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dynare/matlab/DiffuseKalmanSmootherH3_Z.m

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function [alphahat,epsilonhat,etahat,a,P,aK,PK,d,decomp] = DiffuseKalmanSmootherH3_Z(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl)
% function [alphahat,epsilonhat,etahat,a1,P,aK,PK,d,decomp_filt] = DiffuseKalmanSmootherH3(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl)
% Computes the diffuse kalman smoother without measurement error, in the case of a singular var-cov matrix.
% Univariate treatment of multivariate time series.
%
% INPUTS
% T: mm*mm matrix
% Z: pp*mm matrix
% R: mm*rr matrix
% Q: rr*rr matrix
% Pinf1: mm*mm diagonal matrix with with q ones and m-q zeros
% Pstar1: mm*mm variance-covariance matrix with stationary variables
% Y: pp*1 vector
% pp: number of observed variables
% mm: number of state variables
% smpl: sample size
%
% OUTPUTS
% alphahat: smoothed state variables (a_{t|T})
% etahat: smoothed shocks
% epsilonhat:smoothed measurement error
% a: matrix of updated variables (a_{t|t})
% aK: 3D array of k step ahead filtered state variables (a_{t+k|t})
% (meaningless for periods 1:d)
% P: 3D array of one-step ahead forecast error variance
% matrices
% PK: 4D array of k-step ahead forecast error variance
% matrices (meaningless for periods 1:d)
% d: number of periods where filter remains in diffuse part
% (should be equal to the order of integration of the model)
% decomp: decomposition of the effect of shocks on filtered values
%
% SPECIAL REQUIREMENTS
% See "Filtering and Smoothing of State Vector for Diffuse State Space
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
% Analysis, vol. 24(1), pp. 85-98).
% Copyright (C) 2004-2008 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/>.
% Modified by M. Ratto
% New output argument aK: 1-step to nk-stpe ahed predictions)
% New input argument nk: max order of predictions in aK
% New option options_.diffuse_d where the DKF stops (common with
% diffuselikelihood3)
% New icc variable to count number of iterations for Finf steps
% Pstar % Pinf simmetric
% New termination of DKF iterations based on options_.diffuse_d
% Li also stored during DKF iterations !!
% some bugs corrected in the DKF part of the smoother (Z matrix and
% alphahat)
global options_
d = 0;
decomp = [];
nk = options_.nk;
spinf = size(Pinf1);
spstar = size(Pstar1);
v = zeros(pp,smpl);
a = zeros(mm,smpl);
a1 = zeros(mm,smpl+1);
aK = zeros(nk,mm,smpl+nk);
if isempty(options_.diffuse_d),
smpl_diff = 1;
else
smpl_diff=rank(Pinf1);
end
Fstar = zeros(pp,smpl_diff);
Finf = zeros(pp,smpl_diff);
Ki = zeros(mm,pp,smpl);
Li = zeros(mm,mm,pp,smpl);
Linf = zeros(mm,mm,pp,smpl_diff);
L0 = zeros(mm,mm,pp,smpl_diff);
Kstar = zeros(mm,pp,smpl_diff);
P = zeros(mm,mm,smpl+1);
P1 = P;
aK = zeros(nk,mm,smpl+nk);
PK = zeros(nk,mm,mm,smpl+nk);
Pstar = zeros(spstar(1),spstar(2),smpl_diff+1); Pstar(:,:,1) = Pstar1;
Pinf = zeros(spinf(1),spinf(2),smpl_diff+1); Pinf(:,:,1) = Pinf1;
Pstar1 = Pstar;
Pinf1 = Pinf;
crit = options_.kalman_tol;
crit1 = 1.e-6;
steady = smpl;
rr = size(Q,1); % number of structural shocks
QQ = R*Q*transpose(R);
QRt = Q*transpose(R);
alphahat = zeros(mm,smpl);
etahat = zeros(rr,smpl);
epsilonhat = zeros(size(Y));
r = zeros(mm,smpl);
t = 0;
icc=0;
newRank = rank(Pinf(:,:,1),crit1);
while newRank & t < smpl
t = t+1;
a(:,t) = a1(:,t);
Pstar(:,:,t)=tril(Pstar(:,:,t))+tril(Pstar(:,:,t),-1)';
Pinf(:,:,t)=tril(Pinf(:,:,t))+tril(Pinf(:,:,t),-1)';
Pstar1(:,:,t) = Pstar(:,:,t);
Pinf1(:,:,t) = Pinf(:,:,t);
for i=1:pp
Zi = Z(i,:);
v(i,t) = Y(i,t)-Zi*a(:,t);
Fstar(i,t) = Zi*Pstar(:,:,t)*Zi' +H(i,i);
Finf(i,t) = Zi*Pinf(:,:,t)*Zi';
Kstar(:,i,t) = Pstar(:,:,t)*Zi';
if Finf(i,t) > crit & newRank
icc=icc+1;
Kinf(:,i,t) = Pinf(:,:,t)*Zi';
Linf(:,:,i,t) = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
L0(:,:,i,t) = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Zi/Finf(i,t);
a(:,t) = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
Pstar(:,:,t) = Pstar(:,:,t) + ...
