280 lines
9.1 KiB
Matlab
280 lines
9.1 KiB
Matlab
function [alphahat,epsilonhat,etahat,a, aK] = DiffuseKalmanSmootherH3(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
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% function [alphahat,epsilonhat,etahat,a1, aK] = DiffuseKalmanSmootherH3(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
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% Computes the diffuse kalman smoother with measurement error, in the case of a singular var-cov matrix.
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% Univariate treatment of multivariate time series.
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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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% epsilonhat:smoothed measurement error
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% etahat: smoothed shocks
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% a: 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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%
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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 nk-stpe ahed predictions)
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% New input argument nk: max order of predictions in aK
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% New global variable id_ where the DKF stops (common with
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% diffuselikelihood3)
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% New icc variable to count number of iterations for Finf steps
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% Pstar % Pinf simmetric
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% New termination of DKF iterations based on id_
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% Li also stored during DKF iterations !!
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% some bugs corrected in the DKF part of the smoother (Z matrix and
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% alphahat)
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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);
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a1 = zeros(mm,smpl+1);
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aK = zeros(nk,mm,smpl+nk);
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if isempty(options_.diffuse_d),
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smpl_diff = 1;
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else
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smpl_diff=rank(Pinf1);
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end
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Fstar = zeros(pp,smpl_diff);
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Finf = zeros(pp,smpl_diff);
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Ki = zeros(mm,pp,smpl);
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Li = zeros(mm,mm,pp,smpl);
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Linf = zeros(mm,mm,pp,smpl_diff);
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L0 = zeros(mm,mm,pp,smpl_diff);
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Kstar = zeros(mm,pp,smpl_diff);
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P = zeros(mm,mm,smpl+1);
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P1 = P;
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Pstar = zeros(spstar(1),spstar(2),smpl_diff+1); Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl_diff+1); Pinf(:,:,1) = Pinf1;
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Pstar1 = Pstar;
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Pinf1 = Pinf;
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crit = options_.kalman_tol;
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crit1 = 1.e-6;
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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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epsilonhat = zeros(size(Y));
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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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icc=0;
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newRank = rank(Pinf(:,:,1),crit1);
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while newRank & t < smpl
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t = t+1;
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a(:,t) = a1(:,t);
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Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
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Pinf(:,:,t)=tril(Pinf(:,:,t))+transpose(tril(Pinf(:,:,t),-1));
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Pstar1(:,:,t) = Pstar(:,:,t);
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Pinf1(:,:,t) = Pinf(:,:,t);
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for i=1:pp
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v(i,t) = Y(i,t)-a(mf(i),t)-trend(i,t);
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Fstar(i,t) = Pstar(mf(i),mf(i),t) + H(i,i);
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Finf(i,t) = Pinf(mf(i),mf(i),t);
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Kstar(:,i,t) = Pstar(:,mf(i),t);
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if Finf(i,t) > crit & newRank, % original MJ: if Finf(i,t) > crit
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icc=icc+1;
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Kinf(:,i,t) = Pinf(:,mf(i),t);
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Linf(:,:,i,t) = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
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L0(:,:,i,t) = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Z(i,:)/Finf(i,t);
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a(:,t) = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
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Pstar(:,:,t) = Pstar(:,:,t) + ...
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Kinf(:,i,t)*transpose(Kinf(:,i,t))*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - ...
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(Kstar(:,i,t)*transpose(Kinf(:,i,t)) +...
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Kinf(:,i,t)*transpose(Kstar(:,i,t)))/Finf(i,t);
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Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*transpose(Kinf(:,i,t))/Finf(i,t);
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Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
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Pinf(:,:,t)=tril(Pinf(:,:,t))+transpose(tril(Pinf(:,:,t),-1));
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% new terminiation criteria by M. Ratto
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P0=Pinf(:,:,t);
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% newRank = any(diag(P0(mf,mf))>crit);
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% if newRank==0, options_.diffuse_d = i; end,
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if ~isempty(options_.diffuse_d),
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newRank = (icc<options_.diffuse_d);
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%if newRank & any(diag(P0(mf,mf))>crit)==0;
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if newRank & (any(diag(P0(mf,mf))>crit)==0 & rank(P0,crit1)==0);
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disp('WARNING!! Change in OPTIONS_.DIFFUSE_D in univariate DKF')
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options_.diffuse_d = icc;
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newRank=0;
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end
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else
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%newRank = any(diag(P0(mf,mf))>crit);
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newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1));
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if newRank==0,
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options_.diffuse_d = icc;
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end
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end,
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% if newRank==0,
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% options_.diffuse_d=i; % this line is buggy!
