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Fit conditional variance model to data

`EstMdl = estimate(Mdl,y)`

`EstMdl = estimate(Mdl,y,Name,Value)`

```
[EstMdl,EstParamCov,logL,info]
= estimate(___)
```

estimates the unknown parameters of the conditional variance model object
`EstMdl`

= estimate(`Mdl`

,`y`

)`Mdl`

with the observed univariate time series
`y`

, using maximum likelihood. `EstMdl`

is a
fully specified conditional variance model object that stores the results. It is the
same model type as `Mdl`

(see `garch`

, `egarch`

, and `gjr`

).

estimates the conditional variance model with additional options specified by one or
more `EstMdl`

= estimate(`Mdl`

,`y`

,`Name,Value`

)`Name,Value`

pair arguments. For example, you can specify to
display iterative optimization information or presample innovations.

`[`

additionally returns:`EstMdl`

,`EstParamCov`

,`logL`

,`info`

]
= estimate(___)

`EstParamCov`

, the variance-covariance matrix associated with estimated parameters.`logL`

, the optimized loglikelihood objective function.`info`

, a data structure of summary information using any of the input arguments in the previous syntaxes.

Fit a GARCH(1,1) model to simulated data.

Simulate 500 data points from the GARCH(1,1) model

where and

Use the default Gaussian innovation distribution for .

Mdl = garch('Constant',0.0001,'GARCH',0.5,... 'ARCH',0.2); rng default; % For reproducibility [v,y] = simulate(Mdl,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`

.

ToEstMdl = garch(1,1); EstMdl = estimate(ToEstMdl,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} 0.45393 0.11193 4.0557 4.9991e-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)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 9.8911e-05 GARCH: {0.453935} 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.

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).

Mdl = egarch('Constant',0.001,'GARCH',0.7,... 'ARCH',0.5,'Leverage',-0.3); rng default % For reproducibility [v,y] = simulate(Mdl,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`

.

ToEstMdl = egarch(1,1); EstMdl = estimate(ToEstMdl,y)

EGARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant -0.0006387 0.031698 -0.02015 0.98392 GARCH{1} 0.70506 0.067359 10.467 1.222e-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)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: -0.000638702 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.

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 .

Mdl = gjr('Constant',0.001,'GARCH',0.5,... 'ARCH',0.2,'Leverage',0.2); rng default; % For reproducibility [v,y] = simulate(Mdl,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`

.

ToEstMdl = gjr(1,1); EstMdl = estimate(ToEstMdl,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)" 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.

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 1.9986e-06 5.4227e-07 3.6857 0.00022809 GARCH{1} 0.88356 0.0084341 104.76 0 ARCH{1} 0.10903 0.0076471 14.257 4.0403e-46

EstMdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 1.99865e-06 GARCH: {0.883564} at lag [1] ARCH: {0.109026} at lag [1] Offset: 0

EstParamCov =3×310^{-4}× 0.0000 -0.0000 0.0000 -0.0000 0.7113 -0.5343 0.0000 -0.5343 0.5848

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
0.0084
0.0076

These are the standard errors shown in the estimation output display. They correspond (in order) to the constant, GARCH coefficient, and ARCH coefficient.

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 -0.13479 0.022092 -6.101 1.0538e-09 GARCH{1} 0.98391 0.0024221 406.22 0 ARCH{1} 0.19965 0.013966 14.296 2.3323e-46 Leverage{1} -0.060243 0.0056471 -10.668 1.4354e-26

EstMdl = egarch with properties: Description: "EGARCH(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: -0.134785 GARCH: {0.983909} at lag [1] ARCH: {0.199645} at lag [1] Leverage: {-0.0602433} at lag [1] Offset: 0

