estimate
maximizes the loglikelihood function
using fmincon
from Optimization Toolbox™. fmincon
has
many optimization options, such as choice of optimization algorithm
and constraint violation tolerance. Choose optimization options using optimoptions
.
estimate
uses the fmincon
optimization
options by default, with these exceptions. For details, see fmincon
and optimoptions
in Optimization Toolbox.
optimoptions Properties  Description  estimate Settings 

Algorithm  Algorithm for minimizing the negative loglikelihood function  'sqp' 
Display  Level of display for optimization progress  'off' 
Diagnostics  Display for diagnostic information about the function to be minimized  'off' 
ConstraintTolerance  Termination tolerance on constraint violations  1e7 
If you want to use optimization options that differ from the
default, then set your own using optimoptions
.
For example, suppose that you want estimate
to
display optimization diagnostics. The best practice is to set the
namevalue pair argument 'Display','diagnostics'
in estimate
.
Alternatively, you can direct the optimizer to display optimization
diagnostics.
Define a GARCH(1,1) model (Mdl
) and simulate
data from it.
Mdl = garch('ARCH',0.2,'GARCH',0.5,'Constant',0.5); rng(1); y = simulate(Mdl,500);
Mdl
does not have a regression component.
By default, fmincon
does not display the optimization
diagnostics. Use optimoptions
to set it to display
the optimization diagnostics, and set the other fmincon
properties
to the default settings of estimate
listed in the
previous table.
options = optimoptions(@fmincon,'Diagnostics','on','Algorithm',... 'sqp','Display','off','ConstraintTolerance',1e7) % @fmincon is the function handle for fmincon
options = fmincon options: Options used by current Algorithm ('sqp'): (Other available algorithms: 'activeset', 'interiorpoint', 'sqplegacy', 'trustregionreflective') Set properties: Algorithm: 'sqp' ConstraintTolerance: 1.0000e07 Display: 'off' Default properties: CheckGradients: 0 FiniteDifferenceStepSize: 'sqrt(eps)' FiniteDifferenceType: 'forward' MaxFunctionEvaluations: '100*numberOfVariables' MaxIterations: 400 ObjectiveLimit: 1.0000e+20 OptimalityTolerance: 1.0000e06 OutputFcn: [] PlotFcn: [] ScaleProblem: 0 SpecifyConstraintGradient: 0 SpecifyObjectiveGradient: 0 StepTolerance: 1.0000e06 TypicalX: 'ones(numberOfVariables,1)' UseParallel: 0 Options not used by current Algorithm ('sqp') Default properties: FunctionTolerance: 1.0000e06 HessianApproximation: 'not applicable' HessianFcn: [] HessianMultiplyFcn: [] HonorBounds: 1 SubproblemAlgorithm: 'factorization'
The options that you set appear under the Set by user:
heading.
The properties under the Default:
heading are other
options that you can set.
Fit Mdl
to y
using the
new optimization options.
ToEstMdl = garch(1,1);
EstMdl = estimate(ToEstMdl,y,'Options',options);
____________________________________________________________ Diagnostic Information Number of variables: 3 Functions Objective: @(X)Mdl.nLogLikeGaussian(X,V,E,Lags,1,maxPQ,T,nan,trapValue) Gradient: finitedifferencing Hessian: finitedifferencing (or QuasiNewton) Constraints Nonlinear constraints: do not exist Number of linear inequality constraints: 1 Number of linear equality constraints: 0 Number of lower bound constraints: 3 Number of upper bound constraints: 3 Algorithm selected sqp ____________________________________________________________ End diagnostic information GARCH(1,1) Conditional Variance Model:  Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic     Constant 0.431453 0.465647 0.926567 GARCH{1} 0.314345 0.249922 1.25777 ARCH{1} 0.571429 0.326774 1.7487
Note:

The software enforces these constraints while estimating a GARCH model:
Covariancestationarity,
$${\sum}_{i=1}^{P}{\gamma}_{i}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}}<1$$
Positivity of GARCH and ARCH coefficients
Model constant strictly greater than zero
For a t innovation distribution, degrees of freedom strictly greater than two
For GJR models, the constraints enforced during estimation are:
Covariancestationarity constraint,
$${\sum}_{i=1}^{P}{\gamma}_{i}+{\displaystyle {\sum}_{j=1}^{Q}{\alpha}_{j}}+\frac{1}{2}{\displaystyle {\sum}_{j=1}^{Q}{\xi}_{j}}<1$$
Positivity constraints on the GARCH and ARCH coefficients
Positivity on the sum of ARCH and leverage coefficients,
$${\alpha}_{j}+{\xi}_{j}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,Q$$
Model constant strictly greater than zero
For a t innovation distribution, degrees of freedom strictly greater than two
For EGARCH models, the constraints enforced during estimation are:
Stability of the GARCH coefficient polynomial
For a t innovation distribution, degrees of freedom strictly greater than two
estimate
 fmincon
 optimoptions