estimate
Fit conditional variance model to data
Syntax
Description
returns the fully specified, estimated conditional variance model
EstMdl
= estimate(Mdl
,y
)EstMdl
. This model stores the estimated parameter values
resulting from fitting the partially specified conditional variance model
Mdl
to the observed univariate time series
y
by using maximum likelihood.
EstMdl
and Mdl
are the same model
type and have the same structure (see garch
, egarch
, and gjr
).
fits the partially specified conditional variance model EstMdl
= estimate(Mdl
,Tbl1
)Mdl
to response variable in the input table or timetable Tbl1
,
which contains time series data, and returns the fully specified, estimated
conditional variance model EstMdl
.
estimate
selects the response variable named in
Mdl.SeriesName
or the sole variable in
Tbl1
. To select a different response variable in
Tbl1
to fit the model to, use the
ResponseVariable
name-value argument. (since R2023a)
[___] = estimate(___,
specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
Name,Value
)estimate
returns the output argument combination for the
corresponding input arguments. For example, estimate(Mdl,y,Y0=y0)
fits the
conditional variance model Mdl
to the vector of response data
y
, and specifies the vector of presample response data
y0
.
Supply all input data using the same data type. Specifically:
If you specify the numeric vector
y
, optional data sets must be numeric arrays and you must use the appropriate name-value argument. For example, to specify a presample, set theY0
name-value argument to a numeric matrix of presample data.If you specify the table or timetable
Tbl1
, optional data sets must be tables or timetables, respectively, and you must use the appropriate name-value argument. For example, to specify a presample, set thePresample
name-value argument to a table or timetable of presample data.
Examples
Fit GARCH Model to Vector of Response Data
Fit a GARCH(1,1) model to a simulated vector of data.
Simulate 500 data points from the GARCH(1,1) model
where and
Use the default Gaussian innovation distribution for .
Mdl0 = garch(Constant=0.0001,GARCH=0.5,ARCH=0.2); rng("default") % For reproducibility [v,y] = simulate(Mdl0,500);
The output v
contains simulated conditional variances. y
is a column vector of simulated responses (innovations).
Specify a GARCH(1,1) model with unknown coefficients, and fit it to the series y
.
Mdl = garch(1,1); EstMdl = estimate(Mdl,y)
GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 9.8911e-05 3.0726e-05 3.2191 0.001286 GARCH{1} 0.45394 0.11193 4.0557 4.9986e-05 ARCH{1} 0.26374 0.056931 4.6326 3.6111e-06
EstMdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 9.89107e-05 GARCH: {0.453936} at lag [1] ARCH: {0.263739} at lag [1] Offset: 0
The result is a new garch
model called EstMdl
. The parameter estimates in EstMdl
resemble the parameter values that generated the simulated data.
Estimate EGARCH Model Parameters Without Initial Values
Fit an EGARCH(1,1) model to simulated data.
Simulate 500 data points from an EGARCH(1,1) model
where and
(the distribution of is Gaussian).
Mdl0 = egarch(Constant=0.001,GARCH=0.7, ... ARCH=0.5,Leverage=-0.3); rng("default") % For reproducibility [v,y] = simulate(Mdl0,500);
The output v
contains simulated conditional variances. y
is a column vector of simulated responses (innovations).
Specify an EGARCH(1,1) model with unknown coefficients, and fit it to the series y
.
Mdl = egarch(1,1); EstMdl = estimate(Mdl,y)
EGARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue ___________ _____________ __________ __________ Constant -0.00063866 0.031698 -0.020148 0.98392 GARCH{1} 0.70506 0.067359 10.467 1.2221e-25 ARCH{1} 0.56774 0.074746 7.5956 3.063e-14 Leverage{1} -0.32116 0.053345 -6.0204 1.7399e-09
EstMdl = egarch with properties: Description: "EGARCH(1,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: -0.000638661 GARCH: {0.705065} at lag [1] ARCH: {0.567741} at lag [1] Leverage: {-0.321158} at lag [1] Offset: 0
The result is a new egarch
model called EstMdl
. The parameter estimates in EstMdl
resemble the parameter values that generated the simulated data.
