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function [alphahat,epsilonhat,etahat,a,P,aK,PK,decomp,V] = missing_DiffuseKalmanSmootherH3_Z ( T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,diffuse_kalman_tol,decomp_flag,state_uncertainty_flag)
% 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.
%
% INPUTS
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% T: mm*mm matrix state transition matrix
% Z: pp*mm matrix selector matrix for observables in augmented state vector
% R: mm*rr matrix second matrix of the state equation relating the structural innovations to the state variables
% Q: rr*rr matrix covariance matrix of structural errors
% 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
% 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
% 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
% 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
% state estimate
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%
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% OUTPUTS
% alphahat: smoothed state variables (a_{t|T})
% epsilonhat: measurement errors
% etahat: smoothed shocks
% a: 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)
% 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:
% Outputs are stored in decision-rule order, i.e. to get variables in order of declaration
% as in M_.endo_names, ones needs code along the lines of:
% variables_declaration_order(dr.order_var,:) = alphahat
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%
% Algorithm:
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%
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% Uses the univariate filter as described in Durbin/Koopman (2012): "Time
% Series Analysis by State Space Methods", Oxford University Press,
% 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
% Models", in Journal of Time Series Analysis, vol. 21(3), pp. 281-296.
%
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% SPECIAL REQUIREMENTS
% 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
% Analysis, vol. 24(1), pp. 85-98.
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% Copyright (C) 2004-2018 Dynare Team
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%
% 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 nk-stpe ahed predictions)
% 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' )
end
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d = 0 ;
decomp = [ ] ;
spinf = size ( Pinf1 ) ;
spstar = size ( Pstar1 ) ;
v = zeros ( pp , smpl ) ;
a = zeros ( mm , smpl ) ;
a1 = zeros ( mm , smpl + 1 ) ;
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aK = zeros ( nk , mm , smpl + nk ) ;
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Fstar = zeros ( pp , smpl ) ;
Finf = zeros ( pp , smpl ) ;
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Fi = zeros ( pp , smpl ) ;
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 ) ;
P1 = P ;
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 ;
Pinf1 = Pinf ;
rr = size ( Q , 1 ) ; % number of structural shocks
QQ = R * Q * transpose ( R ) ;
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QRt = Q * transpose ( R ) ;
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alphahat = zeros ( mm , smpl ) ;
etahat = zeros ( rr , smpl ) ;
epsilonhat = zeros ( rr , smpl ) ;
r = zeros ( mm , smpl ) ;
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if state_uncertainty_flag
V = zeros ( mm , mm , smpl ) ;
N = zeros ( mm , mm , smpl ) ;
else
V = [ ] ;
end
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t = 0 ;
icc = 0 ;
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if ~ isempty ( Pinf ( : , : , 1 ) )
newRank = rank ( Z * Pinf ( : , : , 1 ) * Z ' , diffuse_kalman_tol ) ;
else
newRank = rank ( Pinf ( : , : , 1 ) , diffuse_kalman_tol ) ;
end
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while newRank && t < smpl
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t = t + 1 ;
a ( : , t ) = a1 ( : , t ) ;
Pstar1 ( : , : , t ) = Pstar ( : , : , t ) ;
Pinf1 ( : , : , t ) = Pinf ( : , : , t ) ;
di = data_index { t } ' ;
for i = di
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
Finf ( i , t ) = Zi * Pinf ( : , : , t ) * Zi ' ; % F_{\infty,t} in 5.7 in DK (2012)
Kstar ( : , i , t ) = Pstar ( : , : , t ) * Zi ' ; % KD (2000), eq. (15)
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 ) + ...
Kinf ( : , i , t ) * Kinf_Finf ' * ( Fstar ( i , t ) / Finf ( i , t ) ) - ...
Kstar ( : , i , t ) * Kinf_Finf ' - ...
