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dynare/matlab/kalman/missing_DiffuseKalmanSmooth...

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function [alphahat,epsilonhat,etahat,a,P1,aK,PK,decomp,V, aalphahat,eetahat,d,alphahat0,aalphahat0,V0,varargout] = 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, filter_covariance_flag, smoother_redux, occbin_)
% function [alphahat,epsilonhat,etahat,a,P1,aK,PK,decomp,V, aalphahat,eetahat,d] = 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, filter_covariance_flag, smoother_redux, occbin_)
% Computes the diffuse kalman smoother in the case of a singular var-cov matrix.
% Univariate treatment of multivariate time series.
%
% INPUTS
% a_initial:mm*1 vector of initial states
% 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
% 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.
% nk: number of forecasting periods
% kalman_tol: tolerance for zero divider
% diffuse_kalman_tol: tolerance for zero divider
% decomp_flag: if true, compute filter decomposition
% state_uncertainty_flag: if true, compute uncertainty about smoothed
% state estimate
% decomp_flag: if true, compute filter decomposition
% filter_covariance_flag: if true, compute filter covariance
% smoother_redux: if true, compute smoother on restricted
% state space, recover static variables from this
%
% 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)
% P1: 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
% V: 3D array of state uncertainty matrices
% aalphahat: filtered states in t-1|t
% eetahat: updated shocks in t|t
% d: number of diffuse periods
%
% 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
%
% Algorithm:
%
% 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
% and
% Koopman/Durbin (2000): "Fast Filtering and Smoothing for Multivariatze State Space
% Models", in Journal of Time Series Analysis, vol. 21(3), pp. 281-296.
%
% SPECIAL REQUIREMENTS
% See "Filtering and Smoothing of State Vector for Diffuse State Space
% Models", S.J. Koopman and J. Durbin (2003), in Journal of Time Series
% Analysis, vol. 24(1), pp. 85-98.
% Copyright © 2004-2023 Dynare Team
%
% 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 <https://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
if size(H,2)>1
error('missing_DiffuseKalmanSmootherH3_Z:: H is not a vector. This must not happens')
end
d = 0;
decomp = [];
spinf = size(Pinf1);
spstar = size(Pstar1);
v = zeros(pp,smpl);
a = zeros(mm,smpl);
a1 = zeros(mm,smpl+1);
a(:,1) = a_initial;
a1(:,1) = a_initial;
aK = zeros(nk,mm,smpl+nk);
Fstar = zeros(pp,smpl);
Finf = zeros(pp,smpl);
Fi = zeros(pp,smpl);
Ki = zeros(mm,pp,smpl);
Kstar = zeros(mm,pp,smpl);
Kinf = zeros(spstar(1),pp,smpl);
P = zeros(mm,mm,smpl+1);
P1 = P;
if filter_covariance_flag
PK = zeros(nk,mm,mm,smpl+nk);
else
PK = [];
end
Pstar = zeros(spstar(1),spstar(2),smpl);
Pstar(:,:,1) = Pstar1;
Pinf = zeros(spinf(1),spinf(2),smpl);
Pinf(:,:,1) = Pinf1;
Pstar1 = Pstar;
Pinf1 = Pinf;
rr = size(Q,1); % number of structural shocks
isqvec = false;
if ndim(Q)>2
Qvec = Q;
Q=Q(:,:,1);
isqvec = true;
end
QQ = R*Q*transpose(R);
QRt = Q*transpose(R);
alphahat = zeros(mm,smpl);
etahat = zeros(rr,smpl);
if smoother_redux
aalphahat = alphahat;
eetahat = etahat;
else
aalphahat = [];
eetahat = [];
end
epsilonhat = zeros(rr,smpl);
r = zeros(mm,smpl);
if state_uncertainty_flag
if smoother_redux
V = zeros(mm+rr,mm+rr,smpl);
else
V = zeros(mm,mm,smpl);
end
N = zeros(mm,mm,smpl);
else
V=[];
end
alphahat0=[];
aalphahat0=[];
V0=[];
if ~occbin_.status
isoccbin = 0;
C=0;
