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dynare/matlab/DiffuseKalmanSmootherH1_Z.m

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Matlab
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function [alphahat,epsilonhat,etahat,atilde,P,aK,PK,d,decomp] = DiffuseKalmanSmootherH1_Z(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl)
% function [alphahat,etahat,atilde, aK] = DiffuseKalmanSmootherH1_Z(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl)
% Computes the diffuse kalman smoother without measurement error, in the case of a non-singular var-cov matrix
%
% INPUTS
% T: mm*mm matrix
% Z: pp*mm matrix
% R: mm*rr matrix
% Q: rr*rr matrix
% H: pp*pp matrix variance of measurement errors
% 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 variables (a_{t|T})
% epsilonhat:smoothed measurement errors
% etahat: smoothed shocks
% atilde: 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) 2004-2011 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 k-step predictions)
% new options_.nk: the max step ahed prediction in aK (default is 4)
% new crit1 value for rank of Pinf
% it is assured that P is symmetric
global options_
d = 0;
decomp = [];
nk = options_.nk;
spinf = size(Pinf1);
spstar = size(Pstar1);
v = zeros(pp,smpl);
a = zeros(mm,smpl+1);
atilde = zeros(mm,smpl);
aK = zeros(nk,mm,smpl+nk);
PK = zeros(nk,mm,mm,smpl+nk);
iF = zeros(pp,pp,smpl);
Fstar = zeros(pp,pp,smpl);
iFinf = zeros(pp,pp,smpl);
K = zeros(mm,pp,smpl);
L = zeros(mm,mm,smpl);
Linf = zeros(mm,mm,smpl);
Kstar = zeros(mm,pp,smpl);
P = zeros(mm,mm,smpl+1);
Pstar = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
Pinf = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
crit = options_.kalman_tol;
crit1 = 1.e-8;
steady = smpl;
rr = size(Q,1);
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+1);
t = 0;
while rank(Pinf(:,:,t+1),crit1) & t<smpl
t = t+1;
v(:,t)= Y(:,t) - Z*a(:,t);
Finf = Z*Pinf(:,:,t)*Z';
if rcond(Finf) < kalman_tol
if ~all(abs(Finf(:)) < kalman_tol)
% The univariate diffuse kalman filter should be used.
return
else
Fstar(:,:,t) = Z*Pstar(:,:,t)*Z' + H;
if rcond(Fstar(:,:,t)) < kalman_tol
if ~all(abs(Fstar(:,:,t))<kalman_tol)
% The univariate diffuse kalman filter should be used.
return
else
a(:,:,t+1) = T*a(:,:,t);
Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
end
else
iFstar = inv(Fstar(:,:,t));
Kstar(:,:,t) = Pstar(:,:,t)*Z'*iFstar(:,:,t);
Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
Pstar(:,:,t+1) = T*(Pstar(:,:,t)-Pstar(:,:,t)*Z'*Kstar(:,:,t)')*T'+QQ;
a(:,:,t+1) = T*(a(:,:,t)+Kstar(:,:,t)*v(:,t));
end
end
else
iFinf(:,:,t) = inv(F);
PZI = Pinf(:,:,t)*Z'*iFinf(:,:,t);
atilde(:,t) = a(:,t) + PZI*v(:,t);
Kinf(:,:,t) = T*PZI;
a(:,t+1) = T*atilde(:,t);
aK(1,:,t+1) = a(:,t+1);
% isn't a meaningless as long as we are in the diffuse part? MJ
for jnk=2:nk
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
Linf(:,:,t) = T - Kinf(:,:,t)*Z;
Fstar(:,:,t) = Z*Pstar(:,:,t)*Z' + H;
Kstar(:,:,t) = (T*Pstar(:,:,t)*Z'-Kinf(:,:,t)*Fstar(:,:,t))*iFinf(:,:,t);
Pstar(:,:,t+1) = T*Pstar(:,:,t)*T'-T*Pstar(:,:,t)*Z'*Kinf(:,:,t)'-T*Pinf(:,:,t)*Z'*Kstar(:,:,t)' + QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*T'-T*Pinf(:,:,t)*Z'*Kinf(:,:,t)';
end
end
d = t;
P(:,:,d+1) = Pstar(:,:,d+1);
iFinf = iFinf(:,:,1:d);
Linf = Linf(:,:,1:d);
Fstar = Fstar(:,:,1:d);
Kstar = Kstar(:,:,1:d);
Pstar = Pstar(:,:,1:d);
Pinf = Pinf(:,:,1:d);
notsteady = 1;
while notsteady & t<smpl
t = t+1;
v(:,t) = Y(:,t) - Z*a(:,t);
P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
F = Z*P(:,:,t)*Z' + H;
if rcond(F) < crit
return
end
iF(:,:,t) = inv(F);
PZI = P(:,:,t)*Z'*iF(:,:,t);
atilde(:,t) = a(:,t) + PZI*v(:,t);
K(:,:,t) = T*PZI;
L(:,:,t) = T-K(:,:,t)*Z;
a(:,t+1) = T*atilde(:,t);
Pf = P(:,:,t);
aK(1,:,t+1) = a(:,t+1);
for jnk=1:nk
Pf = T*Pf*T' + QQ;
PK(jnk,:,:,t+jnk) = Pf;
if jnk>1
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
end
P(:,:,t+1) = T*P(:,:,t)*T'-T*P(:,:,t)*Z'*K(:,:,t)' + QQ;
notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
end
if t<smpl
PZI_s = PZI;
K_s = K(:,:,t);
iF_s = iF(:,:,t);
P_s = P(:,:,t+1);
P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
iF = cat(3,iF(:,:,1:t),repmat(iF_s,[1 1 smpl-t]));
L = cat(3,L(:,:,1:t),repmat(T-K_s*Z,[1 1 smpl-t]));
K = cat(3,K(:,:,1:t),repmat(T*P_s*Z'*iF_s,[1 1 smpl-t]));
end
while t<smpl
t=t+1;
v(:,t) = Y(:,t) - Z*a(:,t);
atilde(:,t) = a(:,t) + PZI*v(:,t);
a(:,t+1) = T*atilde(:,t);
Pf = P(:,:,t);
aK(1,:,t+1) = a(:,t+1);
for jnk=1:nk
Pf = T*Pf*T' + QQ;
PK(jnk,:,:,t+jnk) = Pf;
if jnk>1
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
end
end
t = smpl+1;
while t>d+1
t = t-1;
r(:,t) = Z'*iF(:,:,t)*v(:,t) + L(:,:,t)'*r(:,t+1);
alphahat(:,t) = a(:,t) + P(:,:,t)*r(:,t);
etahat(:,t) = QRt*r(:,t);
end
if d
r0 = zeros(mm,d+1);
r0(:,d+1) = r(:,d+1);
r1 = zeros(mm,d+1);
for t = d:-1:1
r0(:,t) = Linf(:,:,t)'*r0(:,t+1);
r1(:,t) = Z'*(iFinf(:,:,t)*v(:,t)-Kstar(:,:,t)'*r0(:,t+1)) + Linf(:,:,t)'*r1(:,t+1);
alphahat(:,t) = a(:,t) + Pstar(:,:,t)*r0(:,t) + Pinf(:,:,t)*r1(:,t);
etahat(:,t) = QRt*r0(:,t);
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 = Z'*iF(:,:,t)*v(:,t);
% 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) = T^(h-1)*P(:,:,t)*Z'*ZRQinv*Z*R*eta;
end
end
end
end
epsilonhat = Y-Z*alphahat;
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