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Kinf_Finf * Kstar ( : , i , t ) ' ; % KD (2000), eq. (16)
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)
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
% Koopman/Durbin (2000)
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else
% 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
end
if newRank
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if ~ isempty ( Pinf ( : , : , t ) )
oldRank = rank ( Z * Pinf ( : , : , t ) * Z ' , diffuse_kalman_tol ) ;
else
oldRank = rank ( Pinf ( : , : , t ) , diffuse_kalman_tol ) ;
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end
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else
oldRank = 0 ;
end
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
aK ( jnk , : , t + jnk ) = T * dynare_squeeze ( aK ( jnk - 1 , : , t + jnk - 1 ) ) ;
end
Pstar ( : , : , t + 1 ) = T * Pstar ( : , : , t ) * T ' + QQ ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * T ' ;
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if newRank
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if ~ isempty ( Pinf ( : , : , t + 1 ) )
newRank = rank ( Z * Pinf ( : , : , t + 1 ) * Z ' , diffuse_kalman_tol ) ;
else
newRank = rank ( Pinf ( : , : , t + 1 ) , diffuse_kalman_tol ) ;
end
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end
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
end
d = t ;
P ( : , : , d + 1 ) = Pstar ( : , : , d + 1 ) ;
Fstar = Fstar ( : , 1 : d ) ;
Finf = Finf ( : , 1 : d ) ;
Kstar = Kstar ( : , : , 1 : d ) ;
Pstar = Pstar ( : , : , 1 : d ) ;
Pinf = Pinf ( : , : , 1 : d ) ;
Pstar1 = Pstar1 ( : , : , 1 : d ) ;
Pinf1 = Pinf1 ( : , : , 1 : d ) ;
notsteady = 1 ;
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while notsteady && t < smpl
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t = t + 1 ;
a ( : , t ) = a1 ( : , t ) ;
P1 ( : , : , t ) = P ( : , : , t ) ;
di = data_index { t } ' ;
for i = di
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)
P ( : , : , t ) = P ( : , : , t ) - Ki ( : , i , t ) * Ki ( : , i , t ) ' / Fi ( i , t ) ; %filtering according to (6.13) in DK (2012)
else
% do nothing as a_{t,i+1}=a_{t,i} and P_{t,i+1}=P_{t,i}, see
% p. 157, DK (2012)
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end
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
Pf = T * Pf * T ' + QQ ;
PK ( jnk , : , : , t + jnk ) = Pf ;
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
end
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P ( : , : , t + 1 ) = T * P ( : , : , t ) * T ' + QQ ; %transition according to (6.14) in DK (2012)
% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
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end
% $$$ P_s=tril(P(:,:,t))+tril(P(:,:,t),-1)';
% $$$ P1_s=tril(P1(:,:,t))+tril(P1(:,:,t),-1)';
% $$$ Fi_s = Fi(:,t);
% $$$ Ki_s = Ki(:,:,t);
% $$$ L_s =Li(:,:,:,t);
% $$$ if t<smpl
% $$$ P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
% $$$ P1 = cat(3,P1(:,:,1:t),repmat(P1_s,[1 1 smpl-t]));
% $$$ Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t]));
% $$$ Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t]));
% $$$ Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t]));
% $$$ end
% $$$ while t<smpl
% $$$ t=t+1;
% $$$ a(:,t) = a1(:,t);
% $$$ di = data_index{t}';
% $$$ for i=di
% $$$ Zi = Z(i,:);
% $$$ v(i,t) = Y(i,t) - Zi*a(:,t);
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% $$$ if Fi_s(i) > kalman_tol
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% $$$ a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
% $$$ end
% $$$ end
% $$$ a1(:,t+1) = T*a(:,t);
% $$$ Pf = P(:,:,t);
% $$$ for jnk=1:nk,
% $$$ Pf = T*Pf*T' + QQ;
% $$$ aK(jnk,:,t+jnk) = T^jnk*a(:,t);
% $$$ PK(jnk,:,:,t+jnk) = Pf;
% $$$ end
% $$$ end
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%% do backward pass
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ri = zeros ( mm , 1 ) ;
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if state_uncertainty_flag
Ni = zeros ( mm , mm ) ;
end
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t = smpl + 1 ;
while t > d + 1
t = t - 1 ;
di = flipud ( data_index { t } ) ' ;
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 ) ;
ri = Z ( i , : ) ' / Fi ( i , t ) * v ( i , t ) + Li ' * ri ; % DK (2012), 6.15, equation for r_{t,i-1}
if state_uncertainty_flag
Ni = Z ( i , : ) ' / Fi ( i , t ) * Z ( i , : ) + Li ' * Ni * Li ; % KD (2000), eq. (23)
end
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end
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 ) ;
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
if d
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r0 = zeros ( mm , d ) ;
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r0 ( : , d ) = ri ;
r1 = zeros ( mm , d ) ;
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if state_uncertainty_flag
%N_0 at (d+1) is N(d+1), so we can use N for continuing and storing N_0-recursion
N_0 = zeros ( mm , mm , d ) ; %set N_1_{d}=0, below KD (2000), eq. (24)
N_0 ( : , : , d ) = Ni ;
N_1 = zeros ( mm , mm , d ) ; %set N_1_{d}=0, below KD (2000), eq. (24)
N_2 = zeros ( mm , mm , d ) ; %set N_2_{d}=0, below KD (2000), eq. (24)
end
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for t = d : - 1 : 1
di = flipud ( data_index { t } ) ' ;
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
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 ) + ...
