320 lines
11 KiB
Matlab
320 lines
11 KiB
Matlab
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;
|
||
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);
|
||
aK(1,:,t+1) = a1(:,t+1);
|
||
for jnk=2:nk
|
||
aK(jnk,:,t+jnk) = T*squeeze(aK(jnk-1,:,t+jnk-1));
|
||
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);
|
||
aK(1,:,t+1) = a1(:,t+1);
|
||
for jnk=1:nk,
|
||
Pf = T*Pf*T' + QQ;
|
||
PK(jnk,:,:,t+jnk) = Pf;
|
||
if jnk>1
|
||
aK(jnk,:,t+jnk) = T*squeeze(aK(jnk-1,:,t+jnk-1)) ;
|
||
end
|
||
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);
|
||
aK(1,:,t+1) = a1(:,t+1);
|
||
for jnk=1:nk,
|
||
Pf = T*Pf*T' + QQ;
|
||
PK(jnk,:,:,t+jnk) = Pf;
|
||
if jnk>1
|
||
aK(jnk,:,t+jnk) = T*squeeze(aK(jnk-1,:,t+jnk-1));
|
||
end
|
||
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;
|