%% Regression Model with SARIMA Errors and t Innovations
% This example shows how to set the innovation distribution of a regression model with SARIMA errors to a t distribution.
% Specify the regression model with \rm{SARIMA}(1,1,1)\times(1,1,0)_{12} errors:
% \begin{array}{c}{y_t} = {X_t}\beta  + {u_t}\\\left( {1 - 0.2L} \right)\left( {1 - L} \right)\left( {1 - 0.25{L^{12}} - 0.1{L^{24}}} \right)\left( {1 - {L^{12}}} \right){u_t} = \left( {1 + 0.15L} \right){\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,'SAR',{0.25, 0.1},'SARLags',[12 24],...
    'D',1,'Seasonality',12,'MA',0.15,'Intercept',0,...
		'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 by passing Mdl to 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 2015 The MathWorks, Inc.