EstParamCov =4×410^{-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 a GJR(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 GJR(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 = gjr(1,1);
[EstMdl,EstParamCov] = estimate(Mdl,y(2:end),'E0',y(1))
```

GJR(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant 2.4579e-06 5.6867e-07 4.3222 1.545e-05 GARCH{1} 0.88134 0.00949 92.87 0 ARCH{1} 0.064098 0.0091979 6.9688 3.197e-12 Leverage{1} 0.088851 0.0099167 8.9597 3.2551e-19

EstMdl = gjr with properties: Description: "GJR(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 2.4579e-06 GARCH: {0.881336} at lag [1] ARCH: {0.0640985} at lag [1] Leverage: {0.0888512} at lag [1] Offset: 0

EstParamCov =4×410^{-4}× 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.9006 -0.6935 -0.0003 0.0000 -0.6935 0.8460 -0.3606 0.0000 -0.0003 -0.3606 0.9834

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
0.0095
0.0092
0.0099

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.

`Mdl`

— Conditional variance model`garch`

model object | `egarch`

model object | `gjr`

model object`y`

— Single path of response datanumeric column vector

Single path of response data, specified as a numeric column vector. The
software infers the conditional variances from `y`

, i.e.,
the data to which the model is fit.

`y`

is usually an innovation series with mean 0 and
conditional variance characterized by the model specified in
`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 `Offset`

signals the inclusion of
an offset in `Mdl`

.

The last observation of `y`

is the latest
observation.

**Data Types: **`double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as `Name1,Value1,...,NameN,ValueN`

.

`'Display','iter','E0',[0.1; 0.05]`

specifies to display
iterative optimization information, and `[0.05; 0.1]`

as presample
innovations.`'ARCH0'`

— Initial coefficient estimates corresponding to past innovation termsnumeric vector

Initial coefficient estimates corresponding to past innovation terms,
specified as the comma-separated pair consisting of
`'ARCH0'`

and 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`

`'Constant0'`

— Initial conditional variance model constant estimatescalar

Initial conditional variance model constant estimate, specified as the
comma-separated pair consisting of `'Constant0'`

and a
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`

`'Display'`

— Command Window display option`'params'`

(default) | `'diagnostics'`

| `'full'`

| `'iter'`

| `'off'`

| string vector | cell vector of character vectorsCommand Window display option, specified as the comma-separated
pair consisting of `'Display'`

and a value or any
combination of values in this table.

Value | estimate Displays |
---|---|

`'diagnostics'` | Optimization diagnostics |

`'full'` | Maximum likelihood parameter estimates, standard errors, t statistics,
iterative optimization information, and optimization diagnostics |

`'iter'` | Iterative optimization information |

`'off'` | No display in the Command Window |

`'params'` | Maximum likelihood parameter estimates, standard errors, and t statistics |

For example:

To run a simulation where you are fitting many models, and therefore want to suppress all output, use

`'Display','off'`

.To display all estimation results and the optimization diagnostics, use

`'Display',{'params','diagnostics'}`

.

**Data Types: **`char`

| `cell`

| `string`

`'DoF0'`

— Initial `10`

(default) | positive scalarInitial *t*-distribution degrees-of-freedom
parameter estimate, specified as the comma-separated pair consisting
of `'DoF0'`

and a positive scalar. `DoF0`

must
exceed 2.

**Data Types: **`double`

`'E0'`

— Presample innovationsnumeric column vector

Presample innovations, specified as the comma-separated pair
consisting of `'E0'`

and a 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.

`E0`

must contain at least `Mdl.Q`

rows. If `E0`

contains extra rows, then
`estimate`

uses the latest `Mdl.Q`

presample innovations. The last row contains the latest presample
innovation.

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 series`y`

.For EGARCH(

*P*,*Q*) models,`estimate`

sets any necessary presample innovations to zero.

**Data Types: **`double`

`'GARCH0'`

— Initial coefficient estimates for past conditional variance termsnumeric vector

Initial coefficient estimates for past conditional variance terms,
specified as the comma-separated pair consisting of
`'GARCH0'`

and 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`

`'Offset0'`

— Initial innovation mean model offset estimatescalar

Initial innovation mean model offset estimate, specified as the