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% end
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% end new terminiation criteria by M. Ratto
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elseif Fstar(i,t) > crit
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%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
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%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
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%% rank(Pinf)=0. [st<73>phane,11-03-2004].
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Li(:,:,i,t) = eye(mm)-Kstar(:,i,t)*Z(i,:)/Fstar(i,t); % we need to store Li for DKF smoother
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a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
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Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*transpose(Kstar(:,i,t))/Fstar(i,t);
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Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
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end
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end
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a1(:,t+1) = T*a(:,t);
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for jnk=1:nk,
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aK(jnk,:,t+jnk) = T^jnk*a(:,t);
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end
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+ QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
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P0=Pinf(:,:,t+1);
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if newRank,
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%newRank = any(diag(P0(mf,mf))>crit);
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newRank = rank(P0,crit1);
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end
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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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Linf = Linf(:,:,:,1:d);
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L0 = L0(:,:,:,1:d);
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Fstar = Fstar(:,1:d);
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Finf = Finf(:,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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Pstar1 = Pstar1(:,:,1:d);
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Pinf1 = Pinf1(:,:,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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a(:,t) = a1(:,t);
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P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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P1(:,:,t) = P(:,:,t);
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for i=1:pp
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v(i,t) = Y(i,t) - a(mf(i),t) - trend(i,t);
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Fi(i,t) = P(mf(i),mf(i),t)+H(i,i);
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Ki(:,i,t) = P(:,mf(i),t);
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if Fi(i,t) > crit
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Li(:,:,i,t) = eye(mm)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
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a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t);
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P(:,:,t) = P(:,:,t) - Ki(:,i,t)*transpose(Ki(:,i,t))/Fi(i,t);
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P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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end
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end
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a1(:,t+1) = T*a(:,t);
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for jnk=1:nk,
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aK(jnk,:,t+jnk) = T^jnk*a(:,t);
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end
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P(:,:,t+1) = T*P(:,:,t)*transpose(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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P_s=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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Fi_s = Fi(:,t);
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Ki_s = Ki(:,:,t);
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L_s =Li(:,:,:,t);
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if t<smpl
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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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Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t_steady+1]));
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Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t_steady+1]));
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Ki = cat(3,Ki(:,:,1:t),repmat(Ki_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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a(:,t) = a1(:,t);
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for i=1:pp
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v(i,t) = Y(i,t) - a(mf(i),t) - trend(i,t);
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if Fi_s(i) > crit
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a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
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end
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end
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a1(:,t+1) = T*a(:,t);
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for jnk=1:nk,
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aK(jnk,:,t+jnk) = T^jnk*a(:,t);
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end
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end
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ri=zeros(mm,1);
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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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for i=pp:-1:1
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if Fi(i,t) > crit
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ri=Z(i,:)'/Fi(i,t)*v(i,t)+Li(:,:,i,t)'*ri;
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end
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end
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r(:,t) = ri;
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alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t);
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etahat(:,t) = QRt*r(:,t);
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ri = T'*ri;
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end
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if d
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r0 = zeros(mm,d);
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r0(:,d) = ri;
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r1 = zeros(mm,d);
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for t = d:-1:1
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for i=pp:-1:1
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if Finf(i,t) > crit & ~(t==d & i>options_.diffuse_d), % use of options_.diffuse_d to be sure of DKF termination
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r1(:,t) = Z(i,:)'*v(i,t)/Finf(i,t) + ...
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L0(:,:,i,t)'*r0(:,t) + Linf(:,:,i,t)'*r1(:,t);
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r0(:,t) = Linf(:,:,i,t)'*r0(:,t);
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elseif Fstar(i,t) > crit % step needed whe Finf == 0
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r0(:,t)=Z(i,:)'/Fstar(i,t)*v(i,t)+Li(:,:,i,t)'*r0(:,t);
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end
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end
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alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
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r(:,t) = r0(:,t);
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etahat(:,t) = QRt*r(:,t);
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if t > 1
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r0(:,t-1) = transpose(T)*r0(:,t);
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r1(:,t-1) = transpose(T)*r1(:,t);
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end
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end
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end
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epsilonhat = Y-alphahat(mf,:)-trend;
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