Kinf(:,i,t)*Kinf(:,i,t)'*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - ...
(Kstar(:,i,t)*Kinf(:,i,t)' +...
Kinf(:,i,t)*Kstar(:,i,t)')/Finf(i,t);
Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*Kinf(:,i,t)'/Finf(i,t);
Pstar(:,:,t)=tril(Pstar(:,:,t))+tril(Pstar(:,:,t),-1)';
Pinf(:,:,t)=tril(Pinf(:,:,t))+tril(Pinf(:,:,t),-1)';
% new terminiation criteria by M. Ratto
P0=Pinf(:,:,t);
if ~isempty(options_.diffuse_d),
newRank = (icc<options_.diffuse_d);
if newRank & (any(diag(Z*P0*Z')>crit)==0 & rank(P0,crit1)==0);
disp('WARNING!! Change in OPTIONS_.DIFFUSE_D in univariate DKF')
options_.diffuse_d = icc;
newRank=0;
end
else
newRank = (any(diag(Z*P0*Z')>crit) | rank(P0,crit1));
if newRank==0,
options_.diffuse_d = icc;
end
end,
% end new terminiation criteria by M. Ratto
elseif Fstar(i,t) > crit
%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
%% rank(Pinf)=0. [st<73>phane,11-03-2004].
Li(:,:,i,t) = eye(mm)-Kstar(:,i,t)*Z(i,:)/Fstar(i,t); % we need to store Li for DKF smoother
a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*Kstar(:,i,t)'/Fstar(i,t);
Pstar(:,:,t)=tril(Pstar(:,:,t))+tril(Pstar(:,:,t),-1)';
end
end
a1(:,t+1) = T*a(:,t);
for jnk=1:nk,
aK(jnk,:,t+jnk) = T^jnk*a(:,t);
end
Pstar(:,:,t+1) = T*Pstar(:,:,t)*T'+ QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*T';
P0=Pinf(:,:,t+1);
if newRank,
newRank = rank(P0,crit1);
end
end
d = t;
P(:,:,d+1) = Pstar(:,:,d+1);
Linf = Linf(:,:,:,1:d);
L0 = L0(:,:,:,1:d);
Fstar = Fstar(:,1:d);
Finf = Finf(:,1:d);
Kstar = Kstar(:,:,1:d);
Pstar = Pstar(:,:,1:d);
Pinf = Pinf(:,:,1:d);
Pstar1 = Pstar1(:,:,1:d);
Pinf1 = Pinf1(:,:,1:d);
notsteady = 1;
while notsteady & t<smpl
t = t+1;
a(:,t) = a1(:,t);
P(:,:,t)=tril(P(:,:,t))+tril(P(:,:,t),-1)';
P1(:,:,t) = P(:,:,t);
for i=1:pp
Zi = Z(i,:);
v(i,t) = Y(i,t) - Zi*a(:,t);
Fi(i,t) = Zi*P(:,:,t)*Zi' + H(i,i);
Ki(:,i,t) = P(:,:,t)*Zi';
if Fi(i,t) > crit
Li(:,:,i,t) = eye(mm)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t);
P(:,:,t) = P(:,:,t) - Ki(:,i,t)*Ki(:,i,t)'/Fi(i,t);
P(:,:,t)=tril(P(:,:,t))+tril(P(:,:,t),-1)';
end
end
a1(:,t+1) = T*a(:,t);
Pf = P(:,:,t);
for jnk=1:nk,
Pf = T*Pf*T' + QQ;
aK(jnk,:,t+jnk) = T^jnk*a(:,t);
PK(jnk,:,:,t+jnk) = Pf;
end