comma-separated pair consisting of `'Offset0'`

and a
scalar.

By default, `estimate`

sets the initial estimate to
the sample mean of `y`

.

**Data Types: **`double`

`'Options'`

— Optimization options`optimoptions`

optimization controllerOptimization options, specified as the comma-separated pair consisting of
`'Options'`

and an `optimoptions`

optimization
controller. For details on altering 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`

.

`'V0'`

— Presample conditional variancesnumeric column vector with positive entries

Presample conditional variances, specified as the comma-separated pair
consisting of `'V0'`

and numeric column vector with
positive entries. `V0`

provide initial values for
conditional variance process of the conditional variance model
`Mdl`

.

For GARCH(*P*,*Q*) and
GJR(*P*,*Q*) models,
`V0`

must have at least `Mdl.P`

rows.

For EGARCH(*P*,*Q*)
models,`V0`

must have at least
`max(Mdl.P,Mdl.Q)`

rows.

If the number of rows in `V0`

exceeds the necessary
number, only the latest observations are used. The last 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`

`'Leverage0'`

— Initial coefficient estimates past leverage terms`0`

(default) | numeric vectorInitial coefficient estimates past leverage terms, specified as the
comma-separated pair consisting of `'Leverage0'`

and 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`

.

**Data Types: **`double`

`NaN`

s indicate missing values.`estimate`

removes them. The software merges the presample data (`E0`

and`V0`

) separately from the effective sample data (`y`

), and then uses list-wise deletion to remove rows containing at least one`NaN`

. Removing`NaN`

s in the data reduces the sample size, and can also create irregular time series.`estimate`

assumes that you synchronize the presample data such that the latest observations occur simultaneously.If you specify a value for

`Display`

, then it takes precedence over the specifications of the optimization options`Diagnostics`

and`Display`

. Otherwise,`estimate`

honors all selections related to the display of optimization information in the optimization options.If you do not specify

`E0`

and`V0`

, then`estimate`

derives the necessary presample observations from the unconditional, or long-run, variance of the offset-adjusted response process.For all conditional variance models,

`V0`

is the sample average of the squared disturbances of the offset-adjusted response data`y`

.For GARCH(

*P*,*Q*) and GJR(*P*,*Q*) models,`E0`

is the square root of the average squared value of the offset-adjusted response series`y`

.For EGARCH(

*P*,*Q*) models,`E0`

is`0`

.

These specifications minimize initial transient effects.

`EstMdl`

— Conditional variance model containing parameter estimates`garch`

model object | `egarch`

model object | `gjr`

model objectConditional 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 estimatesnumeric 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 valuescalar

Optimized loglikelihood objective function value, returned as a scalar.

**Data Types: **`double`

`info`

— Summary informationstructure array

Summary information, returned as a structure.

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
typing `info.X`

in the Command Window.

**Data Types: **`struct`

Suppose `EstParamCov`

is an estimated parameter
covariance matrix returned by `estimate`

. The software
sets the variances and covariances of parameters fixed during estimation
to `0`

. Enter this command to count the number of
free parameters (`numParams`

) in a fitted model.

numParams = sum(any(EstParamCov))

This command counts the number of columns (or equivalently, rows) with any nonzero values.

[1] Bollerslev, T. “Generalized Autoregressive Conditional
Heteroskedasticity.” *Journal of Econometrics*. Vol. 31,
1986, pp. 307–327.

[2] Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for
Speculative Prices and Rates of Return.” *The Review of Economics and
Statistics*. Vol. 69, 1987, pp. 542–547.

[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, R. F. “Autoregressive Conditional Heteroskedasticity with
Estimates of the Variance of United Kingdom Inflation.”
*Econometrica*. Vol. 50, 1982, pp. 987–1007.

[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, J. D. *Time Series Analysis*. Princeton, NJ:
Princeton University Press, 1994.

- Compare Conditional Variance Models Using Information Criteria
- Likelihood Ratio Test for Conditional Variance Models
- Estimate Conditional Mean and Variance Models
- 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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