regARIMA Model Estimation Using Equality Constraints

estimate requires a regARIMA model and a vector of univariate response data to estimate a regression model with ARIMA errors. Without predictor data, the model specifies the parametric form of an intercept-only regression component with an ARIMA error model. This is not the same as a conditional mean model with a constant. For details, see Alternative ARIMA Model Representations. If you specify a T-by-r matrix of predictor data, then estimate includes a linear regression component for the r series.

estimate returns fitted values for any parameters in the input model with NaN values. For example, if you specify a default regARIMA model and pass a T-by-r matrix of predictor data, then the software sets all parameters to NaN including the r regression coefficients, and estimates them all. If you specify non-NaN values for any parameters, then estimate views these values as equality constraints and honors them during estimation.

For example, suppose residual diagnostics from a linear regression suggest integrated unconditional disturbances. Since the regression intercept is unidentifiable in integrated models, you decide to set the intercept to 0. Specify 'Intercept',0 in the regARIMA model that you pass into estimate. The software views this non-NaN value as an equality constraint, and does not estimate the intercept, its standard error, and its covariance with other estimates. To illustrate further, suppose the true model for a response series yt is

yt=0+utut=εt,

where εt is Gaussian with variance 1. The loglikelihood function for a simulated data set from this model can resemble the surface in the following figure over a grid of variances and intercepts.

rng(1); % For reproducibility
e = randn(100,1);
Variance = 1;
Intercept = 0;
Mdl = regARIMA('Intercept',Intercept,'Variance',Variance);
y = filter(Mdl,e);

gridLength = 50;
intGrid1 = linspace(-1,1,50);
varGrid1 = linspace(0.1,4,50);
[varGrid2,intGrid2] = meshgrid(varGrid1,intGrid1);
LogLGrid = zeros(numel(varGrid1),numel(intGrid1));

for k1 = 1:numel(intGrid1)
    for k2 = 1:numel(varGrid1)
        ToEstMdl = regARIMA('Intercept',...
            intGrid1(k1),'Variance',varGrid1(k2));
        [~,~,LogLGrid(k1,k2)] = estimate(ToEstMdl,y);
    end
end
 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value    StandardError    TStatistic    PValue
                 _____    _____________    __________    ______

    Intercept      -1           0             -Inf         0   
    Variance      0.1           0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.17959          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.25918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.33878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.41837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.49796          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.57755          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.65714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.73673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.81633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.89592          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value     StandardError    TStatistic    PValue
                 _______    _____________    __________    ______

    Intercept         -1          0             -Inf         0   
    Variance     0.97551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.0551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.1347          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.2143          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.2939          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.3735          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.4531          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.5327          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.6122          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.6918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.7714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value    StandardError    TStatistic    PValue
                 _____    _____________    __________    ______

    Intercept       -1          0             -Inf         0   
    Variance     1.851          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     1.9306          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.0102          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.0898          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.1694          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value    StandardError    TStatistic    PValue
                 _____    _____________    __________    ______

    Intercept       -1          0             -Inf         0   
    Variance     2.249          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.3286          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.4082          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.4878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.5673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.6469          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.7265          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.8061          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.8857          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     2.9653          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.0449          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.1245          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.2041          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.2837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.3633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.4429          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.5224          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value    StandardError    TStatistic    PValue
                 _____    _____________    __________    ______

    Intercept       -1          0             -Inf         0   
    Variance     3.602          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.6816          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.7612          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.8408          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value     StandardError    TStatistic    PValue
                 ______    _____________    __________    ______

    Intercept        -1          0             -Inf         0   
    Variance     3.9204          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                 Value    StandardError    TStatistic    PValue
                 _____    _____________    __________    ______

