392 lines
17 KiB
Matlab
392 lines
17 KiB
Matlab
function [alphahat,epsilonhat,etahat,a,P,aK,PK,decomp,V] = missing_DiffuseKalmanSmootherH3_Z(a_initial,T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,diffuse_kalman_tol,decomp_flag,state_uncertainty_flag)
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% function [alphahat,epsilonhat,etahat,a1,P,aK,PK,d,decomp] = missing_DiffuseKalmanSmootherH3_Z(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,decomp_flag,state_uncertainty_flag)
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% Computes the diffuse kalman smoother 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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% a_initial:mm*1 vector of initial states
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% T: mm*mm matrix state transition matrix
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% Z: pp*mm matrix selector matrix for observables in augmented state vector
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% R: mm*rr matrix second matrix of the state equation relating the structural innovations to the state variables
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% Q: rr*rr matrix covariance matrix of structural errors
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% H: pp*1 vector of variance of measurement errors
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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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% 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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% data_index [cell] 1*smpl cell of column vectors of indices.
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% nk number of forecasting periods
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% kalman_tol tolerance for zero divider
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% diffuse_kalman_tol tolerance for zero divider
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% decomp_flag if true, compute filter decomposition
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% state_uncertainty_flag if true, compute uncertainty about smoothed
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% state estimate
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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: measurement errors
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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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% (meaningless for periods 1:d)
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% P: 3D array of one-step ahead forecast error variance
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% matrices
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% PK: 4D array of k-step ahead forecast error variance
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% matrices (meaningless for periods 1:d)
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% decomp: decomposition of the effect of shocks on filtered values
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% V: 3D array of state uncertainty matrices
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%
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% Notes:
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% Outputs are stored in decision-rule order, i.e. to get variables in order of declaration
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% as in M_.endo_names, ones needs code along the lines of:
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% variables_declaration_order(dr.order_var,:) = alphahat
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%
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% Algorithm:
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%
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% Uses the univariate filter as described in Durbin/Koopman (2012): "Time
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% Series Analysis by State Space Methods", Oxford University Press,
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% Second Edition, Ch. 6.4 + 7.2.5
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% and
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% Koopman/Durbin (2000): "Fast Filtering and Smoothing for Multivariatze State Space
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% Models", in Journal of Time Series Analysis, vol. 21(3), pp. 281-296.
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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) 2004-2018 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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if size(H,2)>1
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error('missing_DiffuseKalmanSmootherH3_Z:: H is not a vector. This must not happens')
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end
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d = 0;
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decomp = [];
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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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a(:,1) = a_initial;
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a1(:,1) = a_initial;
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aK = zeros(nk,mm,smpl+nk);
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Fstar = zeros(pp,smpl);
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Finf = zeros(pp,smpl);
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Fi = zeros(pp,smpl);
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Ki = zeros(mm,pp,smpl);
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Kstar = zeros(mm,pp,smpl);
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Kinf = zeros(spstar(1),pp,smpl);
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P = zeros(mm,mm,smpl+1);
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P1 = P;
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PK = zeros(nk,mm,mm,smpl+nk);
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Pstar = zeros(spstar(1),spstar(2),smpl);
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Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl);
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Pinf(:,:,1) = Pinf1;
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Pstar1 = Pstar;
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Pinf1 = Pinf;
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rr = size(Q,1); % number of structural shocks
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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(rr,smpl);
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r = zeros(mm,smpl);
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if state_uncertainty_flag
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V = zeros(mm,mm,smpl);
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N = zeros(mm,mm,smpl);
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else
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V=[];
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end
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t = 0;
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icc=0;
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if ~isempty(Pinf(:,:,1))
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newRank = rank(Z*Pinf(:,:,1)*Z',diffuse_kalman_tol);
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else
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newRank = rank(Pinf(:,:,1),diffuse_kalman_tol);
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end
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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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Pstar1(:,:,t) = Pstar(:,:,t);
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Pinf1(:,:,t) = Pinf(:,:,t);
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di = data_index{t}';
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for i=di
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Zi = Z(i,:);
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v(i,t) = Y(i,t)-Zi*a(:,t); % nu_{t,i} in 6.13 in DK (2012)
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Fstar(i,t) = Zi*Pstar(:,:,t)*Zi' +H(i); % F_{*,t} in 5.7 in DK (2012), relies on H being diagonal
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Finf(i,t) = Zi*Pinf(:,:,t)*Zi'; % F_{\infty,t} in 5.7 in DK (2012)
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Kstar(:,i,t) = Pstar(:,:,t)*Zi'; % KD (2000), eq. (15)
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if Finf(i,t) > diffuse_kalman_tol && newRank % F_{\infty,t,i} = 0, use upper part of bracket on p. 175 DK (2012) for w_{t,i}
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icc=icc+1;
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Kinf(:,i,t) = Pinf(:,:,t)*Zi'; % KD (2000), eq. (15)
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Kinf_Finf = Kinf(:,i,t)/Finf(i,t);
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a(:,t) = a(:,t) + Kinf_Finf*v(i,t); % KD (2000), eq. (16)
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Pstar(:,:,t) = Pstar(:,:,t) + ...
