1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187
|
function [Pstar,Pinf] = compute_Pinf_Pstar(mf,T,R,Q,qz_criterium, restrict_columns)
% [Pstar,Pinf] = compute_Pinf_Pstar(mf,T,R,Q,qz_criterium, restrict_columns)
% Kitagawa transformation of state space system with a quasi-triangular
% transition matrix with unit roots at the top, but excluding zero columns of the transition matrix.
% Computation of Pstar and Pinf for Durbin and Koopman Diffuse filter
%
% The transformed state space is
% y = [ss; z; x];
% s = static variables (zero columns of T)
% z = unit roots
% x = stable roots
% ss = s - z = stationarized static variables
%
% 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
% 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 © 2006-2023 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/>.
np = size(T,1);
if iszero(Q)
% this may happen if users set Q=0 and use heteroskedastic shocks to set
% variances period by period
% this in practice triggers a form of conditional filter where states
% are initialized at st. state with zero variances
Pstar=T*0;
Pinf=T*0;
return
end
if nargin == 6
indx = restrict_columns;
indx0=find(~ismember(1:np,indx));
else
indx=(find(max(abs(T))>=1.e-10));
indx0=(find(max(abs(T))<1.e-10));
end
np0=length(indx0);
T0=T(indx0,indx); % static variables vs. dynamic ones
R0=R(indx0,:); % matrix of shocks for static variables
% Perform Kitagawa transformation only for non-zero columns of T
T=T(indx,indx);
R=R(indx,:);
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);
R1 = QT'*R;
B = R1*Q*R1';
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
if np0
% Now I recover stationarized static variables using
% ss = s-A*z
% and
% z-z(-1) (growth rates of unit roots) only depends on stationary variables
Pstar = blkdiag(zeros(np0),Pstar);
ST = [zeros(length(Pstar),length(indx0)) [T0*QT ;ST]];
R1 = [R0; R1];
% Build the matrix for stationarized variables
STinf = ST(np0+1:np0+nk,np0+1:np0+nk);
iSTinf = inv(STinf);
ST0=ST;
ST0(:,1:np0+nk)=0; % stationarized static + 1st difference only respond to lagged stationary states
ST00 = ST(1:np0,np0+1:np0+nk);
% A\B is the matrix division of A into B, which is roughly the
% same as INV(A)*B
ST0(1:np0,np0+nk+1:end) = ST(1:np0,np0+nk+1:end)-ST00*(iSTinf*ST(np0+1:np0+nk,np0+nk+1:end)); % snip non-stationary part
R10 = R1;
R10(1:np0,:) = R1(1:np0,:)-ST00*(iSTinf*R1(np0+1:np0+nk,:)); % snip non-stationary part
% Kill non-stationary part before projecting Pstar
ST0(np0+1:np0+nk,:)=0;
R10(np0+1:np0+nk,:)=0; % is this questionable???? IT HAS TO in order to match Michel's version!!!
% project Pstar onto static x
Pstar = ST0*Pstar*ST0'+R10*Q*R10';
% QT(1:np0,np0+1:np0+nk) = QT(1:np0,np0+1:np0+nk)+ST(1:np0,np0+1:np0+nk); %%% is this questionable ????
% reorder QT entries
else
STinf = ST(np0+1:np0+nk,np0+1:np0+nk);
end
% stochastic trends with no influence on observed variables are
% arbitrarily initialized to zero
Pinf = zeros(np,np);
Pinf(1:nk,1:nk) = eye(nk);
if np0
STtriu = STinf-eye(nk);
% A\B is the matrix division of A into B, which is roughly the
% same as INV(A)*B
STinf0 = ST00*(eye(nk)-iSTinf*STtriu);
Pinf = blkdiag(zeros(np0),Pinf);
QT = blkdiag(eye(np0),QT);
QTinf = QT;
QTinf(1:np0,np0+1:np0+nk) = STinf0;
QTinf([indx0(:); indx(:)],:) = QTinf;
STinf1 = [zeros(np0+np,np0) [STinf0; eye(nk); zeros(np-nk,nk)] zeros(np0+np,np-nk)];
mf = ismember([indx0(:); indx(:)],mf);
for k = 1:nk
if norm(QTinf(mf,:)*ST([indx0(:); indx(:)],k+np0)) < 1e-8
Pinf(k+np0,k+np0) = 0;
end
end
Pinf = STinf1*Pinf*STinf1';
QT([indx0(:); indx(:)],:) = QT;
else
for k = 1:nk
if norm(QT(mf,:)*ST(:,k)) < 1e-8
Pinf(k+np0,k+np0) = 0;
end
end
end
Pinf = QT*Pinf*QT';
Pstar = QT*Pstar*QT';
|