TT=[];
RR=[];
CC=[];
else
isoccbin = 1;
Qt = repmat(Q,[1 1 3]);
options_=occbin_.info{1};
dr=occbin_.info{2};
endo_steady_state=occbin_.info{3};
exo_steady_state=occbin_.info{4};
exo_det_steady_state=occbin_.info{5};
M_=occbin_.info{6};
occbin_options=occbin_.info{7};
opts_regime = occbin_options.opts_regime;
% first_period_occbin_update = inf;
if isfield(opts_regime,'regime_history') && ~isempty(opts_regime.regime_history)
opts_regime.regime_history=[opts_regime.regime_history(1) opts_regime.regime_history];
else
opts_regime.binding_indicator=zeros(smpl+2,M_.occbin.constraint_nbr);
end
occbin_options.opts_regime = opts_regime;
[~, ~, ~, regimes_] = occbin.check_regimes([], [], [], opts_regime, M_, options_, dr, endo_steady_state, exo_steady_state, exo_det_steady_state);
if length(occbin_.info)>7
if length(occbin_.info)==9 && options_.smoother_redux
TT=repmat(T,1,1,smpl+1);
RR=repmat(R,1,1,smpl+1);
CC=repmat(zeros(mm,1),1,smpl+1);
T0=occbin_.info{8};
R0=occbin_.info{9};
else
TT=occbin_.info{8};
RR=occbin_.info{9};
CC=occbin_.info{10};
% TT = cat(3,TT,T);
% RR = cat(3,RR,R);
% CC = cat(2,CC,zeros(mm,1));
if options_.smoother_redux
my_order_var = dr.restrict_var_list;
CC = CC(my_order_var,:);
RR = RR(my_order_var,:,:);
TT = TT(my_order_var,my_order_var,:);
T0=occbin_.info{11};
R0=occbin_.info{12};
end
if size(TT,3)<(smpl+1)
TT=repmat(T,1,1,smpl+1);
RR=repmat(R,1,1,smpl+1);
CC=repmat(zeros(mm,1),1,smpl+1);
end
end
else
TT=repmat(T,1,1,smpl+1);
RR=repmat(R,1,1,smpl+1);
CC=repmat(zeros(mm,1),1,smpl+1);
end
if ~smoother_redux
T0=T;
R0=R;
end
if ~isinf(occbin_options.first_period_occbin_update)
% initialize state matrices (otherwise they are set to 0 for
% t<first_period_occbin_update!)
TTT=repmat(T0,1,1,smpl+1);
RRR=repmat(R0,1,1,smpl+1);
CCC=repmat(zeros(length(T0),1),1,smpl+1);
end
end
t = 0;
icc=0;
if ~isempty(Pinf(:,:,1))
newRank = rank(Z*Pinf(:,:,1)*Z',diffuse_kalman_tol);
else
newRank = rank(Pinf(:,:,1),diffuse_kalman_tol);
end
if newRank
% add this to get smoothed states in period 0
Pinf_init = Pinf(:,:,1);
Pstar_init = Pstar(:,:,1);
Pstar(:,:,1) = T*Pstar(:,:,1)*T' + QQ;
ainit = a1(:,1);
end
while newRank && t < smpl
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,:);
v(i,t) = Y(i,t)-Zi*a(:,t); % nu_{t,i} in 6.13 in DK (2012)
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}
icc=icc+1;
Kinf(:,i,t) = Pinf(:,:,t)*Zi'; % KD (2000), eq. (15)
Kinf_Finf = Kinf(:,i,t)/Finf(i,t);
a(:,t) = a(:,t) + Kinf_Finf*v(i,t); % KD (2000), eq. (16)
Pstar(:,:,t) = Pstar(:,:,t) + ...
Kinf(:,i,t)*Kinf_Finf'*(Fstar(i,t)/Finf(i,t)) - ...
Kstar(:,i,t)*Kinf_Finf' - ...
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)
elseif Fstar(i,t) > kalman_tol
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)
% Pinf is passed through unaltered, see eq. (17) of
% Koopman/Durbin (2000)
else
% do nothing as a_{t,i+1}=a_{t,i} and P_{t,i+1}=P_{t,i}, see
% p. 157, DK (2012)
end
end
if newRank
if ~isempty(Pinf(:,:,t))
oldRank = rank(Z*Pinf(:,:,t)*Z',diffuse_kalman_tol);
else
oldRank = rank(Pinf(:,:,t),diffuse_kalman_tol);
end
else
oldRank = 0;
end
if isoccbin
TT(:,:,t+1)= T;
RR(:,:,t+1)= R;
end
a1(:,t+1) = T*a(:,t);
aK(1,:,t+1) = a1(:,t+1);
for jnk=2:nk
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
if isqvec
QQ = R*Qvec(:,:,t+1)*transpose(R);
end
Pstar(:,:,t+1) = T*Pstar(:,:,t)*T'+ QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*T';
if newRank
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
end
if oldRank ~= newRank
disp('univariate_diffuse_kalman_filter:: T does influence the rank of Pinf!')
