gjr
GJR conditional variance time series model
Description
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.
Creation
Description
returns a zero-degree conditional variance Mdl
= gjrgjr
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.
Input Arguments
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 degree
nonnegative 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 degree
nonnegative 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 pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
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.
Example: '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 integers
GARCH 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{
. The lengths of
j
}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 integers
ARCH 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{
. The lengths of
j
}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 integers
Leverage 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{
. The
lengths of j
}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
Properties
You can set writable property values when you create the model object by using name-value 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 degree
nonnegative 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
, thenP
is the largest specified lag.If you specify
GARCH
, thenP
is the number of elements of the specified value. If you also specifyGARCHLags
, thengjr
usesGARCHLags
to determineP
instead.Otherwise,
P
is0
.
Data Types: double
Q
— Maximum degree of ARCH and leverage polynomials
nonnegative 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
orLeverageLags
, thenQ
is the maximum between the two specifications.If you specify
ARCH
orLeverage
, thenQ
is the maximum number of elements between the two specifications. If you also specifyARCHLags
orLeverageLags
, thengjr
uses their values to determineQ
instead.Otherwise,
Q
is0
.
Data Types: double
Constant
— Conditional variance model constant
NaN
(default) | positive scalar
Conditional variance model constant, specified as a positive scalar or NaN
value.
Data Types: double
GARCH
— GARCH polynomial coefficients
cell vector of positive scalars or NaN
values
GARCH 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
andGARCHLags
are equal.GARCH{
is the coefficient of lagj
}GARCHLags(
.j
)By default,
GARCH
is anumel(GARCHLags)
-by-1 cell vector ofNaN
values.
Otherwise, the following conditions apply.
The length of
GARCH
isP
.GARCH{
is the coefficient of lagj
}j
.By default,
GARCH
is aP
-by-1 cell vector ofNaN
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 coefficients
cell vector of positive scalars or NaN
values
ARCH 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
andARCHLags
are equal.ARCH{
is the coefficient of lagj
}ARCHLags(
.j
)By default,
ARCH
is aQ
-by-1 cell vector ofNaN
values. For more details, see theQ
property.
Otherwise, the following conditions apply.
The length of
ARCH
isQ
.ARCH{
is the coefficient of lagj
}j
.By default,
ARCH
is aQ
-by-1 cell vector ofNaN
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 coefficients
cell vector of numeric scalars or NaN
values
Leverage 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
andLeverageLags
are equal.Leverage{
is the coefficient of lagj
}LeverageLags(
.j
)By default,
Leverage
is aQ
-by-1 cell vector ofNaN
values. For more details, see theQ
property.
Otherwise, the following conditions apply.
The length of
Leverage
isQ
.Leverage{
is the coefficient of lagj
}j
.By default,
Leverage
is aQ
-by-1 cell vector ofNaN
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 variance
positive scalar
This property is read-only.
The model unconditional variance, specified as a positive scalar.
The unconditional variance is
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 εt
"Gaussian"
(default) | "t"
| structure array
Conditional probability distribution of the innovation process
εt, 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 orDoF
=NaN
.DoF
is estimable.If you specify
"t"
,DoF
isNaN
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 the'DoF'
fields.
Example: Distribution=struct('Name',"t",'DoF',10)
Description
— Model description
string 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
SeriesName
— Response series name
string scalar | character vector | "Y"
Since R2023a
Response series name, specified as a string scalar or character vector.
gjr
stores the value as a string scalar.
Example: "StockReturn"
Data Types: string
| char
Note
All
NaN
-valued model parameters, which include coefficients and the t-innovation-distribution degrees of freedom (if present), are estimable. When you pass the resultinggjr
object and data toestimate
, MATLAB estimates allNaN
-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}
orLeverage{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
orLeverageLags
, or by setting theARCH
orLeverage
property.
In either case,
Q
is the maximum lag between the two polynomials.
Object Functions
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 |
Examples
Create Default GJR 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)" SeriesName: "Y" 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)" SeriesName: "Y" 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 GJR Model Using Shorthand Syntax
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)" SeriesName: "Y" 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 GJR Model Using Longhand Syntax
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)" SeriesName: "Y" 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 GJR Model with Known Coefficients
Create a GJR(1,1) model with mean offset
where
and 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)" SeriesName: "Y" 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 GJR Model Properties
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)" SeriesName: "Y" 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)" SeriesName: "Y" 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)" SeriesName: "Y" 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)" SeriesName: "Y" 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.
Estimate GJR Model
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 GJR Model Observations and Conditional Variances
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 GJR Model Conditional Variances
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');
More About
GJR Model
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
where The GJR(P,Q) conditional variance process, , has the form
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' |
γj | GARCH component coefficients | 'GARCH' |
αj | ARCH component coefficients | 'ARCH' |
ξj | Leverage component coefficients | 'Leverage' |
zt | Series of independent random variables with mean 0 and variance 1 | 'Distribution' |
For stationarity and positivity, GJR models use these constraints:
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.
Tips
You can specify a gjr
model as part of a composition of conditional mean and variance models. For details, see arima
.
References
[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.
Version History
Introduced in R2012aR2023a: Name a GJR model response series
Name the response series of a GJR model by setting the
SeriesName
property of the associated model to a string
scalar. When you supply input response data to model object functions in a table or
timetable, the functions choose the variable with name SeriesName
as the response variable by default.
R2018a: Describe a GJR model
Describe a GJR model by setting the Description
property to a
string scalar.
R2018a: Use indices that are consistent with MATLAB cell array indexing
The indices of cell arrays of lag operator polynomial coefficients follow MATLAB cell array indexing rules.
Affected model properties are GARCH
, ARCH
, and
Leverage
properties.
You cannot access any lag-zero coefficients by using an index of
0
. For example,Mdl.ARCH{0}
issues an error.Remove any instances of such indices of zero from your code. The value of all lag-zero coefficients is
1
except for the lag operator polynomial corresponding to theARCH
property, which has the value0
.You cannot index beyond the maximal lag in the polynomial. For example, if
Mdl.P
is 4, thenMdl.ARCH{p}
issues an error whenp
is greater than4
. For details on the maximal lags of the lag operator polynomials, see the corresponding property descriptions.Remove any instances of such indices beyond the maximal lag from your code. All coefficients beyond the maximal lag are
0
.
R2018a: Models store innovation distribution name as a string scalar
The Name
field of the Distribution
property of
gjr
model objects stores the innovation distribution name
as a string scalar, for example, "Gaussian"
for Gaussian innovations.
Before R2018a, MATLAB stored the innovation distribution name as a character vector, for example
'Gaussian'
for Gaussian innovations. Although most text-data
operations accept character vectors and string scalars for text-data input, the two data
types have some differences. For details, see Text in String and Character Arrays.
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