Constant 0.0045728 0.0044199 1.0346 0.30086 GARCH{1} 0.55808 0.24 2.3253 0.020057 ARCH{1} 0.20461 0.17886 1.144 0.25263 Leverage{1} 0.18066 0.26802 0.67406 0.50027

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

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

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

v = infer(EstMdl,sp); nV = size(v,1); dates = dates((end - nV + 1):end); 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 Returns'); ylabel('Conditional variances'); xlabel('Year'); axis tight; legend({'Estimation Sample Cond. Var.','Forecasted Cond. var.'},... 'Location','NorthWest');

The *Glosten, Jagannathan, and Runkle (GJR) 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.

The GJR model is a generalization of the GARCH model that is appropriate for modeling asymmetric volatility clustering [1]. Specifically, the model posits that the current conditional variance is the sum of these linear processes, with coefficients:

Past conditional variances (the GARCH component or polynomial).

Past squared innovations (the ARCH component or polynomial).

Past squared, negative innovations (the leverage component or polynomial).

Consider the time series

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

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

$${\sigma}_{t}^{2}=\kappa +{\displaystyle \sum _{i=1}^{P}{\gamma}_{i}{\sigma}_{t-i}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\alpha}_{j}}{\epsilon}_{t-j}^{2}+{\displaystyle \sum _{j=1}^{Q}{\xi}_{j}}I\left[{\epsilon}_{t-j}<0\right]{\epsilon}_{t-j}^{2}.$$

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

object. In the table, *I*[*x* < 0] = 1, and 0 otherwise.

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

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

κ > 0 | 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'` |

For stationarity and positivity, GJR models use these constraints:

$$\kappa >0$$

$${\gamma}_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha}_{j}\ge 0$$

$${\alpha}_{j}+{\xi}_{j}\ge 0$$

$${\sum}_{i=1}^{P}{\gamma}_{i}}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}+\frac{1}{2}}{\displaystyle {\sum}_{j=1}^{Q}{\xi}_{j}}<1$$

GJR models are appropriate when negative shocks of contribute more to volatility than positive shocks [2].

If all leverage coefficients are zero, then the GJR model reduces to the GARCH model. Because the GARCH model is nested in the GJR model, you can use likelihood ratio tests to compare a GARCH model fit against a GJR model fit.

You can specify a `gjr`

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

.

[1] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.”
*The Journal of Finance*. Vol. 48, No. 5, 1993, pp. 1779–1801.

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

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