Optimization Settings for Conditional Variance Model Estimation

Optimization Options

estimatemaximizes the loglikelihood function usingfminconfrom Optimization Toolbox™.fminconhas many optimization options, such as choice of optimization algorithm and constraint violation tolerance. Choose optimization options usingoptimoptions.

estimateuses thefminconoptimization options by default, with these exceptions. For details, seefminconandoptimoptionsin 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	`1e-7`

If you want to use optimization options that differ from the default, then set your own usingoptimoptions.

For example, suppose that you wantestimateto display optimization diagnostics. The best practice is to set the name-value pair argument'Display','diagnostics'inestimate. Alternatively, you can direct the optimizer to display optimization diagnostics.

Define a GARCH(1,1) model (Mdl) and simulate data from it.

Mdl0 = garch('ARCH',0.2,'GARCH',0.5,'Constant',0.5); rng(1); y = simulate(Mdl0,500);

Mdldoes not have a regression component. By default,fmincondoes not display the optimization diagnostics. Useoptimoptions将其设置为显示优化诊断方法,and set the otherfminconproperties to the default settings ofestimate前面的表中列出。

options = optimoptions(@fmincon,'Diagnostics','on','Algorithm',...'sqp','Display','off','ConstraintTolerance',1e-7)

options = fmincon options: Options used by current Algorithm ('sqp'): (Other available algorithms: 'active-set', 'interior-point', 'sqp-legacy', 'trust-region-reflective') Set properties: Algorithm: 'sqp' ConstraintTolerance: 1.0000e-07 Display: 'off' Default properties: CheckGradients: 0 FiniteDifferenceStepSize: 'sqrt(eps)' FiniteDifferenceType: 'forward' MaxFunctionEvaluations: '100*numberOfVariables' MaxIterations: 400 ObjectiveLimit: -1.0000e+20 OptimalityTolerance: 1.0000e-06 OutputFcn: [] PlotFcn: [] ScaleProblem: 0 SpecifyConstraintGradient: 0 SpecifyObjectiveGradient: 0 StepTolerance: 1.0000e-06 TypicalX: 'ones(numberOfVariables,1)' UseParallel: 0 Show options not used by current Algorithm ('sqp')

% @fmincon is the function handle for fmincon

The options that you set appear under theSet by user:heading. The properties under theDefault:heading are other options that you can set.

FitMdltoyusing the new optimization options.

Mdl = garch(1,1); EstMdl = estimate(Mdl,y,'Options',options);

____________________________________________________________ Diagnostic Information Number of variables: 3 Functions Objective: @(X)Mdl.nLogLikeGaussian(X,V,E,Lags,1,maxPQ,T,nan,trapValue) Gradient: finite-differencing Hessian: Quasi-Newton 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 (Gaussian Distribution): Value StandardError TStatistic PValue _______ _____________ __________ ________ Constant 0.43145 0.46565 0.92657 0.35415 GARCH{1} 0.31435 0.24992 1.2578 0.20847 ARCH{1} 0.57143 0.32677 1.7487 0.080343

Note

estimatenumerically maximizes the loglikelihood function, potentially using equality, inequality, and lower and upper bound constraints. If you setAlgorithmto anything other thansqp, make sure the algorithm supports similar constraints, such asinterior-point. For example,trust-region-reflectivedoes not support inequality constraints.
estimatesets a constraint level ofConstraintToleranceso constraints are not violated. An estimate with an active constraint has unreliable standard errors because variance-covariance estimation assumes that the likelihood function is locally quadratic around the maximum likelihood estimate.

Conditional Variance Model Constraints

The software enforces these constraints while estimating a GARCH model:

Covariance-stationarity,

$\sum_{i = 1}^{P} γ_{i} + \sum_{j = 1}^{Q} α_{j} < 1$
Positivity of GARCH and ARCH coefficients
Model constant strictly greater than zero
For atinnovation distribution, degrees of freedom strictly greater than two

For GJR models, the constraints enforced during estimation are:

Covariance-stationarity constraint,

$\sum_{i = 1}^{P} γ_{i} + \sum_{j = 1}^{Q} α_{j} + \frac{1}{2} \sum_{j = 1}^{Q} ξ_{j} < 1$
Positivity constraints on the GARCH and ARCH coefficients
Positivity on the sum of ARCH and leverage coefficients,

$α_{j} + ξ_{j} \geq 0, j = 1, \dots, Q$
Model constant strictly greater than zero
For atinnovation distribution, degrees of freedom strictly greater than two

For EGARCH models, the constraints enforced during estimation are:

Stability of the GARCH coefficient polynomial
For atinnovation distribution, degrees of freedom strictly greater than two