189 lines
6.1 KiB
Matlab
189 lines
6.1 KiB
Matlab
function [alphahat,epsilonhat,etahat,atilde,P,aK,PK,decomp] = kalman_smoother(T,R,Q,H,P0,Y,start,mf,kalman_tol,riccati_tol)
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% function [alphahat,epsilonhat,etahat,a,aK,PK,decomp] = kalman_smoother(T,R,Q,H,P,Y,start,mf,kalman_tol,riccati_tol)
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% Computes the kalman smoother of a stationary state space model.
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%
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% INPUTS
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% T [double] mm*mm transition matrix of the state equation.
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% R [double] mm*rr matrix, mapping structural innovations to state variables.
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% Q [double] rr*rr covariance matrix of the structural innovations.
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% H [double] pp*pp (or 1*1 =0 if no measurement error) covariance matrix of the measurement errors.
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% P0 [double] mm*mm variance-covariance matrix with stationary variables
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% Y [double] pp*smpl matrix of (detrended) data, where pp is the maximum number of observed variables.
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% start [integer] scalar, likelihood evaluation starts at 'start'.
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% mf [integer] pp*1 vector of indices.
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% kalman_tol [double] scalar, tolerance parameter (rcond).
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% riccati_tol [double] scalar, tolerance parameter (riccati iteration).
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%
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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) 2004-2010 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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global options_
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option_filter_covariance = options_.filter_covariance;
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option_filter_decomposition = options_.filter_decomposition;
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nk = options_.nk;
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smpl = size(Y,2); % Sample size.
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mm = size(T,2); % Number of state variables.
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pp = size(Y,1); % Maximum number of
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% observed variables.
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rr = size(Q,1);
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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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K = zeros(mm,pp,smpl);
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aK = zeros(nk,mm,smpl+nk);
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iF = zeros(pp,pp,smpl);
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P = zeros(mm,mm,smpl+1);
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QQ = R*Q*R';
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QRt = Q*R';
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alphahat = zeros(mm,smpl);
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etahat = zeros(rr,smpl);
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epsilonhat = zeros(rr,smpl);
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r = zeros(mm,smpl+1);
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oldK = 0;
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if option_filter_covariance
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PK = zeros(nk,mm,mm,smpl+nk);
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else
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PK = [];
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end
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if option_filter_decomposition
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decomp = zeros(nk,mm,rr,smpl+nk);
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else
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decomp = [];
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end
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P(:,:,1) = P0;
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t = 0;
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notsteady = 1;
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F_singular = 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);
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F = P(mf,mf,t) + H;
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if rcond(F) < kalman_tol
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if ~all(abs(F(:))<kalman_tol)
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return
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else
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atilde(:,t) = a(:,t);
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a(:,t+1) = T*a(:,t);
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P(:,:,t+1) = T*P(:,:,t)*T'+QQ;
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end
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else
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F_singular = 0;
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iF(:,:,t) = inv(F);
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K1 = P(:,mf,t)*iF(:,:,t);
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atilde(:,t) = a(:,t) + K1*v(:,t);
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K(:,:,t) = T*K1;
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a(:,t+1) = T*atilde(:,t);
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P(:,:,t+1) = (T*P(:,:,t)-K(:,:,t)*P(mf,:,t))*T'+QQ;
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end
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aK(1,:,t+1) = a(:,t+1);
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if option_filter_covariance
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Pf = P(:,:,t);
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Pf = T*Pf*T' + QQ;
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PK(1,:,:,t+1) = Pf;
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end
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for jnk=2:nk,
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aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
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if option_filter_covariance
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Pf = T*Pf*T' + QQ;
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PK(jnk,:,:,t+jnk) = Pf;
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end
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end
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notsteady = max(max(abs(K(:,:,t)-oldK))) > riccati_tol;
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oldK = K(:,:,t);
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end
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if F_singular
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error('The variance of the forecast error remains singular until the end of the sample')
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end
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if t < smpl
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t0 = t;
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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);
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atilde(:,t) = a(:,t) + K1*v(:,t);
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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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if option_filter_covariance
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Pf = P(:,:,t);
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Pf = T*Pf*T' + QQ;
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PK(1,:,:,t+1) = Pf;
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end
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for jnk=2:nk,
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aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
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if option_filter_covariance
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Pf = T*Pf*T' + QQ;
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PK(jnk,:,:,t+jnk) = Pf;
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end
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end
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end
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K= cat(3,K(:,:,1:t0),repmat(K(:,:,t0),[1 1 smpl-t0+1]));
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P = cat(3,P(:,:,1:t0),repmat(P(:,:,t0),[1 1 smpl-t0+1]));
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iF = cat(3,iF(:,:,1:t0),repmat(iF(:,:,t0),[1 1 smpl-t0+1]));
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end
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t = smpl+1;
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while t>1
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t = t-1;
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r(:,t) = T'*r(:,t+1);
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r(mf,t) = r(mf,t)+iF(:,:,t)*v(:,t) - K(:,:,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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epsilonhat = Y-alphahat(mf,:);
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if option_filter_decomposition
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ZRQinv = inv(QQ(mf,mf));
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for t = 1:smpl
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% calculate eta_tm1t
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eta = QRt(:,mf)*iF(:,:,t)*v(:,t);
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AAA = P(:,mf,t)*ZRQinv*bsxfun(@times,R(mf,:),eta');
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% calculate decomposition
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Ttok = eye(mm,mm);
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decomp(1,:,:,t+1) = AAA;
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for h = 2:nk
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AAA = T*AAA;
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decomp(h,:,:,t+h) = AAA;
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end
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end
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end
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if ~option_filter_covariance
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P = [];
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end
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