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function [A0, A0star, AR, B] = get_companion_matrix(auxiliary_model_name, auxiliary_model_type)
% Gets the companion VAR representation of a PAC auxiliary model.
% Depending on the nature of this auxiliary model the output is
% saved in oo_.{var,trend_component}.(auxiliary_model_name).CompanionMatrix
%
% INPUTS
% - auxiliary_model_name [string] the name of the auxiliary model
% - auxiliary_model_type [string] the type of the auxiliary model
% ('var' or 'trend_component')
%
% OUTPUTS
% - None
% Copyright © 2018-2019 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
global oo_ M_
if nargin < 2
if isfield(M_, 'var') && isfield(M_.var, auxiliary_model_name)
auxiliary_model_type = 'var';
elseif isfield(M_, 'trend_component') && isfield(M_.trend_component, auxiliary_model_name)
auxiliary_model_type = 'trend_component';
else
error('Unknown type of auxiliary model.')
end
end
if strcmp(auxiliary_model_type, 'var')
[AR, ~, Constant] = feval(sprintf('%s.varmatrices', M_.fname), auxiliary_model_name, M_.params, M_.var.(auxiliary_model_name).structural);
isconstant = any(abs(Constant)>0);
M_.var.(auxiliary_model_name).isconstant = isconstant; % FIXME Could be done by preprocessor instead…
elseif strcmp(auxiliary_model_type, 'trend_component')
[AR, A0, A0star] = feval(sprintf('%s.trend_component_ar_a0', M_.fname), auxiliary_model_name, M_.params);
else
error('Unknown type of auxiliary model.')
end
% Get the number of lags
p = size(AR, 3);
% Get the number of variables
n = length(M_.(auxiliary_model_type).(auxiliary_model_name).lhs);
switch auxiliary_model_type
case 'var'
oo_.var.(auxiliary_model_name).CompanionMatrix = zeros(n*p+isconstant);
oo_.var.(auxiliary_model_name).CompanionMatrix(isconstant+(1:n),isconstant+(1:n)) = AR(:,:,1);
for i = 2:p
oo_.var.(auxiliary_model_name).CompanionMatrix(isconstant+(1:n),isconstant+(i-1)*n+(1:n)) = AR(:,:,i);
oo_.var.(auxiliary_model_name).CompanionMatrix(isconstant+(i-1)*n+(1:n),isconstant+(i-2)*n+(1:n)) = eye(n);
end
if isconstant
oo_.var.(auxiliary_model_name).CompanionMatrix(1,1) = 1;
for i=1:n
oo_.var.(auxiliary_model_name).CompanionMatrix(1+i,1) = Constant(i);
end
end
M_.var.(auxiliary_model_name).list_of_variables_in_companion_var = M_.endo_names(M_.var.(auxiliary_model_name).lhs);
if nargout
A0 = [];
A0star = [];
B = [];
end
case 'trend_component'
% Get number of trends.
q = sum(M_.trend_component.(auxiliary_model_name).targets);
% Get the number of equations with error correction.
m = n - q;
% Get the indices of trend and EC equations in the auxiliary model.
target_eqnums_in_auxiliary_model = M_.trend_component.(auxiliary_model_name).target_eqn;
ecm_eqnums_in_auxiliary_model = find(~M_.trend_component.(auxiliary_model_name).targets);
% REMARK It is assumed that the non trend equations are the error correction
% equations. We assume that the model can be cast in the following form:
%
% Δ Xₜ₋₁ = A₀ (Xₜ₋₁ - C₀Zₜ₋₁) + Σᵢ₌₁ᵖ Aᵢ Δ Xₜ₋ᵢ + ϵₜ
%
% Zₜ = Zₜ₋₁ + ηₜ
%
% where Xₜ is a n×1 vector and Zₜ is an m×1 vector, A₀ is a
% n×n matrix, C₀ a n×m matrix, and Aᵢ (i=1,…,p) are n×n
% matrices. Matrix C₀ can be factorized as C₀ = (A₀)⁻¹×Λ,
% where Λ is a n×m matrix.
%
% We rewrite the model in levels (we integrate the first set
% of equations) and rewrite the model as a VAR(1) model. Let
% Yₜ = [Xₜ; Zₜ] be the vertical concatenation of vectors
% Xₜ (variables with EC) and Zₜ (trends). We have
%
% Yₜ = Σᵢ₌₁ᵖ⁺¹ Bᵢ Yₜ₋ᵢ + [εₜ; ηₜ]
%
% with
%
% B₁ = [I+A₀+A₁, -Λ; 0, I]
%
% Bᵢ = [Aᵢ-Aᵢ₋₁, 0; 0, 0] for i = 2,…, p
% and
% Bₚ₊₁ = -[Aₚ, 0; 0, 0]
%
% where the dimensions of I and 0 matrices can easily be
% deduced from the number of EC and trend equations.
% Check that the lhs of candidate ecm equations are at least first differences.
for i = 1:m
if ~get_difference_order(M_.trend_component.(auxiliary_model_name).lhs(ecm_eqnums_in_auxiliary_model(i)))
error([auxiliary_model_name ' is not a trend component model. The LHS variables should be in differences'])
end
end
% Get the EC matrix (the EC term is assumend to be in t-1).
%
% TODO: Check that the EC term is the difference between the
% endogenous variable and the trend variable.
%
% Build B matrices (VAR in levels)
B = zeros(m+q, m+q, p+1);
B(ecm_eqnums_in_auxiliary_model, ecm_eqnums_in_auxiliary_model, 1) = eye(m) + A0 + AR(:,:,1);
B(ecm_eqnums_in_auxiliary_model, target_eqnums_in_auxiliary_model) = -A0star;
B(target_eqnums_in_auxiliary_model, target_eqnums_in_auxiliary_model) = eye(q);
for i = 2:p
B(ecm_eqnums_in_auxiliary_model,ecm_eqnums_in_auxiliary_model,i) = AR(:,:,i) - AR(:,:,i-1);
end
B(ecm_eqnums_in_auxiliary_model,ecm_eqnums_in_auxiliary_model,p+1) = -AR(:,:,p);
% Write Companion matrix
oo_.trend_component.(auxiliary_model_name).CompanionMatrix = zeros(size(B, 1)*size(B, 3));
for i = 1:p
oo_.trend_component.(auxiliary_model_name).CompanionMatrix(1:n, (i-1)*n+(1:n)) = B(:,:,i);
oo_.trend_component.(auxiliary_model_name).CompanionMatrix(i*n+(1:n),(i-1)*n+(1:n)) = eye(n);
end
oo_.trend_component.(auxiliary_model_name).CompanionMatrix(1:n, p*n+(1:n)) = B(:,:,p+1);
M_.trend_component.(auxiliary_model_name).list_of_variables_in_companion_var = M_.endo_names(M_.trend_component.(auxiliary_model_name).lhs);
variables_rewritten_in_levels = M_.trend_component.(auxiliary_model_name).list_of_variables_in_companion_var(ecm_eqnums_in_auxiliary_model);
for i=1:m
id = get_aux_variable_id(variables_rewritten_in_levels{i});
if id
auxinfo = M_.aux_vars(id);
if auxinfo.type==8
M_.trend_component.(auxiliary_model_name).list_of_variables_in_companion_var(ecm_eqnums_in_auxiliary_model(i)) = ...
{M_.endo_names{auxinfo.orig_index}};
else
error('This is a bug. Please contact the Dynare Team.')
end
else
error('This is a bug. Please contact the Dynare Team.')
end
end
end
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