disp('This may happen for models with order of integration >1.')
end
end
if isoccbin
first_period_occbin_update = occbin_options.first_period_occbin_update;
if d>0
first_period_occbin_update = max(t+2,occbin_options.first_period_occbin_update);
% kalman update is not yet robust to accommodate diffuse steps
end
if occbin_options.opts_regime.waitbar
hh_fig = dyn_waitbar(0,'Occbin: Piecewise Kalman Filter');
set(hh_fig,'Name','Occbin: Piecewise Kalman Filter.');
waitbar_indicator=1;
else
waitbar_indicator=0;
end
else
first_period_occbin_update = inf;
waitbar_indicator=0;
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;
while notsteady && t<smpl
t = t+1;
if t==1
Pinit = P(:,:,1);
ainit = a1(:,1);
end
a(:,t) = a1(:,t);
P1(:,:,t) = P(:,:,t);
di = data_index{t}';
if t>=first_period_occbin_update
if waitbar_indicator
dyn_waitbar(t/smpl, hh_fig, sprintf('Period %u of %u', t,smpl));
end
occbin_options.opts_regime.waitbar=0;
if t==1
if isqvec
Qt = cat(3,Q,Qvec(:,:,t:t+1));
end
a0 = a(:,1);
a10 = [a0 a(:,1)];
P0 = P(:,:,1);
P10 = P1(:,:,[1 1]);
data_index0{1}=[];
data_index0(2)=data_index(1);
v0(:,2)=v(:,1);
Y0(:,2)=Y(:,1);
Y0(:,1)=nan;
Fi0 = Fi(:,1);
Ki0 = Ki(:,:,1);
TT01 = cat(3,T,TT(:,:,1));
RR01 = cat(3,R,RR(:,:,1));
CC01 = zeros(size(CC,1),2);
CC01(:,2) = CC(:,1);
[ax, a1x, Px, P1x, vx, Fix, Kix, Tx, Rx, Cx, tmp, error_flag, M_, aha, etaha,TTx,RRx,CCx] = occbin.kalman_update_algo_3(a0,a10,P0,P10,data_index0,Z,v0,Fi0,Ki0,Y0,H,Qt,T0,R0,TT01,RR01,CC01,regimes_(t:t+1),M_,dr,endo_steady_state,exo_steady_state,exo_det_steady_state,options_,occbin_options,kalman_tol,nk);
else
if isqvec
Qt = Qvec(:,:,t-1:t+1);
end
[ax, a1x, Px, P1x, vx, Fix, Kix, Tx, Rx, Cx, tmp, error_flag, M_, aha, etaha,TTx,RRx,CCx] = occbin.kalman_update_algo_3(a(:,t-1),a1(:,t-1:t),P(:,:,t-1),P1(:,:,t-1:t),data_index(t-1:t),Z,v(:,t-1:t),Fi(:,t-1),Ki(:,:,t-1),Y(:,t-1:t),H,Qt,T0,R0,TT(:,:,t-1:t),RR(:,:,t-1:t),CC(:,t-1:t),regimes_(t:t+1),M_,dr,endo_steady_state,exo_steady_state,exo_det_steady_state,options_,occbin_options,kalman_tol,nk);
end
if ~error_flag
regimes_(t:t+2)=tmp;
else
varargout{1} = [];
varargout{2} = [];
varargout{3} = [];
varargout{4} = [];
varargout{5} = [];
varargout{6} = [];
varargout{7} = [];
return
end
if smoother_redux && t>1
aalphahat(:,t-1) = aha(:,1);
end
eetahat(:,t) = etaha(:,2);
a(:,t) = ax(:,1);
a1(:,t) = a1x(:,2);
a1(:,t+1) = ax(:,2);
v(di,t) = vx(di,2);
Fi(di,t) = Fix(di,2);
Ki(:,di,t) = Kix(:,di,2);
TT(:,:,t:t+1) = Tx(:,:,1:2);
RR(:,:,t:t+1) = Rx(:,:,1:2);
CC(:,t:t+1) = Cx(:,1:2);
TTT(:,:,t)=TTx;
RRR(:,:,t)=RRx;
CCC(:,t)=CCx;
P(:,:,t) = Px(:,:,1);
P1(:,:,t) = P1x(:,:,2);
P(:,:,t+1) = Px(:,:,2);
aK(1,:,t+1) = a1(:,t+1);
for jnk=1:nk
PK(jnk,:,:,t+jnk) = Px(:,:,1+jnk);
aK(jnk,:,t+jnk) = ax(:,1+jnk);
end
else
if isoccbin && t==1
if isqvec
QQ = RR(:,:,t)*Qvec(:,:,t)*transpose(RR(:,:,t));
