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GARCH conditional variance time series model
Use garch
to specify a univariate generalized
autoregressive conditional heteroscedastic (GARCH) model. The garch
function returns a garch
object specifying the functional form of
a GARCH(P,Q) model, and stores its
parameters values.
The key components of a garch
model include:
The GARCH polynomial, which is composed of lagged conditional variances. The degree is denoted P.
The ARCH polynomial, which is composed of the composed of lagged squared innovations. The degree is denoted Q.
P and Q are the maximum nonzero lags in the GARCH and ARCH polynomials, respectively. 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 namevalue 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 = garch
Mdl = garch(P,Q)
Mdl = garch(Name,Value)
returns a zerodegree
conditional variance Mdl
= garchgarch
object.
returns a Mdl
= garch(P
,Q
)garch
object that has a GARCH polynomial with a
degree of P
and an ARCH polynomial with a degree of
Q
. The GARCH and ARCH polynomials contain all consecutive
lags from 1 through their degrees, and all coefficients are
NaN
values.
This shorthand syntax allows for easy model template creation 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
namevalue pairs. You can specify multiple values. Enclose each name in single
quotes. For example, Mdl
= garch(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 allows for creating 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 GARCH(1,2) model containing unknown parameter values, enter:
Mdl = garch(1,2);
P
— GARCH polynomial degreeGARCH 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
garch
(P,Q)
shorthand syntax only.
If P
> 0, then you must specify Q
as a positive integer.
Example: garch(1,1)
Data Types: double
Q
— ARCH polynomial degreeARCH polynomial degree, specified as a nonnegative integer. In the
ARCH polynomial and at time t, MATLAB includes all consecutive squared innovation terms from lag
t – 1 through lag t –
Q
.
You can specify this argument using the
garch
(P,Q)
shorthand syntax only.
If P
> 0, then you must specify Q
as a positive integer.
Example: garch(1,1)
Data Types: double
Specify optional
commaseparated 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
GARCH(0,4) model and unknown, but nonzero ARCH coefficient matrices at lags
1
and 4
.'GARCHLags'
— GARCH polynomial lags1:P
(default)  numeric vector of unique positive integersGARCH polynomial lags, specified as the commaseparated 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 integersARCH polynomial lags, specified as the commaseparated pair
consisting of 'ARCHLags'
and a 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 coefficients (specified by the
ARCH
property) are positive or
NaN
values, max(ARCHLags)
determines the value of the Q
property.
Example: 'ARCHLags',[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 commaseparated pair consisting of 'Distribution'
and
a value in this table.
Distribution  Value  Structure Array  

Gaussian  'Gaussian'  struct('Name','Gaussian')  
Student’s t 


The t distribution degrees of freedom parameter is estimable. That
is, if you want estimate
to estimate it along with all other
unknown parameters, then set its value to NaN
.
Example: 'Distribution',struct('Name','t','DoF',10)
Data Types: char
 struct
You can set writable properties when you create a garch
object by
using namevalue pair argument syntax. Or, after you create a model, you can use dot
notation. For example, to create a GARCH(1,1) model with unknown coefficients, and then
specify a t innovation distribution with unknown degrees of freedom,
enter:
Mdl = garch('GARCHLags',1,'ARCHLags',1); Mdl.Distribution = "t";
P
— GARCH polynomial degreeThis property is readonly.
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 namevalue 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 GARCH
, then P
is
the number of elements therein.
If you specify GARCHLags
, then P
is the largest lag therein. This alternative takes precedence over the
others.
Otherwise, P
is 0
.
Data Types: double
Q
— ARCH polynomial degreeThis property is readonly.
ARCH polynomial degree, specified as a nonnegative integer.
Q
is the maximum lag in the ARCH polynomial with a
coefficient that is positive or NaN
. 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 namevalue 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 ARCH
, then
Q
is the number of elements
therein.
If you specify ARCHLags
, then
Q
is the largest lag therein. This
alternative takes precedence over the others.
Otherwise, Q
is
0
.
Data Types: double
Constant
— Conditional variance model constantNaN
(default)  positive scalarConditional variance model constant, specified as a positive scalar or NaN
value.
Data Types: double
GARCH
— GARCH polynomial coefficientsNaN
valuesGARCH polynomial coefficients, specified as a cell vector of positive scalars or NaN
values.
If you set the GARCHLags
namevalue pair argument, 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)
by1 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
by1 cell vector of NaN
values.
Data Types: cell
ARCH
— ARCH polynomial coefficientsNaN
valuesARCH polynomial coefficients, specified as a cell vector of positive
scalars or NaN
values.
If you set the ARCHLags
namevalue pair
argument, 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
numel(ARCHLags)
by1 cell vector
of NaN
values.
