178 lines
5.9 KiB
Matlab
178 lines
5.9 KiB
Matlab
function [alphahat,etahat,atilde, aK] = DiffuseKalmanSmoother1(T,R,Q,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
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% function [alphahat,etahat,a, aK] = DiffuseKalmanSmoother1(T,R,Q,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
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% Computes the diffuse kalman smoother without measurement error, in the case of a non-singular var-cov matrix
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%
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% INPUTS
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% T: mm*mm matrix
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% R: mm*rr matrix
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% Q: rr*rr matrix
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% Pinf1: mm*mm diagonal matrix with with q ones and m-q zeros
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% Pstar1: mm*mm variance-covariance matrix with stationary variables
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% Y: pp*1 vector
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% trend
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% pp: number of observed variables
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% mm: number of state variables
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% smpl: sample size
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% mf: observed variables index in the state vector
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%
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% OUTPUTS
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% alphahat: smoothed state variables (a_{t|T})
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% etahat: smoothed shocks
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% atilde: matrix of updated variables (a_{t|t})
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% aK: 3D array of k step ahead filtered state variables (a_{t+k|t})
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% SPECIAL REQUIREMENTS
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% See "Filtering and Smoothing of State Vector for Diffuse State Space
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% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
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% Analysis, vol. 24(1), pp. 85-98).
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% Copyright (C) 2004-2008 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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% modified by M. Ratto:
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% new output argument aK (1-step to k-step predictions)
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% new options_.nk: the max step ahed prediction in aK (default is 4)
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% new crit1 value for rank of Pinf
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% it is assured that P is symmetric
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global options_
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nk = options_.nk;
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spinf = size(Pinf1);
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spstar = size(Pstar1);
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v = zeros(pp,smpl);
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a = zeros(mm,smpl+1);
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atilde = zeros(mm,smpl);
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aK = zeros(nk,mm,smpl+1);
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iF = zeros(pp,pp,smpl);
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Fstar = zeros(pp,pp,smpl);
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iFinf = zeros(pp,pp,smpl);
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K = zeros(mm,pp,smpl);
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L = zeros(mm,mm,smpl);
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Linf = zeros(mm,mm,smpl);
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Kstar = zeros(mm,pp,smpl);
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P = zeros(mm,mm,smpl+1);
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Pstar = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
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crit = options_.kalman_tol;
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crit1 = 1.e-8;
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steady = smpl;
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rr = size(Q,1);
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QQ = R*Q*transpose(R);
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QRt = Q*transpose(R);
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alphahat = zeros(mm,smpl);
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etahat = zeros(rr,smpl);
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r = zeros(mm,smpl+1);
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Z = zeros(pp,mm);
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for i=1:pp;
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Z(i,mf(i)) = 1;
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end
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t = 0;
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while rank(Pinf(:,:,t+1),crit1) & t<smpl
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t = t+1;
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v(:,t) = Y(:,t) - a(mf,t) - trend(:,t);
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if rcond(Pinf(mf,mf,t)) < crit
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return
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end
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iFinf(:,:,t) = inv(Pinf(mf,mf,t));
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PZI = Pinf(:,mf,t)*iFinf(:,:,t);
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atilde(:,t) = a(:,t) + PZI*v(:,t);
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Kinf(:,:,t) = T*PZI;
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a(:,t+1) = T*atilde(:,t);
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aK(1,:,t+1) = a(:,t+1);
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for jnk=2:nk,
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aK(jnk,:,t+jnk) = T^(jnk-1)*a(:,t+1);
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end
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Linf(:,:,t) = T - Kinf(:,:,t)*Z;
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Fstar(:,:,t) = Pstar(mf,mf,t);
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Kstar(:,:,t) = (T*Pstar(:,mf,t)-Kinf(:,:,t)*Fstar(:,:,t))*iFinf(:,:,t);
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)-T*Pstar(:,mf,t)*transpose(Kinf(:,:,t))-Kinf(:,:,t)*Pinf(mf,mf,t)*transpose(Kstar(:,:,t)) + QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T)-T*Pinf(:,mf,t)*transpose(Kinf(:,:,t));
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end
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d = t;
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P(:,:,d+1) = Pstar(:,:,d+1);
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iFinf = iFinf(:,:,1:d);
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Linf = Linf(:,:,1:d);
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Fstar = Fstar(:,:,1:d);
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Kstar = Kstar(:,:,1:d);
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Pstar = Pstar(:,:,1:d);
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Pinf = Pinf(:,:,1:d);
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notsteady = 1;
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while notsteady & t<smpl
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t = t+1;
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v(:,t) = Y(:,t) - a(mf,t) - trend(:,t);
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P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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if rcond(P(mf,mf,t)) < crit
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return
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end
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iF(:,:,t) = inv(P(mf,mf,t));
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PZI = P(:,mf,t)*iF(:,:,t);
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atilde(:,t) = a(:,t) + PZI*v(:,t);
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K(:,:,t) = T*PZI;
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L(:,:,t) = T-K(:,:,t)*Z;
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a(:,t+1) = T*atilde(:,t);
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aK(1,:,t+1) = a(:,t+1);
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for jnk=2:nk,
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aK(jnk,:,t+jnk) = T^(jnk-1)*a(:,t+1);
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end
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P(:,:,t+1) = T*P(:,:,t)*transpose(T)-T*P(:,mf,t)*transpose(K(:,:,t)) + QQ;
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notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
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end
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if t<smpl
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PZI_s = PZI;
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K_s = K(:,:,t);
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iF_s = iF(:,:,t);
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P_s = P(:,:,t+1);
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t_steady = t+1;
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P = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
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iF = cat(3,iF(:,:,1:t),repmat(inv(P_s(mf,mf)),[1 1 smpl-t_steady+1]));
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L = cat(3,L(:,:,1:t),repmat(T-K_s*Z,[1 1 smpl-t_steady+1]));
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K = cat(3,K(:,:,1:t),repmat(T*P_s(:,mf)*iF_s,[1 1 smpl-t_steady+1]));
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end
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while t<smpl
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t=t+1;
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v(:,t) = Y(:,t) - a(mf,t) - trend(:,t);
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atilde(:,t) = a(:,t) + PZI*v(:,t);
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a(:,t+1) = T*a(:,t) + K_s*v(:,t);
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aK(1,:,t+1) = a(:,t+1);
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for jnk=2:nk,
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aK(jnk,:,t+jnk) = T^(jnk-1)*a(:,t+1);
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end
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end
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t = smpl+1;
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while t>d+1
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t = t-1;
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r(:,t) = Z'*iF(:,:,t)*v(:,t) + L(:,:,t)'*r(:,t+1);
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alphahat(:,t) = a(:,t) + P(:,:,t)*r(:,t);
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etahat(:,t) = QRt*r(:,t);
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end
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if d
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r0 = zeros(mm,d+1);
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r0(:,d+1) = r(:,d+1);
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r1 = zeros(mm,d+1);
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for t = d:-1:1
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r0(:,t) = Linf(:,:,t)'*r0(:,t+1);
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r1(:,t) = Z'*(iFinf(:,:,t)*v(:,t)-Kstar(:,:,t)'*r0(:,t+1)) + Linf(:,:,t)'*r1(:,t+1);
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alphahat(:,t) = a(:,t) + Pstar(:,:,t)*r0(:,t) + Pinf(:,:,t)*r1(:,t);
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etahat(:,t) = QRt*r0(:,t);
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end
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end
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