else
QQ = RR(:,:,t)*Q*transpose(RR(:,:,t));
end
T = TT(:,:,t);
C = CC(:,t);
a1(:,t) = T*a(:,t)+C; %transition according to (6.14) in DK (2012)
P(:,:,t) = T*P(:,:,t)*T' + QQ; %transition according to (6.14) in DK (2012)
P1(:,:,t) = P(:,:,t);
end
for i=di
Zi = Z(i,:);
v(i,t) = Y(i,t) - Zi*a(:,t); % nu_{t,i} in 6.13 in DK (2012)
Fi(i,t) = Zi*P(:,:,t)*Zi' + H(i); % F_{t,i} in 6.13 in DK (2012), relies on H being diagonal
Ki(:,i,t) = P(:,:,t)*Zi'; % K_{t,i}*F_(i,t) in 6.13 in DK (2012)
if Fi(i,t) > kalman_tol
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)
end
end
if isqvec
QQ = R*Qvec(:,:,t)*transpose(R);
end
if smoother_redux
ri=zeros(mm,1);
for st=t:-1:max(d+1,t-1)
di = flipud(data_index{st})';
for i = di
if Fi(i,st) > kalman_tol
Li = eye(mm)-Ki(:,i,st)*Z(i,:)/Fi(i,st);
ri = Z(i,:)'/Fi(i,st)*v(i,st)+Li'*ri; % DK (2012), 6.15, equation for r_{t,i-1}
end
end
if st==t-1
aalphahat(:,st) = a1(:,st) + P1(:,:,st)*ri;
else
if isoccbin
if isqvec
QRt = Qvec(:,:,st)*transpose(RR(:,:,st));
else
QRt = Q*transpose(RR(:,:,st));
end
T = TT(:,:,st);
else
if isqvec
QRt = Qvec(:,:,st)*transpose(R);
end
end
eetahat(:,st) = QRt*ri;
end
ri = T'*ri; % KD (2003), eq. (23), equation for r_{t-1,p_{t-1}}
end
if t==1
aalphahat0 = P1(:,:,st)*ri;
end
end
if isoccbin
if isqvec
QQ = RR(:,:,t+1)*Qvec(:,:,t+1)*transpose(RR(:,:,t+1));
else
QQ = RR(:,:,t+1)*Q*transpose(RR(:,:,t+1));
end
T = TT(:,:,t+1);
C = CC(:,t+1);
else
if isqvec
QQ = R*Qvec(:,:,t+1)*transpose(R);
end
end
a1(:,t+1) = T*a(:,t)+C; %transition according to (6.14) in DK (2012)
P(:,:,t+1) = T*P(:,:,t)*T' + QQ; %transition according to (6.14) in DK (2012)
if filter_covariance_flag
Pf = P(:,:,t+1);
end
aK(1,:,t+1) = a1(:,t+1);
if ~isempty(nk) && nk>1 && isoccbin && (t>=first_period_occbin_update || isinf(first_period_occbin_update))
opts_simul = occbin_options.opts_regime;
opts_simul.SHOCKS = zeros(nk,M_.exo_nbr);
if smoother_redux
tmp=zeros(M_.endo_nbr,1);
tmp(dr.restrict_var_list)=a(:,t);
opts_simul.endo_init = tmp(dr.inv_order_var);
else
opts_simul.endo_init = a(dr.inv_order_var,t);
end
opts_simul.init_regime = []; %regimes_(t);
opts_simul.waitbar=0;
options_.occbin.simul=opts_simul;
[~, out, ss] = occbin.solver(M_,options_,dr,steady_state,exo_steady_state,exo_det_steady_state);
end
for jnk=1:nk
if filter_covariance_flag
if jnk>1
Pf = T*Pf*T' + QQ;
end
PK(jnk,:,:,t+jnk) = Pf;
end
if jnk>1
if isoccbin && (t>=first_period_occbin_update || isinf(first_period_occbin_update))
if smoother_redux
aK(jnk,:,t+jnk) = out.piecewise(jnk,dr.order_var(dr.restrict_var_list)) - out.ys(dr.order_var(dr.restrict_var_list))';
else
aK(jnk,dr.inv_order_var,t+jnk) = out.piecewise(jnk,:) - out.ys';
end
else
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
end
end
end
end
if waitbar_indicator
dyn_waitbar_close(hh_fig);