Estimate GJR Model Parameters Without Initial Values
Fit a GJR(1,1) model to simulated data.
Simulate 500 data points from a GJR(1,1) model.
where and
Use the default Gaussian innovation distribution for .
Mdl0 = gjr(Constant=0.001,GARCH=0.5, ... ARCH=0.2,Leverage=0.2); rng("default") % For reproducibility [v,y] = simulate(Mdl0,500);
The output v
contains simulated conditional variances. y
is a column vector of simulated responses (innovations).
Specify a GJR(1,1) model with unknown coefficients, and fit it to the series y
.
Mdl = gjr(1,1); EstMdl = estimate(Mdl,y)
GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 0.00097382 0.00025135 3.8743 0.00010694 GARCH{1} 0.46056 0.071793 6.4151 1.4077e-10 ARCH{1} 0.24126 0.063409 3.8047 0.00014196 Leverage{1} 0.25051 0.11265 2.2237 0.026171
EstMdl = gjr with properties: Description: "GJR(1,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 0.000973819 GARCH: {0.460555} at lag [1] ARCH: {0.241256} at lag [1] Leverage: {0.250507} at lag [1] Offset: 0
The result is a new gjr
model called EstMdl
. The parameter estimates in EstMdl
resemble the parameter values that generated the simulated data.
Estimate GARCH Model Parameters Using Presample Data
Fit a GARCH(1,1) model to the daily close NASDAQ Composite Index returns.
Load the NASDAQ data included with the toolbox. Convert the index to returns.
load Data_EquityIdx nasdaq = DataTable.NASDAQ; y = price2ret(nasdaq); T = length(y); figure plot(y) xlim([0,T]) title("NASDAQ Returns")
The returns exhibit volatility clustering.
Specify a GARCH(1,1) model, and fit it to the series. One presample innovation is required to initialize this model. Use the first observation of y
as the necessary presample innovation.
Mdl = garch(1,1); [EstMdl,EstParamCov] = estimate(Mdl,y(2:end),E0=y(1))
GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 2.0419e-06 5.4779e-07 3.7275 0.00019337 GARCH{1} 0.88074 0.0086252 102.11 0 ARCH{1} 0.11193 0.0078755 14.212 7.6874e-46
EstMdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 2.04192e-06 GARCH: {0.880735} at lag [1] ARCH: {0.111929} at lag [1] Offset: 0
EstParamCov = 3×3
10-4 ×
0.0000 -0.0000 0.0000
-0.0000 0.7439 -0.5640
0.0000 -0.5640 0.6202
The output EstMdl
is a new garch
model with estimated parameters.
Use the output variance-covariance matrix to calculate the estimate standard errors.
se = sqrt(diag(EstParamCov))
se = 3×1
0.0000
0.0086
0.0079
These are the standard errors shown in the estimation output display. They correspond (in order) to the constant, GARCH coefficient, and ARCH coefficient.
Estimate EGARCH Model Parameters Using Presample Data
Fit an EGARCH(1,1) model to the daily close NASDAQ Composite Index returns.
Load the NASDAQ data included with the toolbox. Convert the index to returns.
load Data_EquityIdx nasdaq = DataTable.NASDAQ; y = price2ret(nasdaq); T = length(y); figure plot(y) xlim([0,T]) title("NASDAQ Returns")
The returns exhibit volatility clustering.
Specify an EGARCH(1,1) model, and fit it to the series. One presample innovation is required to initialize this model. Use the first observation of y
as the necessary presample innovation.
Mdl = egarch(1,1); [EstMdl,EstParamCov] = estimate(Mdl,y(2:end),E0=y(1))
EGARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue _________ _____________ __________ __________ Constant -0.13478 0.022092 -6.101 1.0539e-09 GARCH{1} 0.98391 0.0024221 406.22 0 ARCH{1} 0.19964 0.013966 14.296 2.3323e-46 Leverage{1} -0.060243 0.005647 -10.668 1.4356e-26
EstMdl = egarch with properties: Description: "EGARCH(1,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: -0.134785 GARCH: {0.983909} at lag [1] ARCH: {0.199645} at lag [1] Leverage: {-0.0602432} at lag [1] Offset: 0
EstParamCov = 4×4
10-3 ×
0.4881 0.0533 -0.1018 0.0106
0.0533 0.0059 -0.0118 0.0017
-0.1018 -0.0118 0.1950 0.0016
0.0106 0.0017 0.0016 0.0319
The output EstMdl
is a new egarch
model with estimated parameters.