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
by1 cell vector of
NaN
values.
Data Types: cell
UnconditionalVariance
— Model unconditional varianceThis property is readonly.
The model unconditional variance, specified as a positive scalar.
The unconditional variance is
$${\sigma}_{\epsilon}^{2}=\frac{\kappa}{(1{\displaystyle {\sum}_{i=1}^{P}{\gamma}_{i}}{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}})}.$$
κ is the conditional variance model constant
(Constant
).
Data Types: double
Offset
— Innovation mean model offset0
(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 processConditional 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 tdistribution 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
namevalue pair argument, then the DoF
field is
NaN
by default.
Data Types: struct
Description
— Model descriptionModel description, specified as a string scalar or character vector. By default, this property describes the parametric form of the model, for example,
"GARCH(1,1) Conditional Variance Model (Gaussian
Distribution)"
.
Example: 'Description','Model 1'
Data Types: string
 char
All NaN
valued model parameters, which include coefficients and the tinnovationdistribution degrees of freedom (if present), are estimable. That is, when you pass the resulting garch
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
and ARCH
coefficients
are subject to a nearzero 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, 1e12
.
Includes a coefficient in the model if its magnitude is
greater than 1e12
, and excludes the
coefficient otherwise. In other words, the software considers
excluded coefficients to be sufficiently close to zero.
For details, see LagOp
.
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 garch
model object and specify its parameter values using dot notation.
Create a GARCH(0,0) model.
Mdl = garch
Mdl = garch with properties: Description: "GARCH(0,0) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 Q: 0 Constant: NaN GARCH: {} ARCH: {} Offset: 0
Mdl
is a garch
model. It contains an unknown constant, its offset is 0
, and the innovation distribution is 'Gaussian'
. The model does not have a GARCH or ARCH polynomial.
Specify two unknown ARCH coefficients for lags one and two using dot notation.
Mdl.ARCH = {NaN NaN}
Mdl = garch with properties: Description: "GARCH(0,2) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 Q: 2 Constant: NaN GARCH: {} ARCH: {NaN NaN} at lags [1 2] Offset: 0
The Q
and ARCH
properties are updated to 2
and {NaN NaN}
. The two ARCH coefficients are associated with lags 1 and 2.
Create a garch
model using the shorthand notation garch(P,Q)
, where P
is the degree of the GARCH polynomial and Q
is the degree of the ARCH polynomial.
Create a GARCH(3,2) model.
Mdl = garch(3,2)
Mdl = garch with properties: Description: "GARCH(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] Offset: 0
Mdl
is a garch
model object. All properties of Mdl
, except P
, Q
, and Distribution
, are NaN
values. By default, the software:
Includes a conditional variance model constant
Excludes a conditional mean model offset (i.e., the offset is 0
)
Includes all lag terms in the ARCH and GARCH lagoperator polynomials up to lags Q
and P
, respectively
Mdl
specifies only the functional form of a GARCH model. Because it contains unknown parameter values, you can pass Mdl
and the timeseries data to estimate
to estimate the parameters.
Create a garch
model using namevalue pair arguments.
Specify a GARCH(1,1) model. By default, the conditional mean model offset is zero. Specify that the offset is NaN
.
Mdl = garch('GARCHLags',1,'ARCHLags',1,'Offset',NaN)
Mdl = garch with properties: Description: "GARCH(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] Offset: NaN
Mdl
is a garch
model object. The software sets all parameters (the properties of the model object) to NaN
, except P
, Q
, and Distribution
.
Since Mdl
contains NaN
values, Mdl
is only appropriate for estimation only. Pass Mdl
and timeseries data to estimate
.
Create a GARCH(1,1) model with mean offset,
where
and is an independent and identically distributed standard Gaussian process.
Mdl = garch('Constant',0.0001,'GARCH',0.75,... 'ARCH',0.1,'Offset',0.5)
Mdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 0.0001 GARCH: {0.75} at lag [1] ARCH: {0.1} at lag [1] Offset: 0.5
garch
assigns default values to any properties you do not specify with namevalue pair arguments.
Access the properties of a garch
model object using dot notation.
Create a garch
model object.
Mdl = garch(3,2)
Mdl = garch with properties: Description: "GARCH(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] 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 = garch with properties: Description: "GARCH(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] 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 I.I.D. disturbances have a t distribution with five degrees of freedom.
Mdl.Distribution = struct('Name','t','DoF',5)
Mdl = garch with properties: Description: "GARCH(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] 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 = garch with properties: Description: "GARCH(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] 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 GARCH model to an annual time series of Danish nominal stock returns from 19221999.
Load the Data_Danish
data set. Plot the nominal returns (nr
).
