Specify GJR Models
Default GJR Model
The default GJR(P,Q) model in Econometrics Toolbox™ is of the form
with Gaussian innovation distribution and
The indicator function equals 1 if and 0 otherwise. The default model has no mean offset, and the lagged variances and squared innovations are at consecutive lags.
You can specify a model of this form using the shorthand syntaxgjr(P,Q)
. For the input argumentsP
andQ
, enter the number of lagged variances (GARCH terms),P, and lagged squared innovations (ARCH and leverage terms),Q, respectively. The following restrictions apply:
PandQmust be nonnegative integers.
IfP> 0, then you must also specifyQ> 0
When you use this shorthand syntax,gjr
creates agjr
model with these default property values.
Property | Default Value |
---|---|
P |
Number of GARCH terms,P |
Q |
Number of ARCH and leverage terms,Q |
Offset |
0 |
Constant |
NaN |
GARCH |
Cell vector ofNaN s |
ARCH |
Cell vector ofNaN s |
Leverage |
Cell vector ofNaN s |
Distribution |
"Gaussian" |
To assign nondefault values to any properties, you can modify the created model using dot notation.
To illustrate, consider specifying the GJR(1,1) model
with Gaussian innovation distribution and
Mdl = gjr(1,1)
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
The created model,Mdl
, hasNaN
s for all model parameters. ANaN
value signals that a parameter needs to be estimated or otherwise specified by the user. All parameters must be specified to forecast or simulate the model.
To estimate parameters, input the model (along with data) toestimate
. This returns a new fittedgjr
model. The fitted model has parameter estimates for each inputNaN
value.
Callinggjr
without any input arguments returns a GJR(0,0) model specification with default property values:
DefaultMdl = gjr
DefaultMdl = 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
Specify Default GJR Model
This example shows how to use the shorthandgjr(P,Q)
syntax to specify the default GJR(P,Q) model,
with a Gaussian innovation distribution and
By default, all parameters in the created model have unknown values.
指定默认GJR(1,1)模型:
Mdl = gjr(1,1)
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
The output shows that the created model,Mdl
, hasNaN
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) toestimate
.
Using Name-Value Pair Arguments
The most flexible way to specify GJR models is using name-value pair arguments. You do not need, nor are you able, to specify a value for every model property.gjr
assigns default values to any model properties you do not (or cannot) specify.
The general GJR(P,Q) model is of the form
where and
The innovation distribution can be Gaussian or Student’st. The default distribution is Gaussian.
In order to estimate, forecast, or simulate a model, you must specify the parametric form of the model (e.g., which lags correspond to nonzero coefficients, the innovation distribution) and any known parameter values. You can set any unknown parameters equal toNaN
, and then input the model toestimate
(along with data) to get estimated parameter values.
gjr
(andestimate
) returns a model corresponding to the model specification. You can modify models to change or update the specification. Input models (with noNaN
values) toforecast
orsimulate
for forecasting and simulation, respectively. Here are some example specifications using name-value arguments.
Model | Specification |
---|---|
|
gjr('GARCH',NaN,'ARCH',NaN,... orgjr(1,1) |
|
gjr('Offset',NaN,'GARCH',NaN,... |
|
gjr('Constant',0.1,'GARCH',0.6,... |
Here is a full description of the name-value arguments you can use to specify GJR models.
Note
You cannot assign values to the propertiesP
andQ
.egarch
setsP
equal to the largest GARCH lag, andQ
equal to the largest lag with a nonzero squared innovation coefficient, including ARCH and leverage coefficients.
Name-Value Arguments for GJR Models
Name | Corresponding GJR Model Term(s) | When to Specify |
---|---|---|
Offset |
Mean offset,μ | To include a nonzero mean offset. For example,'Offset',0.2 . If you plan to estimate the offset term, specify'Offset',NaN .By default, Offset has value0 (meaning, no offset). |
Constant |
Constant in the conditional variance model,κ | To set equality constraints forκ. For example, if a model has known constant 0.1, specify'Constant',0.1 .By default, Constant has valueNaN . |
GARCH |
GARCH coefficients, | To set equality constraints for the GARCH coefficients. For example, to specify a GJR(1,1) model with
specify'GARCH',0.6 .You only need to specify the nonzero elements of GARCH . If the nonzero coefficients are at nonconsecutive lags, specify the corresponding lags usingGARCHLags .Any coefficients you specify must satisfy all stationarity constraints. |
GARCHLags |
Lags corresponding to the nonzero GARCH coefficients | GARCHLags is not a model property.Use this argument as a shortcut for specifying GARCH when the nonzero GARCH coefficients correspond to nonconsecutive lags. For example, to specify nonzero GARCH coefficients at lags 1 and 3, e.g., nonzero
and
specify'GARCHLags',[1,3] .Use GARCH andGARCHLags together to specify known nonzero GARCH coefficients at nonconsecutive lags. For example, if
and
specify'GARCH',{0.3,0.1},'GARCHLags',[1,3] |
ARCH |
ARCH coefficients, | To set equality constraints for the ARCH coefficients. For example, to specify a GJR(1,1) model with
specify'ARCH',0.3 .You only need to specify the nonzero elements of ARCH . If the nonzero coefficients are at nonconsecutive lags, specify the corresponding lags usingARCHLags . |
ARCHLags |
Lags corresponding to nonzero ARCH coefficients |
Use this argument as a shortcut for specifying Use |
Leverage |
Leverage coefficients, | To set equality constraints for the leverage coefficients. For example, to specify a GJR(1,1) model with
specify You only need to specify the nonzero elements of |
LeverageLags |
Lags corresponding to nonzero leverage coefficients |
Use this argument as a shortcut for specifying Use |
Distribution |
Distribution of the innovation process | Use this argument to specify a Student’stinnovation distribution. By default, the innovation distribution is Gaussian. For example, to specify atdistribution with unknown degrees of freedom, specify To specify atinnovation distribution with known degrees of freedom, assign |
Specify GJR Model Using Econometric Modeler App
您可以指定滞后结构,创新区域ribution, and leverages of GJR models using theEconometric Modelerapp. The app treats all coefficients as unknown and estimable, including the degrees of freedom parameter for atinnovation distribution.
