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dynare/matlab/DiffuseKalmanSmootherH3corr.m

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function [alphahat,epsilonhat,etahat,a,aK] = DiffuseKalmanSmootherH3corr(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
% function [alphahat,epsilonhat,etahat,a1] = DiffuseKalmanSmootherH3corr(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
% Computes the diffuse kalman smoother with measurement error, in the case of a singular var-cov matrix.
% Univariate treatment of multivariate time series.
%
% INPUTS
% T: mm*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
% trend
% pp: number of observed variables
% mm: number of state variables
% smpl: sample size
% mf: observed variables index in the state vector
%
% OUTPUTS
% alphahat: smoothed state variables (a_{t|T})
% epsilonhat:smoothed measurement error
% etahat: smoothed shocks
% a: matrix of updated variables (a_{t|t})
% aK: matrix of one step ahead filtered state variables (a_{t+k|t})
% SPECIAL REQUIREMENTS
% See "Fast Filtering and Smoothing for Multivariate State Space
% Models", S.J. Koopman and J. Durbin (2000, in Journal of Time Series
% Analysis, vol. 21(3), pp. 281-296).
% 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/>.
global options_;
nk = options_.nk;
rr = size(Q,1);
T = cat(1,cat(2,T,zeros(mm,pp)),zeros(pp,mm+pp));
R = cat(1,cat(2,R,zeros(mm,pp)),cat(2,zeros(pp,rr),eye(pp)));
Q = cat(1,cat(2,Q,zeros(rr,pp)),cat(2,zeros(pp,rr),H));
if size(Pinf1,1) % Otherwise Pinf = 0 (no unit root)
Pinf1 = cat(1,cat(2,Pinf1,zeros(mm,pp)),zeros(pp,mm+pp));
end
Pstar1 = cat(1,cat(2,Pstar1,zeros(mm,pp)),cat(2,zeros(pp,mm),H));
spinf = size(Pinf1);
spstar = size(Pstar1);
Pstar = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
Pinf = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
Pstar1 = Pstar;
Pinf1 = Pinf;
v = zeros(pp,smpl);
a = zeros(mm+pp,smpl);
a1 = zeros(mm+pp,smpl+1);
aK = zeros(nk,mm,smpl+nk);
Fstar = zeros(pp,smpl);
Finf = zeros(pp,smpl);
Fi = zeros(pp,smpl);
Ki = zeros(mm+pp,pp,smpl);
Li = zeros(mm+pp,mm+pp,pp,smpl);
Linf = zeros(mm+pp,mm+pp,pp,smpl);
L0 = zeros(mm+pp,mm+pp,pp,smpl);
Kstar = zeros(mm+pp,pp,smpl);
Kinf = zeros(mm+pp,pp,smpl);
P = zeros(mm+pp,mm+pp,smpl+1);
P1 = zeros(mm+pp,mm+pp,smpl+1);
crit = options_.kalman_tol;
steady = smpl;
QQ = R*Q*transpose(R);
QRt = Q*transpose(R);
alphahat = zeros(mm+pp,smpl);
etahat = zeros(rr,smpl);
epsilonhat = zeros(pp,smpl);
r = zeros(mm+pp,smpl+1);
Z = zeros(pp,mm+pp);
for i=1:pp;
Z(i,mf(i)) = 1;
Z(i,mm+i) = 1;
end
%% [1] Kalman filter...
t = 0;
newRank = rank(Pinf(:,:,1),crit);
while newRank & t < smpl
t = t+1;
a(:,t) = a1(:,t);
Pstar1(:,:,t) = Pstar(:,:,t);
Pinf1(:,:,t) = Pinf(:,:,t);
for i=1:pp
v(i,t) = Y(i,t)-a(mf(i),t)-a(mm+i,t)-trend(i,t);
Fstar(i,t) = Pstar(mf(i),mf(i),t)+Pstar(mm+i,mm+i,t);
Finf(i,t) = Pinf(mf(i),mf(i),t);
Kstar(:,i,t) = Pstar(:,mf(i),t)+Pstar(:,mm+i,t);
if Finf(i,t) > crit
Kinf(:,i,t) = Pinf(:,mf(i),t);
Linf(:,:,i,t) = eye(mm+pp) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
L0(:,:,i,t) = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Z(i,:)/Finf(i,t);
a(:,t) = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
Pstar(:,:,t) = Pstar(:,:,t) + ...
Kinf(:,i,t)*transpose(Kinf(:,i,t))*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - ...
(Kstar(:,i,t)*transpose(Kinf(:,i,t)) +...
Kinf(:,i,t)*transpose(Kstar(:,i,t)))/Finf(i,t);
Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*transpose(Kinf(:,i,t))/Finf(i,t);
else %% 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].
a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*transpose(Kstar(:,i,t))/Fstar(i,t);
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)*transpose(T)+ QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
P0=Pinf(:,:,t+1);
newRank = ~all(abs(P0(:))<crit);
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))+transpose(tril(P(:,:,t),-1));
P1(:,:,t) = P(:,:,t);
for i=1:pp
v(i,t) = Y(i,t) - a(mf(i),t) - a(mm+i,t) - trend(i,t);
Fi(i,t) = P(mf(i),mf(i),t)+P(mm+i,mm+i,t);
Ki(:,i,t) = P(:,mf(i),t)+P(:,mm+i,t);
if Fi(i,t) > crit
Li(:,:,i,t) = eye(mm+pp)-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)*transpose(Ki(:,i,t))/Fi(i,t);
P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
end
end
a1(:,t+1) = T*a(:,t);
for jnk=1:nk,
aK(jnk,:,t+jnk) = T^jnk*a(:,t);
end
P(:,:,t+1) = T*P(:,:,t)*transpose(T) + QQ;
notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
end
P_s=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
Fi_s = Fi(:,t);
Ki_s = Ki(:,:,t);
L_s =Li(:,:,:,t);
if t<smpl
t_steady = t+1;
P = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t_steady+1]));
Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t_steady+1]));
Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t_steady+1]));
end
while t<smpl
t=t+1;
a(:,t) = a1(:,t);
for i=1:pp
v(i,t) = Y(i,t) - a(mf(i),t) - a(mm+i,t) - trend(i,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);
for jnk=1:nk,
aK(jnk,:,t+jnk) = T^jnk*a(:,t);
end
end
%% [2] Kalman smoother...
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(:,t);
alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t);
tmp = QRt*r(:,t);
etahat(:,t) = tmp(1:rr);
epsilonhat(:,t) = tmp(rr+(1:pp));
ri = T'*ri;
end
if d
r0 = zeros(mm+pp,d);
r0(:,d) = ri;
r1 = zeros(mm+pp,d);
for t = d:-1:1
for i=pp:-1:1
if Finf(i,t) > crit
r1(:,t) = transpose(Z)*v(:,t)/Finf(i,t) + ...
L0(:,:,i,t)'*r0(:,t) + Linf(:,:,i,t)'*r1(:,t);
r0(:,t) = transpose(Linf(:,:,i,t))*r0(:,t);
end
end
alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
r(:,t-1) = r0(:,t);
tmp = QRt*r(:,t);
etahat(:,t) = tmp(1:rr);
epsilonhat(:,t) = tmp(rr+(1:pp));
if t > 1
r0(:,t-1) = T'*r0(:,t);
r1(:,t-1) = T'*r1(:,t);
end
end
end
alphahat = alphahat(1:mm,:);
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