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 a GARCH(1,1) model is appropriate for this series.
Create a GARCH(1,1) model. The conditional mean offset is zero by default. To estimate the offset, specify that it is NaN
.
Mdl = garch('GARCHLags',1,'ARCHLags',1,'Offset',NaN);
Fit the GARCH(1,1) model to the data.
EstMdl = estimate(Mdl,nr);
GARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ _________ Constant 0.0044476 0.007814 0.56918 0.56923 GARCH{1} 0.84932 0.26495 3.2056 0.0013477 ARCH{1} 0.07325 0.14953 0.48986 0.62423 Offset 0.11227 0.039214 2.8629 0.0041974
EstMdl
is a fully specified garch
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 garch
model object. That is, simulate from an estimated garch
model or a known garch
model in which you specify all parameter values.
Load the Data_Danish
data set.
load Data_Danish;
nr = DataTable.RN;
Create a GARCH(1,1) model with an unknown conditional mean offset. Fit the model to the annual nominal return series.
Mdl = garch('GARCHLags',1,'ARCHLags',1,'Offset',NaN); EstMdl = estimate(Mdl,nr);
GARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ _________ Constant 0.0044476 0.007814 0.56918 0.56923 GARCH{1} 0.84932 0.26495 3.2056 0.0013477 ARCH{1} 0.07325 0.14953 0.48986 0.62423 Offset 0.11227 0.039214 2.8629 0.0041974
Simulate 100 paths of conditional variances and responses for each period from the estimated GARCH model.
numObs = numel(nr); % 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.
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 garch
model object. That is, forecast from an estimated garch
model or a known garch
model in which you specify all parameter values. The example follows from Estimate GARCH Model.
Load the Data_Danish
data set.
load Data_Danish;
nr = DataTable.RN;
Create a GARCH(1,1) model with an unknown conditional mean offset, and fit the model to the annual, nominal return series.
Mdl = garch('GARCHLags',1,'ARCHLags',1,'Offset',NaN); EstMdl = estimate(Mdl,nr);
GARCH(1,1) Conditional Variance Model with Offset (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ _________ Constant 0.0044476 0.007814 0.56918 0.56923 GARCH{1} 0.84932 0.26495 3.2056 0.0013477 ARCH{1} 0.07325 0.14953 0.48986 0.62423 Offset 0.11227 0.039214 2.8629 0.0041974
Forecast the conditional variance of the nominal return series 10 years into the future using the estimated GARCH 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,'Y0',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');
A GARCH 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.
A GARCH model posits that the current conditional variance is the sum of these linear processes, with coefficients for each term:
Past conditional variances (the GARCH component or polynomial)
Past squared innovations (the ARCH component or polynomial)
Constant offsets for the innovation mean and conditional variance models
Consider the time series
$${y}_{t}=\mu +{\epsilon}_{t},$$
where $${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$ The GARCH(P,Q) conditional variance process, $${\sigma}_{t}^{2}$$, has the form
$${\sigma}_{t}^{2}=\kappa +{\gamma}_{1}{\sigma}_{t1}^{2}+\dots +{\gamma}_{P}{\sigma}_{tP}^{2}+{\alpha}_{1}{\epsilon}_{t1}^{2}+\dots +{\alpha}_{Q}{\epsilon}_{tQ}^{2}.$$
In lag operator notation, the model is
$$\left(1{\gamma}_{1}L\dots {\gamma}_{P}{L}^{P}\right){\sigma}_{t}^{2}=\kappa +\left({\alpha}_{1}L+\dots +{\alpha}_{Q}{L}^{Q}\right){\epsilon}_{t}^{2}.$$
The table shows how
the variables correspond to the properties of the garch
model
object.
Variable  Description  Property 

μ  Innovation mean model constant offset  'Offset' 
κ > 0  Conditional variance model constant  'Constant' 
$${\gamma}_{i}\ge 0$$  GARCH component coefficients  'GARCH' 
$${\alpha}_{j}\ge 0$$  ARCH component coefficients  'ARCH' 
z_{t}  Series of independent random variables with mean 0 and variance 1  'Distribution' 
For stationarity and positivity, GARCH models use these constraints:
$$\kappa >0$$
$${\gamma}_{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\alpha}_{j}\ge 0$$
$${\sum}_{i=1}^{P}{\gamma}_{i}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}}<1$$
Engle’s original ARCH(Q) model is equivalent to a GARCH(0,Q) specification.
GARCH models are appropriate when positive and negative shocks of equal magnitude contribute equally to volatility [1].
You can specify a garch
model as part of a composition of
conditional mean and variance models. For details, see arima
.
[1] Tsay, R. S. Analysis of Financial Time Series. 3rd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2010.
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