    Intercept     -1            0             -Inf         0   
    Variance       4            0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance          0.1          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.17959          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.25918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.33878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.41837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.49796          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.57755          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.65714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.73673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.81633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.89592          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance      0.97551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.0551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.1347          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.2143          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.2939          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.3735          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.4531          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.5327          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.6122          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.6918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.7714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance        1.851          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       1.9306          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.0102          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.0898          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.1694          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance        2.249          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.3286          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.4082          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.4878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.5673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.6469          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.7265          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.8061          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.8857          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       2.9653          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.0449          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.1245          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.2041          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.2837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.3633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.4429          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.5224          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance        3.602          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.6816          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.7612          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.8408          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance       3.9204          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.95918          0             -Inf         0   
    Variance            4          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance          0.1          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.17959          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.25918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.33878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.41837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.49796          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.57755          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.65714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.73673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.81633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.89592          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance      0.97551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.0551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.1347          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.2143          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.2939          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.3735          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.4531          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.5327          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.6122          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.6918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.7714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance        1.851          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       1.9306          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.0102          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.0898          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.1694          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance        2.249          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.3286          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.4082          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.4878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.5673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.6469          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.7265          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.8061          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.8857          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       2.9653          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.0449          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.1245          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.2041          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.2837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.3633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.4429          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.5224          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance        3.602          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.6816          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.7612          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.8408          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance       3.9204          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.91837          0             -Inf         0   
    Variance            4          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance          0.1          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.17959          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.25918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.33878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.41837          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.49796          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.57755          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.65714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.73673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.81633          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.89592          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance      0.97551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.0551          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.1347          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.2143          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.2939          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.3735          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.4531          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.5327          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.6122          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.6918          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.7714          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance        1.851          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       1.9306          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.0102          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.0898          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.1694          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance        2.249          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.3286          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.4082          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.4878          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.5673          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.6469          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.7265          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf         0   
    Variance       2.8061          0              Inf         0   

 
    ARMA(0,0) Error Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic    PValue
                 ________    _____________    __________    ______

    Intercept    -0.87755          0             -Inf        ...
surf(intGrid2,varGrid2,LogLGrid) % 3D loglikelihood plot        
xlabel 'Intercept';
ylabel 'Variance';
zlabel 'Loglikelihood';
shading interp

Notice that the maximum (darkest red region ) occurs around Intercept = 0 and Variance = 1. If you apply an equality constraint, then the optimizer views a two-dimensional slice (in this example) of the loglikelihood function at that constraint. The following plots display the loglikelihood at several different Intercept equality constraints.

intValue = [intGrid1(10), intGrid1(20),...
    intGrid1(30), intGrid1(40)];
for k = 1:4
    subplot(2,2,k)
%    plot(varGrid1,LogLGrid(find(intGrid2 == intValue(k))))
    plot(varGrid1,LogLGrid(intGrid2 == intValue(k)))
    title(sprintf('Loglikelihood, Intercpet = %.3f',intValue(k)))
    xlabel 'Variance';
    ylabel 'Loglikelihood';
    hold on
    h1 = gca;
    plot([Variance Variance],h1.YLim,'r:')
    hold off
end

In each case, Variance = 1 (its true value) occurs very close to the maximum of the loglikelihood function. Rather than constrain Intercept, the following plots display the likelihood function using several Variance equality constraints.

varValue = [varGrid1(10),varGrid1(20),varGrid1(30),varGrid1(40)];
for k = 1:4
    subplot(2,2,k)
%    plot(intGrid1,LogLGrid(find(varGrid2 == varValue(k))))
    plot(intGrid1,LogLGrid(varGrid2 == varValue(k)))
    title(sprintf('Loglikelihood, Variance = %.3f',varValue(k)))
    xlabel('Intercept')
    ylabel('Loglikelihood')
    hold on
    h2 = gca;
    plot([Intercept Intercept],h2.YLim,'r:')
    hold off
end

In each case, Intercept = 0 (its true value) occurs very close to the maximum of the loglikelihood function.

estimate also honors a subset of equality constraints while estimating all other parameters set to NaN. For example, suppose r = 3, and you know that β2 = 5. Specify Beta = [NaN; 5; NaN] in the regARIMA model, and pass this model with the data to estimate.

estimate optionally returns the estimated variance-covariance matrix for the estimated parameters. The parameter order in this matrix is:

  • Intercept

  • Nonzero AR coefficients at positive lags

  • Nonzero SAR coefficients at positive lags

  • Nonzero MA coefficients at positive lags

  • Nonzero SMA coefficients at positive lags

  • Regression coefficients (when you specify X in estimate)

  • Innovation variance

  • Degrees of freedom for the t distribution

If any parameter known to the optimizer has an equality constraint, then the corresponding row and column of the variance-covariance matrix has all zeros.

See Also

|

Related Topics