Use the output variance-covariance matrix to calculate the estimate standard errors.
se = sqrt(diag(EstParamCov))
se = 4×1
0.0221
0.0024
0.0140
0.0056
These are the standard errors shown in the estimation output display. They correspond (in order) to the constant, GARCH coefficient, ARCH coefficient, and leverage coefficient.
Fit GARCH Model to Response Variable in Timetable
Since R2023a
Fit a GARCH(1,1) model to the average weekly closing NASDAQ returns. Supply a timetable of data and specify the series for the fit.
Load the U.S. equity indices data Data_EquityIdx.mat
.
load Data_EquityIdx
The timetable DataTimeTable
contains the daily NASDAQ closing prices, among other indices.
Compute the weekly average closing prices of all timetable variables.
DTTW = convert2weekly(DataTimeTable,Aggregation="mean");
Compute the weekly returns.
DTTRet = price2ret(DTTW); T = height(DTTRet)
T = 626
Plot the weekly NASDAQ returns.
figure
plot(DTTRet.Time,DTTRet.NASDAQ)
title("NASDAQ Weekly Returns")
The returns exhibit volatility clustering.
When you plan to supply a timetable, you must ensure it has all the following characteristics:
The selected response variable is numeric and does not contain any missing values.
The timestamps in the
Time
variable are regular, and they are ascending or descending.
Remove all missing values from the timetable, relative to the NASDAQ returns series.
DTTRet = rmmissing(DTTRet,DataVariables="NASDAQ");
numobs = height(DTTRet)
numobs = 626
Because all sample times have observed NASDAQ returns, rmmissing
does not remove any observations.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTTRet,"weeks")
areTimestampsRegular = logical
1
areTimestampsSorted = issorted(DTTRet.Time)
areTimestampsSorted = logical
1
areTimestampsRegular = 1
indicates that the timestamps of DTTRet
represent a regular weekly sample. areTimestampsSorted = 1
indicates that the timestamps are sorted.
Specify a GARCH(1,1) model, and fit it to the series. Specify the entire timetable of returns and select the variable containing the NASDAQ returns.
Mdl = garch(1,1);
[EstMdl,EstParamCov] = estimate(Mdl,DTTRet,ResponseVariable="NASDAQ")
GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 1.7406e-06 8.9077e-07 1.9541 0.050694 GARCH{1} 0.65947 0.059314 11.118 1.0229e-28 ARCH{1} 0.33773 0.079595 4.2431 2.2044e-05
EstMdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 1.74062e-06 GARCH: {0.659467} at lag [1] ARCH: {0.337731} at lag [1] Offset: 0
EstParamCov = 3×3
0.0000 -0.0000 0.0000
-0.0000 0.0035 -0.0039
0.0000 -0.0039 0.0063
Specify Presample Data In Timetable
Since R2023a
Fit a GJR(1,1) model to the average weekly closing NASDAQ returns. Specify in-sample and presample data in a timetable.
Load the U.S. equity indices data Data_EquityIdx.mat
.
load Data_EquityIdx
The timetable DataTimeTable
contains the daily NASDAQ closing prices, among other indices.
Compute the weekly average closing prices of all timetable variables.
DTTW = convert2weekly(DataTimeTable,Aggregation="mean");
Compute the weekly returns.
DTTRet = price2ret(DTTW); T = height(DTTRet)
T = 626
Plot the weekly NASDAQ returns.
figure
plot(DTTRet.Time,DTTRet.NASDAQ)
title("NASDAQ Weekly Returns")
The returns exhibit volatility clustering.
When you plan to supply a timetable, you must ensure it has all the following characteristics:
The selected response variable is numeric and does not contain any missing values.
