%% Regression Model with ARMA Errors and t Innovations
% This example shows how to set the innovation distribution of a regression model with ARMA errors to a t distribution.
% Specify the regression model with ARMA(3,2) errors:
% \begin{array}{l}{y_t} = {X_t}\left[ \begin{array}{l}2.5\\ - 0.6\end{array} \right] + {u_t}\\{u_t} = 0.7{u_{t - 1}} - 0.3{u_{t - 2}} + 0.1{u_{t - 3}} + {\varepsilon _t} + 0.5{\varepsilon _{t - 1}} + 0.2{\varepsilon _{t - 2}},\end{array}
% where \varepsilon_t has a t distribution with the default degrees of freedom and unit variance.

Mdl = regARIMA('Intercept',0,'Beta',[2.5; -0.6],...
    'AR',{0.7, -0.3, 0.1},'MA',{0.5, 0.2},'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_5 distribution.

Mdl.Distribution = struct('Name','t','DoF',5)

% You can simulate or forecast responses from Mdl 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 2015 The MathWorks, Inc.