## 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

`$\begin{array}{l}{y}_{t}=0+{u}_{t}\\ {u}_{t}={\epsilon }_{t},\end{array}$`

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; Mdl0 = regARIMA('Intercept',Intercept,'Variance',Variance); y = filter(Mdl0,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) Mdl = regARIMA('Intercept',... intGrid1(k1),'Variance',varGrid1(k2)); [~,~,LogLGrid(k1,k2)] = estimate(Mdl,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.

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