The timestamps in the
Time
variable are regular, and they are ascending or descending.
Remove all missing values from the timetable, relative to the NASDAQ returns series.
DTTRet = rmmissing(DTTRet,DataVariables="NASDAQ");
numobs = height(DTTRet)
numobs = 626
Because all sample times have observed NASDAQ returns, rmmissing
does not remove any observations.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTTRet,"weeks")
areTimestampsRegular = logical
1
areTimestampsSorted = issorted(DTTRet.Time)
areTimestampsSorted = logical
1
areTimestampsRegular = 1
indicates that the timestamps of DTTRet
represent a regular weekly sample. areTimestampsSorted = 1
indicates that the timestamps are sorted.
Because , one presample innovation is required to initialize this model. Partition the timetable of data into sets containing required presample and insample observations.
numpreobs = 1; DTTRetPresample = DTTRet(1:numpreobs,:); DTTRetInsample = DTTRet((numpreobs+1):end,:);
Specify a GJR(1,1) model, and fit it to the series. Specify the timetables of NASDAQ weekly returns for the required presample innovations and in-sample data.
Mdl = gjr(1,1); [EstMdl,EstParamCov] = estimate(Mdl,DTTRetInsample,Presample=DTTRetPresample, ... ResponseVariable="NASDAQ",PresampleInnovationVariable="NASDAQ");
GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ ___________ Constant 7.7296e-07 5.3835e-07 1.4358 0.15106 GARCH{1} 0.77143 0.033165 23.26 1.1149e-119 ARCH{1} 0.092918 0.033665 2.7601 0.0057785 Leverage{1} 0.19222 0.050339 3.8185 0.00013428
The output EstMdl
is a new gjr
model with estimated parameters.
Use the output variance-covariance matrix to calculate the estimate standard errors.
se = sqrt(diag(EstParamCov))
se = 4×1
0.0000
0.0332
0.0337
0.0503
These are the standard errors shown in the estimation output display. They correspond (in order) to the constant, GARCH coefficient, ARCH coefficient, and leverage coefficient.
Input Arguments
Mdl
— Partially specified conditional variance model
garch
model object | egarch
model object | gjr
model object
Partially specified conditional variance model containing unknown
parameters, specified as a garch
, egarch
, or gjr
model object. Properties of
Mdl
describe the model structure and can specify
parameter values.
estimate
fits unspecified
(NaN
-valued) parameters to the data
y
.
estimate
treats specified parameters as equality
constraints during estimation.
y
— Single path of observed response data yt
numeric column vector
Single path of observed response data
yt, to which the conditional
variance model Mdl
is fit, specified as a
numobs
-by-1 numeric column vector.
y
represents numobs
observations
of a univariate time series.
y
is usually a mean 0 series of innovations or
residuals and conditional variance characterized by the conditional variance
model Mdl
. In this case, y
is a
continuation of the innovation series E0
.
y
can also represent an innovation series with mean 0
plus an offset. A nonzero value of Mdl.Offset
signals the
inclusion of an offset in Mdl
.
The last observation of y
is the latest
observation.
Data Types: double
Tbl1
— Time series data
table | timetable
Since R2023a
Time series data containing response variable
yt, to which
estimate
fits the conditional variance model
Mdl
, specified as a table or timetable with
numvars
variables and numobs
rows.
You can optionally select a response variable by using the
ResponseVariable
name-value argument.
The selected response variable in Tbl1
is a numeric
vector representing a single path of numobs
observations.
Each row is an observation, and measurements in each row occur simultaneously.
If Tbl1
is a timetable, it must represent a sample
with a regular datetime time step (see isregular
), and the datetime
vector Tbl1.Time
must be strictly ascending or
descending.
If Tbl1
is a table, the last row contains the latest
observation.
Name-Value Arguments
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.
Example: Display="iter",E0=[0.1; 0.05]
specifies to display
iterative optimization information, and [0.05; 0.1]
as presample
innovations.
