# estimate

Fit conditional variance model to data

## Syntax

• EstMdl = estimate(Mdl,y) example
• EstMdl = estimate(Mdl,y,Name,Value) example
• [EstMdl,EstParamCov,logL,info] = estimate(___) example

## Description

example

EstMdl = estimate(Mdl,y) estimates the unknown parameters of the conditional variance model object 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).

example

EstMdl = estimate(Mdl,y,Name,Value) estimates the conditional variance model with additional options specified by one or more Name,Value pair arguments. For example, you can specify to display iterative optimization information or presample innovations.

example

[EstMdl,EstParamCov,logL,info] = estimate(___) additionally returns: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.

## Examples

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### Estimate GARCH Model Parameters Without Initial Values

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: ---------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 9.89111e-05 3.07264e-05 3.21909 GARCH{1} 0.453934 0.111926 4.05567 ARCH{1} 0.263739 0.0569312 4.63259 EstMdl = GARCH(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: 9.89111e-05 GARCH: {0.453934} at Lags [1] ARCH: {0.263739} at Lags [1] 

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).

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: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant -0.000638689 0.0316977 -0.0201494 GARCH{1} 0.705065 0.0673594 10.4672 ARCH{1} 0.567741 0.0747457 7.59563 Leverage{1} -0.321158 0.0533449 -6.0204 EstMdl = EGARCH(1,1) Conditional Variance Model: ----------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: -0.000638689 GARCH: {0.705065} at Lags [1] ARCH: {0.567741} at Lags [1] Leverage: {-0.321158} at Lags [1] 

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 .

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: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 0.000973819 0.000251354 3.87429 GARCH{1} 0.460555 0.0717928 6.41505 ARCH{1} 0.241255 0.0634092 3.80474 Leverage{1} 0.250508 0.112655 2.22368 EstMdl = GJR(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: 0.000973819 GARCH: {0.460555} at Lags [1] ARCH: {0.241255} at Lags [1] Leverage: {0.250508} at Lags [1] 

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: ---------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 1.99864e-06 5.42273e-07 3.68567 GARCH{1} 0.883564 0.00843403 104.762 ARCH{1} 0.109026 0.00764706 14.2573 EstMdl = GARCH(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: 1.99864e-06 GARCH: {0.883564} at Lags [1] ARCH: {0.109026} at Lags [1] EstParamCov = 1.0e-04 * 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 = 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.

### 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: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant -0.134783 0.022092 -6.10101 GARCH{1} 0.983909 0.00242211 406.22 ARCH{1} 0.199644 0.0139654 14.2955 Leverage{1} -0.0602429 0.00564702 -10.6681 EstMdl = EGARCH(1,1) Conditional Variance Model: ----------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: -0.134783 GARCH: {0.983909} at Lags [1] ARCH: {0.199644} at Lags [1] Leverage: {-0.0602429} at Lags [1] EstParamCov = 1.0e-03 * 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 = 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.

### Estimate GJR Model Parameters Using Presample Data

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: -------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 2.45647e-06 5.68527e-07 4.32076 GARCH{1} 0.881379 0.00948646 92.9092 ARCH{1} 0.0640741 0.00919501 6.96836 Leverage{1} 0.0888268 0.0099137 8.96 EstMdl = GJR(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: 2.45647e-06 GARCH: {0.881379} at Lags [1] ARCH: {0.0640741} at Lags [1] Leverage: {0.0888268} at Lags [1] EstParamCov = 1.0e-04 * 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.8999 -0.6930 -0.0002 0.0000 -0.6930 0.8455 -0.3605 0.0000 -0.0002 -0.3605 0.9828 

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 = 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.

## Input Arguments

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### Mdl — Conditional variance modelgarch model object | egarch model object | gjr model object

Conditional variance model containing unknown parameters, specified as a garch, egarch, or gjr model object.

estimate treats non-NaN elements in Mdl as equality constraints, and does not estimate the corresponding parameters.

### 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

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Display','iter','E0',[0.1; 0.05] specifies to display iterative optimization information, and [0.05; 0.1] as presample innovations.

## For GARCH, EGARCH, and GJR Models

### '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' | cell vector of strings

Command Window display option, specified as the comma-separated pair consisting of 'Display' and a string or cell vector of strings.

Set Display using any combination of values in this table.

Valueestimate 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

### 'DoF0' — Initial t-distribution degrees-of-freedom parameter estimate10 (default) | positive scalar

Initial 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 optionsoptimoptions optimization controller | optimset optimization controller

Optimization options, specified as the comma-separated pair consisting of 'Options' and an optimoptions or optimset optimization controller. For details on altering the default values of the optimizer, see optimoptions, optimset, or fmincon in Optimization Toolbox™.

Suppose that you want to change the constraint tolerance to 1e-6. Set Options = optimoptions(@fmincon,'TolCon',1e-6,'Algorithm','sqp'), and then pass Options into estimate using 'Options',Options.

By default, estimate uses the same default options as fmincon, except Algorithm = sqp and TolCon = 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

## For EGARCH and GJR Models

### 'Leverage0' — Initial coefficient estimates past leverage terms0 (default) | numeric vector

Initial 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

 Notes   NaNs 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 NaNs 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.

## Output Arguments

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### EstMdl — Conditional variance model containing parameter estimatesgarch 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 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 0s.

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.

FieldDescription
exitflagOptimization exit flag (see fmincon in Optimization Toolbox)
optionsOptimization options controller (see optimoptions and fmincon in Optimization Toolbox)
XVector of final parameter estimates
X0Vector 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

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### Tips

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.

## References

[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.