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Kinf(:,i,t)*Kinf_Finf'*(Fstar(i,t)/Finf(i,t)) - ...
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Kstar(:,i,t)*Kinf_Finf' - ...
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Kinf_Finf*Kstar(:,i,t)'; % KD (2000), eq. (16)
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Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*Kinf(:,i,t)'/Finf(i,t); % KD (2000), eq. (16)
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elseif Fstar(i,t) > kalman_tol
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a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t); % KD (2000), eq. (17)
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Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*Kstar(:,i,t)'/Fstar(i,t); % KD (2000), eq. (17)
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% Pinf is passed through unaltered, see eq. (17) of
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% Koopman/Durbin (2000)
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else
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% do nothing as a_{t,i+1}=a_{t,i} and P_{t,i+1}=P_{t,i}, see
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% p. 157, DK (2012)
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end
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end
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if newRank
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if ~isempty(Pinf(:,:,t))
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oldRank = rank(Z*Pinf(:,:,t)*Z',diffuse_kalman_tol);
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else
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oldRank = rank(Pinf(:,:,t),diffuse_kalman_tol);
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end
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else
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oldRank = 0;
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end
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a1(:,t+1) = T*a(:,t);
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aK(1,:,t+1) = a1(:,t+1);
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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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end
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*T'+ QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*T';
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if newRank
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if ~isempty(Pinf(:,:,t+1))
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newRank = rank(Z*Pinf(:,:,t+1)*Z',diffuse_kalman_tol);
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else
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newRank = rank(Pinf(:,:,t+1),diffuse_kalman_tol);
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end
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end
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if oldRank ~= newRank
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disp('univariate_diffuse_kalman_filter:: T does influence the rank of Pinf!')
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disp('This may happen for models with order of integration >1.')
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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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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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P1(:,:,t) = P(:,:,t);
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di = data_index{t}';
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for i=di
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Zi = Z(i,:);
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v(i,t) = Y(i,t) - Zi*a(:,t); % nu_{t,i} in 6.13 in DK (2012)
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Fi(i,t) = Zi*P(:,:,t)*Zi' + H(i); % F_{t,i} in 6.13 in DK (2012), relies on H being diagonal
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Ki(:,i,t) = P(:,:,t)*Zi'; % K_{t,i}*F_(i,t) in 6.13 in DK (2012)
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if Fi(i,t) > kalman_tol
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a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t); %filtering according to (6.13) in DK (2012)
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P(:,:,t) = P(:,:,t) - Ki(:,i,t)*Ki(:,i,t)'/Fi(i,t); %filtering according to (6.13) in DK (2012)
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else
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% do nothing as a_{t,i+1}=a_{t,i} and P_{t,i+1}=P_{t,i}, see
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% p. 157, DK (2012)
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end
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end
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a1(:,t+1) = T*a(:,t); %transition according to (6.14) in DK (2012)
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Pf = P(:,:,t);
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aK(1,:,t+1) = a1(:,t+1);
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for jnk=1:nk
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Pf = T*Pf*T' + QQ;
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PK(jnk,:,:,t+jnk) = Pf;
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if jnk>1
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aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
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end
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end
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P(:,:,t+1) = T*P(:,:,t)*T' + QQ; %transition according to (6.14) in DK (2012)
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% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
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end
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% $$$ P_s=tril(P(:,:,t))+tril(P(:,:,t),-1)';
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% $$$ P1_s=tril(P1(:,:,t))+tril(P1(:,:,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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% $$$ P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
