Commit e97e5c34 by Marco Ratto Committed by Stéphane Adjemian (Charybdis)

### Exclude zero columns of T from Kitagawa transformation, since ordschur is...

```Exclude zero columns of T from Kitagawa transformation, since ordschur is extremely noisy for multiple zero eigenvalues.
This can make a lot of difference for large models that have hundreds of definitions.```
parent 52978365
 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) function [Z,ST,R1,QT,Pstar,Pinf] = schur_statespace_transformation(mf,T,R,Q,qz_criterium, restrict_columns) % function [Z,ST,QT,R1,Pstar,Pinf] = schur_statespace_transformation(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. Computation of Pstar and % Pinf for Durbin and Koopman Diffuse filter % 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 ... ... @@ -44,6 +51,20 @@ function [Z,ST,R1,QT,Pstar,Pinf] = schur_statespace_transformation(mf,T,R,Q,qz_c % You should have received a copy of the GNU General Public License % along with Dynare. If not, see . np = size(T,1); if nargin == 6, indx = restrict_columns; indx0=find(~ismember([1:np],indx)); else indx=(find(max(abs(T))>0)); indx0=(find(max(abs(T))==0)); end 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; ... ... @@ -93,14 +114,40 @@ if i == nk+1 Pstar(nk1,nk1)=(B(nk1,nk1)+c)/(1-ST(nk1,nk1)*ST(nk1,nk1)); end Z = QT(mf,:); R1 = QT'*R; ST1=ST; % now I recover stationarized static variables % using % ss = s-z and % z-z(-1) (growth rates of unit roots) only depends on stationary variables np0=length(indx0); Pstar = blkdiag(zeros(np0),Pstar); ST = [zeros(length(Pstar),length(indx0)) [T0*QT ;ST]]; R1 = [R0; R1]; ST0=ST; ST0(:,1:np0+nk)=0; ST0(np0+1:np0+nk,:)=0; ST0(1:np0,np0+nk+1:end) = ST(1:np0,np0+nk+1:end)-ST(1:np0,np0+1:np0+nk)*ST(np0+1:np0+nk,np0+nk+1:end); R10 = R1; R10(np0+1:np0+nk,:)=0; R10(1:np0,:) = R1(1:np0,:)-ST(1:np0,np0+1:np0+nk)*R1(np0+1:np0+nk,:); Pstar = ... ST0*Pstar*ST0'+ ... R10*Q*R10'; QT = blkdiag(eye(np0),QT); QT(1:np0,np0+1:np0+nk) = QT(1:np0,np0+1:np0+nk)+ST(1:np0,np0+1:np0+nk); % reorder QT entries QT([indx0(:); indx(:)],:) = QT; Z = QT(mf,:); ST(1:np0,:) = ST0(1:np0,:); R1(1:np0,:) = R10(1:np0,:); % 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); [QQ,RR,EE] = qr(Z*ST(:,1+np0:nk+np0),0); k = find(abs(diag([RR; zeros(nk-size(Z,1),size(RR,2))])) < 1e-8); if length(k) > 0 k1 = EE(:,k); ... ... @@ -108,3 +155,5 @@ if length(k) > 0 dd(k1) = zeros(length(k1),1); Pinf(1:nk,1:nk) = diag(dd); end Pinf = blkdiag(zeros(np0),Pinf);
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