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function [Z,ST,R1,QT,Pstar,Pinf] = schur_statespace_transformation(mf,T,R,Q,qz_criterium)
% function [Z,ST,QT,R1,Pstar,Pinf] = schur_statespace(mf,T,R,Q,qz_criterium)
% Kitagawa transformation of state space system with a quasi-triangular
% transition matrix with unit roots at the top. Computation of Pstar and
% Pinf for Durbin and Koopman Diffuse filter
%
% INPUTS
% mf [integer] vector of indices of observed variables in
% state vector
% T [double] matrix of transition
% R [double] matrix of structural shock effects
% Q [double] matrix of covariance of structural shocks
% qz_criterium [double] numerical criterium for unit roots
%
% OUTPUTS
% Z [double] transformed matrix of measurement equation
% ST [double] tranformed matrix of transition
% R1 [double] tranformed matrix of structural shock effects
% QT [double] matrix of Schur vectors
% Pstar [double] matrix of covariance of stationary part
% Pinf [double] matrix of covariance initialization for
% nonstationary part
%
% ALGORITHM
% Real Schur transformation of transition equation
%
% SPECIAL REQUIREMENTS
% None
% Copyright (C) 2006-2011 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 <http://www.gnu.org/licenses/>.
np = size(T,1);
[QT,ST] = schur(T);
e1 = abs(ordeig(ST)) > 2-qz_criterium;
[QT,ST] = ordschur(QT,ST,e1);
k = find(abs(ordeig(ST)) > 2-qz_criterium);
nk = length(k);
nk1 = nk+1;
Pstar = zeros(np,np);
B = QT'*R*Q*R'*QT;
i = np;
while i >= nk+2
if ST(i,i-1) == 0
if i == np
c = zeros(np-nk,1);
else
c = ST(nk1:i,:)*(Pstar(:,i+1:end)*ST(i,i+1:end)')+...
ST(i,i)*ST(nk1:i,i+1:end)*Pstar(i+1:end,i);
end
q = eye(i-nk)-ST(nk1:i,nk1:i)*ST(i,i);
Pstar(nk1:i,i) = q\(B(nk1:i,i)+c);
Pstar(i,nk1:i-1) = Pstar(nk1:i-1,i)';
i = i - 1;
else
if i == np
c = zeros(np-nk,1);
c1 = zeros(np-nk,1);
else
c = ST(nk1:i,:)*(Pstar(:,i+1:end)*ST(i,i+1:end)')+...
ST(i,i)*ST(nk1:i,i+1:end)*Pstar(i+1:end,i)+...
ST(i,i-1)*ST(nk1:i,i+1:end)*Pstar(i+1:end,i-1);
c1 = ST(nk1:i,:)*(Pstar(:,i+1:end)*ST(i-1,i+1:end)')+...
ST(i-1,i-1)*ST(nk1:i,i+1:end)*Pstar(i+1:end,i-1)+...
ST(i-1,i)*ST(nk1:i,i+1:end)*Pstar(i+1:end,i);
end
q = [eye(i-nk)-ST(nk1:i,nk1:i)*ST(i,i) -ST(nk1:i,nk1:i)*ST(i,i-1);...
-ST(nk1:i,nk1:i)*ST(i-1,i) eye(i-nk)-ST(nk1:i,nk1:i)*ST(i-1,i-1)];
z = q\[B(nk1:i,i)+c;B(nk1:i,i-1)+c1];
Pstar(nk1:i,i) = z(1:(i-nk));
Pstar(nk1:i,i-1) = z(i-nk+1:end);
Pstar(i,nk1:i-1) = Pstar(nk1:i-1,i)';
Pstar(i-1,nk1:i-2) = Pstar(nk1:i-2,i-1)';
i = i - 2;
end
end
if i == nk+1
c = ST(nk+1,:)*(Pstar(:,nk+2:end)*ST(nk1,nk+2:end)')+ST(nk1,nk1)*ST(nk1,nk+2:end)*Pstar(nk+2:end,nk1);
Pstar(nk1,nk1)=(B(nk1,nk1)+c)/(1-ST(nk1,nk1)*ST(nk1,nk1));
end
Z = QT(mf,:);
R1 = QT'*R;
% stochastic trends with no influence on observed variables are
% arbitrarily initialized to zero
Pinf = zeros(np,np);
Pinf(1:nk,1:nk) = eye(nk);
[QQ,RR,EE] = qr(Z*ST(:,1:nk),0);
k = find(abs(diag([RR; zeros(nk-size(Z,1),size(RR,2))])) < 1e-8);
if length(k) > 0
k1 = EE(:,k);
dd =ones(nk,1);
dd(k1) = zeros(length(k1),1);
Pinf(1:nk,1:nk) = diag(dd);
end
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