At the command line, open theEconometric Modelerapp.
econometricModeler
Alternatively, open the app from the apps gallery (seeEconometric Modeler).
In the app, you can see all supported models by selecting a time series variable for the response in theTime Seriespane. Then, on theEconometric Modelertab, in theModelssection, click the arrow to display the models gallery.
TheGARCH Models部分包含ns all supported conditional variance models. To specify a GJR model, clickGJR
. TheGJR Model Parametersdialog box appears.
Adjustable parameters include:
GARCH Degree– The order of the GARCH polynomial.
ARCH Degree– The order of the ARCH polynomial. The value of this parameter also specifies the order of the leverage polynomial.
Include Offset– The inclusion of a model offset.
酒店ovation Distribution– The innovation distribution.
As you adjust parameter values, the equation in theModel Equationsection changes to match your specifications. Adjustable parameters correspond to input and name-value pair arguments described in the previous sections and in thegjr
reference page.
For more details on specifying models using the app, seeFitting Models to DataandSpecifying Univariate Lag Operator Polynomials Interactively.
Specify GJR Model with Mean Offset
This example shows how to specify a GJR(P,Q) model with a mean offset. Use name-value pair arguments to specify a model that differs from the default model.
Specify a GJR(1,1) model with a mean offset,
where and
Mdl = gjr('Offset',NaN,'GARCHLags',1,'ARCHLags',1,...'LeverageLags',1)
Mdl = gjr with properties: Description: "GJR(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
The mean offset appears in the output as an additional parameter to be estimated or otherwise specified.
Specify GJR Model with Nonconsecutive Lags
This example shows how to specify a GJR model with nonzero coefficients at nonconsecutive lags.
Specify a GJR(3,1) model with nonzero GARCH terms at lags 1 and 3. Include a mean offset.
Mdl = gjr('Offset',NaN,'GARCHLags',[1,3],'ARCHLags',1,...'LeverageLags',1)
Mdl = gjr with properties: Description: "GJR(3,1) Conditional Variance Model with Offset (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 3 Q: 1 Constant: NaN GARCH: {NaN NaN} at lags [1 3] ARCH: {NaN} at lag [1] Leverage: {NaN} at lag [1] Offset: NaN
The unknown nonzero GARCH coefficients correspond to lagged variances at lags 1 and 3. The output shows only the nonzero coefficients.
Display the value ofGARCH
:
Mdl.GARCH
ans=1×3 cell array{[NaN]} {[0]} {[NaN]}
TheGARCH
cell array returns three elements. The first and third elements have valueNaN
, indicating these coefficients are nonzero and need to be estimated or otherwise specified. By default,gjr
sets the interim coefficient at lag 2 equal to zero to maintain consistency with MATLAB® cell array indexing.
Specify GJR Model with Known Parameter Values
This example shows how to specify a GJR model with known parameter values. You can use such a fully specified model as an input tosimulate
orforecast
.
Specify the GJR(1,1) model
with a Gaussian innovation distribution.
Mdl = gjr('Constant',0.1,'GARCH',0.6,'ARCH',0.2,...'Leverage',0.1)
Mdl = gjr with properties: Description: "GJR(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 0.1 GARCH: {0.6} at lag [1] ARCH: {0.2} at lag [1] Leverage: {0.1} at lag [1] Offset: 0
Because all parameter values are specified, the created model has noNaN
values. The functionssimulate
andforecast
don't accept input models withNaN
values.
Specify GJR Model with t Innovation Distribution
This example shows how to specify a GJR model with a Student's t innovation distribution.
Specify a GJR(1,1) model with a mean offset,
where and
Assume follows a Student's t innovation distribution with 10 degrees of freedom.
tDist = struct('Name','t','DoF',10); Mdl = gjr('Offset',NaN,'GARCHLags',1,'ARCHLags',1,...'LeverageLags',1,'Distribution',tDist)
Mdl = gjr with properties: Description: "GJR(1,1) Conditional Variance Model with Offset (t Distribution)" Distribution: Name = "t", DoF = 10 P: 1 Q: 1 Constant: NaN GARCH: {NaN} at lag [1] ARCH: {NaN} at lag [1] Leverage: {NaN} at lag [1] Offset: NaN
The value ofDistribution
is astruct
array with fieldName
equal to't'
and fieldDoF
equal to10
. When you specify the degrees of freedom, they aren't estimated if you input the model toestimate
.
See Also
Objects
Functions
Related Examples
- Econometric Modeler App Overview
- Specifying Univariate Lag Operator Polynomials Interactively
- Specify the Conditional Variance Model Innovation Distribution
- Modify Properties of Conditional Variance Models