ResponseVariable
— Variable to select from Tbl1
to treat as response variable yt
string scalar | character vector | integer | logical vector
Since R2023a
Variable to select from Tbl1
to treat as the response variable
yt, specified as one of the following
data types:
String scalar or character vector containing a variable name in
Tbl1.Properties.VariableNames
Variable index (integer) to select from
Tbl1.Properties.VariableNames
A length
numvars
logical vector, whereResponseVariable(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
, andsum(ResponseVariable)
is1
The selected variable must be a numeric vector and cannot contain missing values (NaN
).
If Tbl1
has one variable, the default specifies that variable. Otherwise, the default matches the variable to name in Mdl.SeriesName
.
Example: ResponseVariable="StockRate2"
Example: ResponseVariable=[false false true false]
or ResponseVariable=3
selects the third table variable as the response variable.
Data Types: double
| logical
| char
| cell
| string
E0
— Presample innovations
numeric column vector
Presample innovations εt,
specified as a numpreobs
-by-1 numeric column vector.
The presample innovations provide initial values for the innovations
process of the conditional variance model Mdl
. The
presample innovations derive from a distribution with mean 0. Use
E0
only when you supply the vector of response
data y
.
numpreobs
is the number of presample observations.
Each row is a presample observation. The last row contains the latest
presample observation. numpreobs
must be at least
Mdl.Q
. If numpreobs
>
Mdl.Q
, estimate
uses the
latest required number of observations only. The last element or row
contains the latest observation.
The defaults are:
For GARCH(P,Q) and GJR(P,Q) models,
estimate
sets any necessary presample innovations to the square root of the average squared value of the offset-adjusted response seriesy
.For EGARCH(P,Q) models,
estimate
sets any necessary presample innovations to zero.
Data Types: double
V0
— Presample conditional variances
positive numeric column vector
Presample conditional variances
σt2,
specified as a numpreobs
-by-1 positive column vector.
V0
provides initial values for conditional
variance process of the conditional variance model
Mdl
. Use V0
only when you
supply the vector of response data y
.
Each row is a presample observation. The last row contains the latest presample observation.
For GARCH(P,Q) and GJR(P,Q) models,
numpreobs
must be at leastMdl.P
.For EGARCH(P,Q) models,
numpreobs
must be at leastmax([Mdl.P Mdl.Q])
.
If numpreobs
exceeds the minimum
number, estimate
uses only the latest
observations. The last element or row contains the latest
observation.
By default, estimate
sets the necessary
presample conditional variances to the average squared value of the
offset-adjusted response series y
.
Data Types: double
Presample
— Presample data
table | timetable
Since R2023a
Presample data for innovations
εt or conditional
variances
σt2
to initialize the model for estimation, specified as a table or
timetable, the same type as Tbl1
, with
numprevars
variables and
numpreobs
rows. Use
Presample
only when you supply a table or
timetable of data Tbl1
.
Each selected variable is a single path of
numpreobs
observations representing the presample
of innovations or conditional variances for the selected response
variable in Tbl1
.
Each row is a presample observation, and measurements in each row
occur simultaneously. numpreobs
must be one of the
following values:
Mdl.P
whenPresample
provides only presample conditional variancesMdl.Q
whenPresample
provides only presample innovationsmax([Mdl.P Mdl.Q])
whenPresample
provides presample innovations and conditional variances
If you supply more rows than necessary,
estimate
uses the latest required number of
observations only.
If Presample
is a timetable, all the following
conditions must be true:
Presample
must represent a sample with a regular datetime time step (seeisregular
).The inputs
Tbl1
andPresample
must be consistent in time such thatPresample
immediately precedesTbl1
with respect to the sampling frequency and order.The datetime vector of sample timestamps
Presample.Time
must be ascending or descending.
If Presample
is a table, the last row contains
the latest presample observation.
By default:
For GARCH(P,Q) and GJR(P,Q) models,
estimate
sets any necessary presample innovations to the square root of the average squared value of the offset-adjusted response seriesy
.For EGARCH(P,Q) models,
estimate
sets any necessary presample innovations to zero.estimate
sets the necessary presample conditional variances to the average squared value of the offset-adjusted response series.