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% $$$ P1 = cat(3,P1(:,:,1:t),repmat(P1_s,[1 1 smpl-t]));
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% $$$ Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t]));
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% $$$ Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t]));
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% $$$ Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t]));
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% $$$ end
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% $$$ while t<smpl
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% $$$ t=t+1;
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% $$$ a(:,t) = a1(:,t);
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% $$$ di = data_index{t}';
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% $$$ for i=di
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% $$$ Zi = Z(i,:);
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% $$$ v(i,t) = Y(i,t) - Zi*a(:,t);
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% $$$ if Fi_s(i) > kalman_tol
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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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% $$$ Pf = P(:,:,t);
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% $$$ for jnk=1:nk,
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% $$$ Pf = T*Pf*T' + QQ;
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% $$$ aK(jnk,:,t+jnk) = T^jnk*a(:,t);
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% $$$ PK(jnk,:,:,t+jnk) = Pf;
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% $$$ end
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% $$$ end
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%% do backward pass
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ri=zeros(mm,1);
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if state_uncertainty_flag
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Ni=zeros(mm,mm);
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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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di = flipud(data_index{t})';
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for i = di
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if Fi(i,t) > kalman_tol
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Li = eye(mm)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
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ri = Z(i,:)'/Fi(i,t)*v(i,t)+Li'*ri; % DK (2012), 6.15, equation for r_{t,i-1}
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if state_uncertainty_flag
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Ni = Z(i,:)'/Fi(i,t)*Z(i,:)+Li'*Ni*Li; % KD (2000), eq. (23)
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end
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end
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end
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r(:,t) = ri; % DK (2012), below 6.15, r_{t-1}=r_{t,0}
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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; % KD (2003), eq. (23), equation for r_{t-1,p_{t-1}}
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if state_uncertainty_flag
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N(:,:,t) = Ni; % DK (2012), below 6.15, N_{t-1}=N_{t,0}
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V(:,:,t) = P1(:,:,t)-P1(:,:,t)*N(:,:,t)*P1(:,:,t); % KD (2000), eq. (7) with N_{t-1} stored in N(:,:,t)
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Ni = T'*Ni*T; % KD (2000), eq. (23), equation for N_{t-1,p_{t-1}}
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end
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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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if state_uncertainty_flag
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%N_0 at (d+1) is N(d+1), so we can use N for continuing and storing N_0-recursion
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N_0=zeros(mm,mm,d); %set N_1_{d}=0, below KD (2000), eq. (24)
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N_0(:,:,d) = Ni;
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N_1=zeros(mm,mm,d); %set N_1_{d}=0, below KD (2000), eq. (24)
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N_2=zeros(mm,mm,d); %set N_2_{d}=0, below KD (2000), eq. (24)
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end
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for t = d:-1:1
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di = flipud(data_index{t})';
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for i = di
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if Finf(i,t) > diffuse_kalman_tol
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% recursions need to be from highest to lowest term in order to not
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% overwrite lower terms still needed in this step
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Linf = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
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L0 = (Kinf(:,i,t)*(Fstar(i,t)/Finf(i,t))-Kstar(:,i,t))*Z(i,:)/Finf(i,t);
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r1(:,t) = Z(i,:)'*v(i,t)/Finf(i,t) + ...
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L0'*r0(:,t) + ...
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Linf'*r1(:,t); % KD (2000), eq. (25) for r_1
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r0(:,t) = Linf'*r0(:,t); % KD (2000), eq. (25) for r_0
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if state_uncertainty_flag
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N_2(:,:,t)=Z(i,:)'/Finf(i,t)^2*Z(i,:)*Fstar(i,t) ...
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+ Linf'*N_2(:,:,t)*Linf...
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+ Linf'*N_1(:,:,t)*L0...
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+ L0'*N_1(:,:,t)'*Linf...
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+ L0'*N_0(:,:,t)*L0; % DK (2012), eq. 5.29
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N_1(:,:,t)=Z(i,:)'/Finf(i,t)*Z(i,:)+Linf'*N_1(:,:,t)*Linf...