Linf ' * r1 ( : , t ) ; % KD (2000), eq. (25) for r_1
r0 ( : , t ) = Linf ' * r0 ( : , t ) ; % KD (2000), eq. (25) for r_0
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if state_uncertainty_flag
N_2 ( : , : , t ) = Z ( i , : ) ' / Finf ( i , t ) ^2 * Z ( i , : ) * Fstar ( i , t ) ...
+ Linf ' * N_2 ( : , : , t ) * Linf ...
+ Linf ' * N_1 ( : , : , t ) * L0 ...
+ L0 ' * N_1 ( : , : , t ) ' * Linf ...
+ L0 ' * N_0 ( : , : , t ) * L0 ; % DK (2012), eq. 5.29
N_1 ( : , : , t ) = Z ( i , : ) ' / Finf ( i , t ) * Z ( i , : ) + Linf ' * N_1 ( : , : , t ) * Linf ...
+ 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
N_0 ( : , : , t ) = Linf ' * N_0 ( : , : , t ) * Linf ; % DK (2012), eq. 5.19, noting that L^(0) is named Linf
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
if state_uncertainty_flag
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
end
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end
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
V ( : , : , t ) = Pstar ( : , : , t ) - Pstar ( : , : , t ) * N_0 ( : , : , t ) * Pstar ( : , : , t ) ...
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- ( Pinf ( : , : , t ) * N_1 ( : , : , t ) * Pstar ( : , : , t ) ) ' ...
- Pinf ( : , : , t ) * N_1 ( : , : , t ) * Pstar ( : , : , t ) ...
- 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}
r1 ( : , t - 1 ) = T ' * r1 ( : , t ) ; % KD (2000), below eq. (25) r_{t-1,p_{t-1}}=T'*r_{t,0}
if state_uncertainty_flag
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
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
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
end
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end
end
end
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if decomp_flag
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decomp = zeros ( nk , mm , rr , smpl + nk ) ;
ZRQinv = inv ( Z * QQ * Z ' ) ;
for t = max ( d , 1 ) : smpl
ri_d = zeros ( mm , 1 ) ;
di = flipud ( data_index { t } ) ' ;
for i = di
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if Fi ( i , t ) > kalman_tol
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ri_d = Z ( i , : ) ' / Fi ( i , t ) * v ( i , t ) + ri_d - Ki ( : , i , t ) ' * ri_d / Fi ( i , t ) * Z ( i , : ) ' ;
end
end
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% calculate eta_tm1t
eta_tm1t = QRt * ri_d ;
% calculate decomposition
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Ttok = eye ( mm , mm ) ;
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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
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epsilonhat = Y - Z * alphahat ;
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if ( d == smpl )
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warning ( [ ' missing_DiffuseKalmanSmootherH3_Z:: There isn' ' t enough information to estimate the initial conditions of the nonstationary variables' ] ) ;
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return
end