If you specify the Presample
, you must specify
the presample innovations or conditional variance variable names by
using the PresampleInnovationVariable
or
PresampleVarianceVariable
name-value
argument.
PresampleInnovationVariable
— Variable of Presample
containing data for the presample innovations εt
string scalar | character vector | integer | logical vector
Since R2023a
Variable of Presample
containing data for the
presample innovations εt,
specified as one of the following data types:
String scalar or character vector containing the variable name to select from
Presample.Properties.VariableNames
Variable index (positive integer) to select from
Presample.Properties.VariableNames
A logical vector, where
PresampleInnovationVariable(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain
missing values (NaN
s).
If you specify presample innovations data by using the
Presample
name-value argument, you must specify
PresampleInnovationVariable
.
Example: PresampleInnovationVariable="StockRateInnov0"
Example: PresampleInnovationVariable=[false false true
false]
or PresampleInnovationVariable=3
selects the third table variable as the presample innovations
variable.
Data Types: double
| logical
| char
| cell
| string
PresampleVarianceVariable
— Variable of Presample
containing data for the presample conditional variances σt2
string scalar | character vector | integer | logical vector
Since R2023a
Variable of Presample
containing data for the presample conditional
variances
σt2,
specified as one of the following data types:
String scalar or character vector containing a variable name in
Presample.Properties.VariableNames
Variable index (positive integer) to select from
Presample.Properties.VariableNames
A logical vector, where
PresampleVarianceVariable(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values
(NaN
s).
If you specify presample conditional variance data by using the Presample
name-value argument, you must specify PresampleVarianceVariable
.
Example: PresampleVarianceVariable="StockRateVar0"
Example: PresampleVarianceVariable=[false false true false]
or PresampleVarianceVariable=3
selects the third table variable as the presample conditional variance variable.
Data Types: double
| logical
| char
| cell
| string
Options
— Optimization options
optimoptions
optimization controller
Optimization options, specified as an optimoptions
optimization
controller. For details on modifying the default values of the optimizer, see optimoptions
or fmincon
in Optimization Toolbox™.
For example, to change the constraint tolerance to 1e-6
, set
options =
optimoptions(@fmincon,ConstraintTolerance=1e-6,Algorithm="sqp")
. Then,
pass Options
into estimate
using
Options=options
.
By default, estimate
uses the same default options as
fmincon
, except Algorithm
is
"sqp"
and ConstraintTolerance
is
1e-7
.
Constant0
— Initial conditional variance model constant estimate
numeric scalar
Initial conditional variance model constant estimate, specified as a numeric scalar.
For GARCH(P,Q) and
GJR(P,Q) models,
Constant0
must be a positive scalar.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
GARCH0
— Initial coefficient estimates for past conditional variance terms
numeric vector
Initial coefficient estimates for past conditional variance terms, specified as a numeric vector.
For GARCH(P,Q) and GJR(P,Q) models:
GARCH0
must be a numeric vector containing nonnegative elements.GARCH0
contains the initial coefficient estimates associated with the past conditional variance terms that compose the GARCH polynomial.
For EGARCH(P,Q) models,
GARCH0
contains the initial coefficient estimates associated with past log conditional variance terms that compose the GARCH polynomial.
The number of coefficients in GARCH0
must equal the
number of lags associated with nonzero coefficients in the GARCH
polynomial, as specified in the GARCHLags
property of
Mdl
.
By default, estimate
derives initial estimates
using standard time series techniques.
Data Types: double
ARCH0
— Initial coefficient estimates corresponding to past innovation terms
numeric vector
Initial coefficient estimates corresponding to past innovation terms, specified as a numeric vector.
For GARCH(P,Q) and GJR(P,Q) models:
ARCH0
must be a numeric vector containing nonnegative elements.ARCH0
contains the initial coefficient estimates associated with the past squared innovation terms that compose the ARCH polynomial.By default,
estimate
derives initial estimates using standard time series techniques.
For EGARCH(P,Q) models:
ARCH0
contains the initial coefficient estimates associated with the magnitude of the past standardized innovations that compose the ARCH polynomial.By default,
estimate
sets the initial coefficient estimate associated with the first nonzero lag in the model to a small positive value. All other values are zero.
