Optimization Settings for Conditional Mean Model Estimation

Optimization Options

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 PropertiesDescriptionestimate Settings
AlgorithmAlgorithm for minimizing the negative loglikelihood function'sqp'
DisplayLevel of display for optimization progress'off'
DiagnosticsDisplay for diagnostic information about the function to be minimized'off'
ConstraintToleranceTermination tolerance on constraint violations1e-7

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 name-value pair argument 'Display','diagnostics' in estimate. Alternatively, you can direct the optimizer to display optimization diagnostics.

Define an AR(1) model Mdl and simulate data from it.

Mdl = arima('AR',0.5,'Constant',0,'Variance',1);
rng(1); % For reproducibility
y = simulate(Mdl,25);

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',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:
FiniteDifferenceStepSize: 'sqrt(eps)'
FiniteDifferenceType: 'forward'
MaxFunctionEvaluations: '100*numberOfVariables'
MaxIterations: 400
ObjectiveLimit: -1.0000e+20
OptimalityTolerance: 1.0000e-06
OutputFcn: []
PlotFcn: []
ScaleProblem: 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 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.

Mdl = arima(1,0,0);
EstMdl = estimate(Mdl,y,'Options',options);
____________________________________________________________
Diagnostic Information

Number of variables: 3

Functions
Objective:                            @(X)nLogLike(X,YData,XData,E,V,Mdl,AR.Lags,MA.Lags,maxPQ,T,isDistributionT,options,userSpecifiedY0,userSpecifiedE0,userSpecifiedV0,trapValue)
Hessian:                              finite-differencing (or Quasi-Newton)
Nonlinear constraints:                @(x)internal.econ.arimaNonLinearConstraints(x,LagsAR,LagsSAR,LagsMA,LagsSMA,tolerance)

Constraints
Number of nonlinear inequality constraints: 1
Number of nonlinear equality constraints:   0

Number of linear inequality constraints:    0
Number of linear equality constraints:      0
Number of lower bound constraints:          3
Number of upper bound constraints:          3

Algorithm selected
sqp

____________________________________________________________
End diagnostic information

ARIMA(1,0,0) Model (Gaussian Distribution):

Value      StandardError    TStatistic     PValue
_________    _____________    __________    _________

Constant    -0.064857       0.23456        -0.2765        0.78217
AR{1}         0.46386       0.15781         2.9393      0.0032895
Variance       1.2308       0.47275         2.6035      0.0092266

Note

• estimate numerically maximizes the loglikelihood function, potentially using equality, inequality, and lower and upper bound constraints. If you set Algorithm to anything other than sqp, make sure the algorithm supports similar constraints, such as interior-point. For example, trust-region-reflective does not support inequality constraints.

• estimate sets a constraint level of ConstraintTolerance so 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 Mean Model Constraints

The software enforces these constraints while estimating an ARIMA model:

• Stability of nonseasonal and seasonal AR operator polynomials

• Invertibility of nonseasonal and seasonal MA operator polynomials

• Innovation variance strictly greater than zero

• Degrees of freedom strictly greater than two for a t innovation distribution