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    %% Convert VAR Model to VEC Model Using Cell Arrays
% Consider converting the following VAR(3) model to a VEC(2) model.
%
% $${y_t} = \left[ {\begin{array}{*{20}{c}}
% {0.5}\\
% {1}\\
% { - 2}
% \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
% {0.54}&{0.86}&{ - 0.43}\\
% {1.83}&{0.32}&{0.34}\\
% { - 2.26}&{ - 1.31}&{3.58}
% \end{array}} \right]{y_{t - 1}} + \left[ {\begin{array}{*{20}{c}}
% {0.14}&{ - 0.12}&{0.05}\\
% {0.14}&{0.07}&{0.10}\\
% {0.07}&{0.16}&{0.07}
% \end{array}} \right]{y_{t - 3}} + {\varepsilon _t}.$$
%
%%
% Specify the coefficient matrices ($A_1$, $A_2$, and $A_3$) of the VAR(3)
% model terms $y_{t-1}$, $y_{t-2}$, and $y_{t-3}$.

% Copyright 2015 The MathWorks, Inc.

A1 = [0.54  0.86 -0.43;
      1.83  0.32  0.34;
     -2.26 -1.31  3.58];
A2 = zeros(3); 
A3 = [0.14 -0.12 0.05;
      0.14  0.07 0.10;
      0.07  0.16 0.07];
%%
% Pack the matrices into separate cells of a 3 dimensional cell vector.  Put |A1|
% into the first cell, |A2| into the second cell, and |A3| into the third
% cell.
VAR = {A1 A2 A3};
%%
% Compute the coefficient matrices of $\Delta y_{t-1}$ and $\Delta
% y_{t-2}$, and error-correction coefficient matrix of the equivalent
% VEC(2) model.
[VEC,C] = var2vec(VAR);
size(VEC)
%%
% The specification of a cell array of matrices for the input argument
% indicates that the VAR(3) model is a reduced-form model expressed as a
% difference equation. |VAR{1}| is the coefficient of $y_{t-1}$, and
% subsequent elements correspond to subsequent lags.
%%
% |VEC| is a 1-by-2 cell vector of 3-by-3 coefficient matrices for the
% VEC(2) equivalent of the VAR(3) model.  Because the VAR(3) model is in
% reduced form, the equivalent VEC model is also.  That is, |VEC{1}| is the
% coefficient of $\Delta y_{t-1}$, and subsequent elements correspond to
% subsequent lags.  The orientation of |VEC| corresponds to the orientation
% of |VAR|.
%%
% Display the VEC(2) model coefficients.
B1 = VEC{1}
B2 = VEC{2}
C
%%
% Since the constant offsets between the models are equivalent, the
% resulting VEC(2) model is
%
% $$\begin{array}{rcl}\Delta {y_t} &=& \left[ {\begin{array}{*{20}{c}}
% {0.5}\\
% {1}\\
% { - 2}
% \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
% { - 0.14}&{0.12}&{ - 0.05}\\
% { - 0.14}&{ - 0.07}&{ - 0.10}\\
% { - 0.07}&{ - 0.16}&{ - 0.07}
% \end{array}} \right]\Delta {y_{t - 1}} + \left[ {\begin{array}{*{20}{c}}
% { - 0.14}&{0.12}&{ - 0.05}\\
% { - 0.14}&{ - 0.07}&{ - 0.10}\\
% { - 0.07}&{ - 0.16}&{ - 0.07}
% \end{array}} \right]\Delta {y_{t - 2}}\\ &+& \left[ {\begin{array}{*{20}{c}}
% { - 0.32}&{0.74}&{ - 0.38}\\
% {1.97}&{ - 0.61}&{0.44}\\
% { - 2.19}&{ - 1.15}&{2.65}
% \end{array}} \right]{y_{t - 1}} + {\varepsilon _t}\end{array}.$$
%