The number of coefficients in ARCH0
must equal the
number of lags associated with nonzero coefficients in the ARCH
polynomial, as specified in the ARCHLags
property of
Mdl
.
Data Types: double
Offset0
— Initial innovation mean model offset estimate
scalar
Initial innovation mean model offset estimate, specified as a scalar.
By default, estimate
sets the initial estimate
to the sample mean of y
.
Data Types: double
DoF0
— Initial estimate of t-distribution degrees-of-freedom parameter
10
(default) | positive scalar
Initial estimate of the t-distribution degrees-of-freedom parameter
ν, specified as a positive scalar. DoF0
must
exceed 2.
Data Types: double
Display
— Command Window display option
"params"
(default) | "diagnostics"
| "full'"
| "iter"
| "off"
| string vector | cell vector of character vectors
Command Window display option, specified as one or more of the values in this table.
Value | Information Displayed |
---|---|
"diagnostics" | Optimization diagnostics |
"full" | Maximum likelihood parameter estimates, standard errors, t statistics, iterative optimization information, and optimization diagnostics |
"iter" | Iterative optimization information |
"off" | None |
"params" | Maximum likelihood parameter estimates, standard errors, and t statistics and p-values of coefficient significance tests |
Example: Display="off"
is well suited for running a simulation that
estimates many models.
Example: Display=["params" "diagnostics"]
displays all estimation
results and the optimization diagnostics.
Data Types: char
| cell
| string
Leverage0
— Initial coefficient estimates past leverage terms ξ
numeric vector
Initial coefficient estimates past leverage terms, specified as a numeric vector.
For EGARCH(P,Q) models,
Leverage0
contains the initial coefficient
estimates associated with past standardized innovation terms that
compose the leverage polynomial.
For GJR(P,Q) models,
Leverage0
contains the initial coefficient
estimates associated with past, squared, negative innovations that
compose the leverage polynomial.
The number of coefficients in Leverage0
must equal
the number of lags associated with nonzero coefficients in the leverage
polynomial (Leverage
), as specified in
LeverageLags
.
By default, estimate
sets the initial estimate
of the coefficient associated with the first non-zero lag in the model
to a small negative value, and it sets all others initial coefficients
to zero.
Data Types: double
Note
NaN
values inY
,E0
, andV0
indicate missing values.estimate
removes missing values from specified data by list-wise deletion.For the presample,
estimate
horizontally concatenatesE0
andV0
, and then it removes any row of the concatenated matrix containing at least oneNaN
.For the estimation sample,
estimate
removes any row ofY
containing at least oneNaN
.
This type of data reduction reduces the effective sample size and can create an irregular time series.
For numeric data inputs,
estimate
assumes that you synchronize the presample data such that the latest observations occur simultaneously.estimate
issues an error when any table or timetable input contains missing values.
Output Arguments
EstMdl
— Conditional variance model containing parameter estimates
garch
model object | egarch
model object | gjr
model object
Conditional variance model containing parameter estimates, returned as a
garch
, egarch
, or gjr
model object.
estimate
uses maximum likelihood to calculate all
parameter estimates not constrained by Mdl
(i.e.,
constrained parameters have known values).
EstMdl
is a fully specified conditional variance model.
To infer conditional variances for diagnostic checking, pass
EstMdl
to infer
. To simulate or forecast
conditional variances, pass EstMdl
to simulate
or forecast
, respectively.
EstParamCov
— Variance-covariance matrix of maximum likelihood estimates
numeric matrix
Variance-covariance matrix of maximum likelihood estimates of model parameters known to the optimizer, returned as a numeric matrix.
The rows and columns associated with any parameters estimated by maximum likelihood contain the covariances of estimation error. The standard errors of the parameter estimates are the square root of the entries along the main diagonal.
