estimate
Class: egarch
Fit EGARCH conditional variance model to data
Syntax
EstMdl = estimate(Mdl,y)
[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y)
[EstMdl,EstParamCov,logL,info] = estimate(Mdl,y,Name,Value)
Description
EstMdl = estimate(Mdl,y) uses
maximum likelihood to estimate the parameters of the EGARCH(P,Q)
model Mdl given the observed univariate time series y. EstMdl is
a new egarch model that stores
the results.
[EstMdl,EstParamCov,logL,info]
= estimate(Mdl,y) additionally returns EstParamCov,
the variancecovariance matrix associated with estimated parameters, logL,
the optimized loglikelihood objective function, and info,
a data structure of summary information.
[EstMdl,EstParamCov,logL,info] = estimate(Mdl,y,Name,Value) estimates
the model with additional options specified by one or more Name,Value pair
arguments.
Output Arguments
EstMdl 
Model containing the parameter estimates, returned as an egarch model. estimate uses
maximum likelihood to calculate all parameter estimates not constrained
by Mdl (i.e., all parameters in Mdl that
you set to NaN).

EstParamCov 
Variancecovariance matrix of maximum likelihood estimates of
the model parameters known to the optimizer, returned as a 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 held fixed
as equality constraints contain zeros.
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 Nonzero leverage coefficients at positive lags Degrees of freedom (t innovation
distribution only) Offset (models with nonzero offset only)

logL 
Optimized loglikelihood objective function value.

info 
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.

Examples
expand all
Fit an EGARCH(1,1) model to simulated data.
Simulate 500 data points from an EGARCH(1,1) model
where
and
(the distribution of z_{t} is
Gaussian).
Mdl = egarch('Constant',0.001,'GARCH',0.7,...
'ARCH',0.5,'Leverage',0.3);
rng('default');
[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)
EstMdl = estimate(ToEstMdl,y)
EGARCH(1,1) Conditional Variance Model:

Conditional Probability Distribution: Gaussian
Standard t
Parameter Value Error Statistic
   
Constant 0.000638686 0.0316977 0.0201493
GARCH{1} 0.705065 0.0673595 10.4672
ARCH{1} 0.56774 0.0747457 7.59563
Leverage{1} 0.321158 0.0533449 6.02041
EstMdl =
EGARCH(1,1) Conditional Variance Model:

Distribution: Name = 'Gaussian'
P: 1
Q: 1
Constant: 0.000638686
GARCH: {0.705065} at Lags [1]
ARCH: {0.56774} 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.
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 = Dataset.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 0.134782 0.0220919 6.10095
GARCH{1} 0.983909 0.0024221 406.221
ARCH{1} 0.199644 0.0139655 14.2956
Leverage{1} 0.0602427 0.00564703 10.668
EstMdl =
EGARCH(1,1) Conditional Variance Model:

Distribution: Name = 'Gaussian'
P: 1
Q: 1
Constant: 0.134782
GARCH: {0.983909} at Lags [1]
ARCH: {0.199644} at Lags [1]
Leverage: {0.0602427} at Lags [1]
EstParamCov =
1.0e03 *
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 variancecovariance 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.
Tip
Suppose EstParamCov is an estimated parameter
covariance matrix returned by estimate. Since the
software sets the variances and covariances of parameters fixed during
estimation to 0, one way to count the number of
free parameters (numParams) in a fitted model is
to enter the following command.
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.
See Also
egarch  filter  forecast  infer  print  simulate
More About