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+L0'*N_0(:,:,t)*Linf; % DK (2012), eq. 5.29; note that, compared to DK (2003) this drops the term (L_1'*N(:,:,t+1)*Linf(:,:,t))' in the recursion due to it entering premultiplied by Pinf when computing V, and Pinf*Linf'*N=0
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N_0(:,:,t)=Linf'*N_0(:,:,t)*Linf; % DK (2012), eq. 5.19, noting that L^(0) is named Linf
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end
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elseif Fstar(i,t) > kalman_tol % step needed whe Finf == 0
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L_i=eye(mm) - Kstar(:,i,t)*Z(i,:)/Fstar(i,t);
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r0(:,t) = Z(i,:)'/Fstar(i,t)*v(i,t)+L_i'*r0(:,t); % propagate r0 and keep r1 fixed
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if state_uncertainty_flag
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N_0(:,:,t)=Z(i,:)'/Fstar(i,t)*Z(i,:)+L_i'*N_0(:,:,t)*L_i; % propagate N_0 and keep N_1 and N_2 fixed
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end
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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); % KD (2000), eq. (26)
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r(:,t) = r0(:,t);
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etahat(:,t) = QRt*r(:,t); % KD (2000), eq. (27)
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if state_uncertainty_flag
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V(:,:,t)=Pstar(:,:,t)-Pstar(:,:,t)*N_0(:,:,t)*Pstar(:,:,t)...
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-(Pinf(:,:,t)*N_1(:,:,t)*Pstar(:,:,t))'...
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- Pinf(:,:,t)*N_1(:,:,t)*Pstar(:,:,t)...
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- Pinf(:,:,t)*N_2(:,:,t)*Pinf(:,:,t); % DK (2012), eq. 5.30
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end
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if t > 1
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r0(:,t-1) = T'*r0(:,t); % KD (2000), below eq. (25) r_{t-1,p_{t-1}}=T'*r_{t,0}
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r1(:,t-1) = T'*r1(:,t); % KD (2000), below eq. (25) r_{t-1,p_{t-1}}=T'*r_{t,0}
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if state_uncertainty_flag
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N_0(:,:,t-1)= T'*N_0(:,t)*T; % KD (2000), below eq. (25) N_{t-1,p_{t-1}}=T'*N_{t,0}*T
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N_1(:,:,t-1)= T'*N_1(:,t)*T; % KD (2000), below eq. (25) N^1_{t-1,p_{t-1}}=T'*N^1_{t,0}*T
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N_2(:,:,t-1)= T'*N_2(:,t)*T; % KD (2000), below eq. (25) N^2_{t-1,p_{t-1}}=T'*N^2_{t,0}*T
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|
end
|
|
end
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|
end
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|
end
|
|
|
|
if decomp_flag
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|
decomp = zeros(nk,mm,rr,smpl+nk);
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|
ZRQinv = inv(Z*QQ*Z');
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|
for t = max(d,1):smpl
|
|
ri_d = zeros(mm,1);
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|
di = flipud(data_index{t})';
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|
for i = di
|
|
if Fi(i,t) > kalman_tol
|
|
ri_d = Z(i,:)'/Fi(i,t)*v(i,t)+ri_d-Ki(:,i,t)'*ri_d/Fi(i,t)*Z(i,:)';
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|
end
|
|
end
|
|
|
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% calculate eta_tm1t
|
|
eta_tm1t = QRt*ri_d;
|
|
% calculate decomposition
|
|
Ttok = eye(mm,mm);
|
|
AAA = P1(:,:,t)*Z'*ZRQinv*Z*R;
|
|
for h = 1:nk
|
|
BBB = Ttok*AAA;
|
|
for j=1:rr
|
|
decomp(h,:,j,t+h) = eta_tm1t(j)*BBB(:,j);
|
|
end
|
|
Ttok = T*Ttok;
|
|
end
|
|
end
|
|
end
|
|
|
|
epsilonhat = Y - Z*alphahat;
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|
|
|
|
|
if (d==smpl)
|
|
warning(['missing_DiffuseKalmanSmootherH3_Z:: There isn''t enough information to estimate the initial conditions of the nonstationary variables']);
|
|
return
|
|
end
|