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    %% Impulse Response Function of VAR Model
% Compute the generalized impulse response function of the two-dimensional
% VAR(3) model
%
% $${y_t} = \left[ {\begin{array}{*{20}{c}}
% {1}&{ - 0.2}\\
% { - 0.1}&{0.3}
% \end{array}} \right]{y_{t - 1}} - \left[ {\begin{array}{*{20}{c}}
% {0.75}&{ - 0.1}\\
% { - 0.05}&{0.15}
% \end{array}} \right]{y_{t - 2}} + \left[ {\begin{array}{*{20}{c}}
% {0.55}&{ - 0.02}\\
% { - 0.01}&{0.03}
% \end{array}} \right]{y_{t - 3}} + {\varepsilon _t}.$$
%
% In the equation, $y_t = [y_{1,t}\;\;\;y_{2,t}]'$, $\varepsilon_t =
% [\varepsilon_{1,t}\;\;\;
% \varepsilon_{2,t}]'$, and, for all _t_, $\varepsilon_t$ is Gaussian with
% mean zero and covariance matrix
%
% $$\Sigma  = \left[ {\begin{array}{*{20}{c}}
% {0.5}&{ - 0.1}\\
% { - 0.1}&{0.25}
% \end{array}} \right].$$
%
%%
% Create a cell vector of matrices for the autoregressive coefficients as
% you encounter them in the model expressed in difference-equation
% notation.  Specify the innovation covariance matrix.
AR1 = [1 -0.2; -0.1 0.3];
AR2 = -[0.75 -0.1; -0.05 0.15];
AR3 = [0.55 -0.02; -0.01 0.03];
ar0 = {AR1 AR2 AR3};

InnovCov = [0.5 -0.1; -0.1 0.25];
%%
% Compute the entire, generalized impulse response function of $y_t$.
% Because no MA terms exist, specify an empty array (|[]|) for the second input
% argument. 
Y = armairf(ar0,[],'Method','generalized','InnovCov',InnovCov);
size(Y)
%%
% |Y| is a 31-by-2-2 array of impulse responses.  Rows correspond to
% periods, columns correspond to variables, and pages correspond to the
% variable that |armairf| shocks.  |armairf| satisfies the stopping
% criterion after 31 periods.  You can specify to stop sooner using the
% |'NumObs'| name-value pair argument.  This practice is beneficial when
% the system has many variables.
%%
% Compute and display the generalized impulse responses for the first 10
% periods.
Y20 = armairf(ar0,[],'Method','generalized','InnovCov',InnovCov,...
    'NumObs',10)
%%
% The impulse responses appear to die out with increasing time, 
% suggesting a stable system.