%% Regression Model with AR Errors and t Innovations
% This example shows how to set the innovation distribution of a regression model with AR errors to a t distribution.
% Specify the regression model with AR(4) errors:
% \begin{array}{l}{y_t} = {X_t}\left[ \begin{array}{l} - 2\\0.5\end{array} \right] + {u_t}\\{u_t} = 0.2{u_{t - 1}} + 0.1{u_{t - 4}} + {\varepsilon _t},\end{array}
% where \varepsilon_t has a t distribution with the default degrees of freedom and unit variance.

Mdl = regARIMA('AR',{0.2,0.1},'ARLags',[1,4],...
    'Intercept',0,'Beta',[-2;0.5],'Variance',1,...
    'Distribution','t')

% The default degrees of freedom is NaN. If you don't know the degrees of freedom, then you can estimate it by passing Mdl and the data to estimate.
%
% Specify a t_{10} distribution.

Mdl.Distribution = struct('Name','t','DoF',10)

% You can simulate or forecast responses using simulate or forecast because Mdl is completely specified.
% In applications, such as simulation, the software normalizes the random t innovations. In other words, Variance overrides the theoretical variance of the t random variable (which is DoF/(DoF - 2)), but preserves the kurtosis of the distribution.
% Copyright 2017 The MathWorks, Inc.