P(:,:,t+1) = T*P(:,:,t)*T' + QQ;
notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
end
P_s=tril(P(:,:,t))+tril(P(:,:,t),-1)';
P1_s=tril(P1(:,:,t))+tril(P1(:,:,t),-1)';
Fi_s = Fi(:,t);
Ki_s = Ki(:,:,t);
L_s =Li(:,:,:,t);
if t<smpl
P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
P1 = cat(3,P1(:,:,1:t),repmat(P1_s,[1 1 smpl-t]));
Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t]));
Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t]));
Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t]));
end
while t<smpl
t=t+1;
a(:,t) = a1(:,t);
for i=1:pp
Zi = Z(i,:);
v(i,t) = Y(i,t) - Zi*a(:,t);
if Fi_s(i) > crit
a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
end
end
a1(:,t+1) = T*a(:,t);
Pf = P(:,:,t);
for jnk=1:nk,
Pf = T*Pf*T' + QQ;
aK(jnk,:,t+jnk) = T^jnk*a(:,t);
PK(jnk,:,:,t+jnk) = Pf;
end
end
ri=zeros(mm,1);
t = smpl+1;
while t>d+1
t = t-1;
for i=pp:-1:1
if Fi(i,t) > crit
ri = Z(i,:)'/Fi(i,t)*v(i,t)+Li(:,:,i,t)'*ri;
end
end
r(:,t) = ri;
alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t);
etahat(:,t) = QRt*r(:,t);
ri = T'*ri;
end
if d
r0 = zeros(mm,d);
r0(:,d) = ri;
r1 = zeros(mm,d);
for t = d:-1:2
for i=pp:-1:1
% if Finf(i,t) > crit & ~(t==d & i>options_.diffuse_d), % use of options_.diffuse_d to be sure of DKF termination
if Finf(i,t) > crit
r1(:,t) = Z(i,:)'*v(i,t)/Finf(i,t) + ...
L0(:,:,i,t)'*r0(:,t) + Linf(:,:,i,t)'*r1(:,t);
r0(:,t) = Linf(:,:,i,t)'*r0(:,t);
elseif Fstar(i,t) > crit % step needed whe Finf == 0
r0(:,t) = Z(i,:)'/Fstar(i,t)*v(i,t)+Li(:,:,i,t)'*r0(:,t);
end
end
alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
r(:,t) = r0(:,t);
etahat(:,t) = QRt*r(:,t);
if t > 1
r0(:,t-1) = T'*r0(:,t);
r1(:,t-1) = T'*r1(:,t);
end
end
end
if nargout > 7
decomp = zeros(nk,mm,rr,smpl+nk);
ZRQinv = inv(Z*QQ*Z');
for t = max(d,1):smpl
ri_d = zeros(mm,1);
for i=pp:-1:1
if Fi(i,t) > crit
ri_d = Z(i,:)'/Fi(i,t)*v(i,t)+Li(:,:,i,t)'*ri_d;
end
end
% calculate eta_tm1t
eta_tm1t = QRt*ri_d;
% calculate decomposition
Ttok = eye(mm,mm);
for h = 1:nk
for j=1:rr
eta=zeros(rr,1);
eta(j) = eta_tm1t(j);
decomp(h,:,j,t+h) = Ttok*P1(:,:,t)*Z'*ZRQinv*Z*R*eta;
end
Ttok = T*Ttok;
end
end
end
epsilonhat = Y-Z*alphahat;
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