The rows and columns associated with any parameters that are held fixed as
equality constraints contain 0
s.
estimate
uses the outer product of
gradients (OPG) method to perform covariance matrix
estimation.
estimate
orders the parameters in
EstParamCov
as follows:
Constant
Nonzero GARCH coefficients at positive lags
Nonzero ARCH coefficients at positive lags
For EGARCH and GJR models, nonzero leverage coefficients at positive lags
Degrees of freedom (t innovation distribution only)
Offset (models with nonzero offset only)
Data Types: double
logL
— Optimized loglikelihood objective function value
scalar
Optimized loglikelihood objective function value, returned as a scalar.
Data Types: double
info
— Optimization summary
structure array
Optimization summary, returned as a structure array with the fields described in this table.
Field | Description |
---|---|
exitflag | Optimization exit flag (see fmincon in Optimization Toolbox) |
options | Optimization options controller (see optimoptions and fmincon in Optimization Toolbox) |
X | Vector of final parameter estimates |
X0 | Vector of initial parameter estimates |
For example, you can display the vector of final estimates by entering info.X
in the Command Window.
Data Types: struct
Tips
Algorithms
If you do not specify the presample data (
E0
andV0
, orPresample
),estimate
derives the necessary presample observations from the unconditional, or long-run, variance of the offset-adjusted response process.For all conditional variance models, presample conditional variances are the sample average of the squared disturbances of the offset-adjusted response data.
For GARCH(P,Q) and GJR(P,Q) models, presample innovations are the square root of the average squared value of the offset-adjusted response data.
For EGARCH(P,Q) models, presample innovations are
0
.
These specifications minimize initial transient effects.
If you specify a value for the
Display
name-value argument, it takes precedence over the specifications of the optimization optionsDiagnostics
andDisplay
. Otherwise,estimate
honors all selections related to the display of optimization information in the optimization options.
References
[1] Bollerslev, Tim. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (April 1986): 307–27. https://doi.org/10.1016/0304-4076(86)90063-1.
[2] Bollerslev, Tim. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics 69 (August 1987): 542–47. https://doi.org/10.2307/1925546.
[3] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[4] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, 1995.
[5] Engle, Robert. F. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (July 1982): 987–1007. https://doi.org/10.2307/1912773.
[6] 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.
[7] Greene, W. H. Econometric Analysis. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1997.
[8] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
Version History
Introduced in R2012aR2023a: estimate
accepts input data in tables and timetables
In addition to accepting input data (in-sample and presample data) in numeric
arrays, estimate
accepts input data in tables or regular
timetables. When you supply data in a table or timetable,
estimate
chooses the default series on which to operate,
but you can use the specified optional name-value argument to select a different
series.
Name-value arguments to support tabular workflows include:
ResponseVariable
specifies the variable name of the response series in the input dataTbl1
, to which the model is fit.Presample
specifies the input table or timetable of presample innovations and conditional variance data.PresampleInnovationVariable
specifies the variable name of the innovations series to select fromPresample
.PresampleVarianceVariable
specifies the variable name of the conditional variance series to select fromPresample
.
R2019b: estimate
includes the final lag in all estimated univariate time series model polynomials
estimate
includes the final polynomial lag as specified in the input model template for estimation. In other words, the specified polynomial degrees of an input model template returned by an object creation function and the corresponding polynomial degrees of the estimated model returned by estimate are equal.
Before R2019b, estimate
removed trailing lags estimated below the tolerance of 1e-12
.
Polynomial degrees require minimum presample observations for operations downstream of estimation, such as model forecasting and simulation. If a model template in your code does not describe the data generating process well, then the polynomials in the estimated model can have higher degrees than in previous releases. Consequently, you must supply additional presample responses for operations on the estimated model; otherwise, the function issues an error. For more details, see the Y0
name-value argument.
See Also
Objects
Functions
Topics
- Compare Conditional Variance Models Using Information Criteria
- Likelihood Ratio Test for Conditional Variance Models
- Estimate Conditional Mean and Variance Model
- Model Exchange Rate Volatility
- Maximum Likelihood Estimation for Conditional Variance Models
- Conditional Variance Model Estimation with Equality Constraints
- Presample Data for Conditional Variance Model Estimation
- Initial Values for Conditional Variance Model Estimation
- Optimization Settings for Conditional Variance Model Estimation
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