estimate

Fit conditional variance model to data

Syntax

EstMdl = estimate(Mdl,y)

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

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

Description

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).

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.

[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

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

$$y_t={\epsilon }_t,$$

where $${\epsilon }_t={\sigma }_t{z}_t$$ and

$${\sigma }_t^2=0.0001+0.5{\sigma }_{t-1}^2+0.2{\epsilon }_{t-1}^2.$$

Use the default Gaussian innovation distribution for $$z_t$$.

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      4.9988e-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: 9.89108e-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.

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

$$y_t={\epsilon }_t,$$

where $${\epsilon }_t={\sigma }_t{z}_t,$$ and

$$\log {\sigma }_t^2=0.001+0.7\log {\sigma }_{t-1}^2+0.5\left[\frac{\left|{\epsilon }_{t-1}\right|}{{\sigma }_{t-1}}-\sqrt{\frac{2}{\pi }}\right]-0.3\left(\frac{{\epsilon }_{t-1}}{{\sigma }_{t-1}}\right)$$

(the distribution of $$z_t$$ 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       -0.00063867      0.031698       -0.020149        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)"
    Distribution: Name = "Gaussian"
               P: 1
               Q: 1
        Constant: -0.000638665
           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.

$$y_t={\epsilon }_t,$$

where $${\epsilon }_t={\sigma }_t{z}_t$$ and

$${\sigma }_t^2=0.001+0.5{\sigma }_{t-1}^2+0.2{\epsilon }_{t-1}^2+0.2I\left[{\epsilon }_{t-1}<0\right]{\epsilon }_{t-1}^2.$$

Use the default Gaussian innovation distribution for $$z_t$$.

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         0.02617

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.

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    1.9987e-06     5.4228e-07        3.6857      0.00022807
    GARCH{1}       0.88356      0.0084341        104.76               0
    ARCH{1}        0.10903      0.0076472        14.257       4.041e-46

EstMdl = 
  garch with properties:

     Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 1
               Q: 1
        Constant: 1.99867e-06
           GARCH: {0.883563} at lag [1]
            ARCH: {0.109027} at lag [1]
          Offset: 0

EstParamCov = 3×3
10-4 ×

    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.

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        -0.13479       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 = 4×4
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.

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 (Gaussian Distribution):
 
                     Value       StandardError    TStatistic      PValue  
                   __________    _____________    __________    __________

    Constant       2.4571e-06     5.6866e-07        4.3209      1.5542e-05
    GARCH{1}          0.88132       0.009492        92.849               0
    ARCH{1}          0.064153      0.0092048        6.9695      3.1798e-12
    Leverage{1}      0.088795      0.0099197        8.9514      3.5108e-19

EstMdl = 
  gjr with properties:

     Description: "GJR(1,1) Conditional Variance Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 1
               Q: 1
        Constant: 2.45712e-06
           GARCH: {0.881321} at lag [1]
            ARCH: {0.0641529} at lag [1]
        Leverage: {0.0887949} at lag [1]
          Offset: 0

EstParamCov = 4×4
10-4 ×

    0.0000   -0.0000    0.0000    0.0000
   -0.0000    0.9010   -0.6942    0.0002
    0.0000   -0.6942    0.8473   -0.3615
    0.0000    0.0002   -0.3615    0.9840

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.

Input Arguments

Mdl — Conditional variance model

garch 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 data

numeric 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 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 terms

numeric 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 estimate

numeric scalar

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

'Display' — Command Window display option

'params' (default) | 'diagnostics' | 'full' | 'iter' | 'off' | string vector | cell vector of character vectors

Command Window display option, specified as the comma-separated pair consisting of 'Display' and 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

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

'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 the comma-separated pair consisting of 'DoF0' and a positive scalar. DoF0 must exceed 2.

Data Types: double

'E0' — Presample innovations

numeric 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 terms

numeric 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 estimate

scalar

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 controller

Optimization options, specified as the comma-separated pair consisting of 'Options' and 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.

'V0' — Presample conditional variances

numeric 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 terms

0 (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 in the presample or estimation data indicate missing data, and 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

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 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 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

• To access values of the estimation results, including the number of free parameters in the model, pass EstMdl to summarize.

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.