end
P1(:,:,t+1) = P(:,:,t+1);
if ~isinf(first_period_occbin_update) && isoccbin
regimes_ = regimes_(1:smpl+1);
else
regimes_ = struct();
TTT=TT;
RRR=RR;
CCC=CC;
% return
end
varargout{1} = regimes_;
varargout{2} = TTT;
varargout{3} = RRR;
varargout{4} = CCC;
varargout{5} = TT;
varargout{6} = RR;
varargout{7} = CC;
% $$$ P_s=tril(P(:,:,t))+tril(P(:,:,t),-1)';
% $$$ 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,:);
% $$$ v(i,t) = Y(i,t) - Zi*a(:,t);
% $$$ if Fi_s(i) > kalman_tol
% $$$ a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
% $$$ 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
%% do backward pass
ri=zeros(mm,1);
if state_uncertainty_flag
Ni=zeros(mm,mm);
end
t = smpl+1;
while t > d+1
t = t-1;
di = flipud(data_index{t})';
for i = di
if Fi(i,t) > kalman_tol
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
end
end
r(:,t) = ri; % DK (2012), below 6.15, r_{t-1}=r_{t,0}
alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t);
if isoccbin
if isqvec
QRt = Qvec(:,:,t)*transpose(RR(:,:,t));
else
QRt = Q*transpose(RR(:,:,t));
end
R = RR(:,:,t);
T = TT(:,:,t);
else
if isqvec
QRt = Qvec(:,:,t)*transpose(R);
end
end
etahat(:,t) = QRt*r(:,t);
ri = T'*ri; % KD (2003), eq. (23), equation for r_{t-1,p_{t-1}}
if state_uncertainty_flag
N(:,:,t) = Ni; % DK (2012), below 6.15, N_{t-1}=N_{t,0}
if smoother_redux
ptmp = [P1(:,:,t) R*Q; (R*Q)' Q];
ntmp = [N(:,:,t) zeros(mm,rr); zeros(rr,mm+rr)];
V(:,:,t) = ptmp - ptmp*ntmp*ptmp;
else
V(:,:,t) = P1(:,:,t)-P1(:,:,t)*N(:,:,t)*P1(:,:,t); % KD (2000), eq. (7) with N_{t-1} stored in N(:,:,t)
end
Ni = T'*Ni*T; % KD (2000), eq. (23), equation for N_{t-1,p_{t-1}}
end
end
if d==0 % recover states in period t=0
a0 = ainit;
r0 = ri;
P0 = Pinit;
% if OCCBIN, P1 in t=1 must be consistent with the regime in 1
alphahat0 = a0 + P0*r0;
% we do NOT need eps(0)!
% alphahat is smoothed state in t=0, so that S(1)=T*s(0)+R*eps(1);
if state_uncertainty_flag
N0 = Ni; % DK (2012), below 6.15, N_{t-1}=N_{t,0}
if smoother_redux
ptmp = [P0 R*Q; (R*Q)' Q];
ntmp = [N0 zeros(mm,rr); zeros(rr,mm+rr)];
V0 = ptmp - ptmp*ntmp*ptmp;
else
V0 = P0-P0*N0*P0; % KD (2000), eq. (7) with N_{t-1} stored in N(:,:,t)
end
end
else % diffuse filter
r0 = zeros(mm,d);
r0(:,d) = ri;
r1 = zeros(mm,d);
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
for t = d:-1:1
di = flipud(data_index{t})';
for i = di
if Finf(i,t) > diffuse_kalman_tol
% recursions need to be from highest to lowest term in order to not
% overwrite lower terms still needed in this step
Linf = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
L0 = (Kinf(:,i,t)*(Fstar(i,t)/Finf(i,t))-Kstar(:,i,t))*Z(i,:)/Finf(i,t);
r1(:,t) = Z(i,:)'*v(i,t)/Finf(i,t) + ...
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
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
elseif Fstar(i,t) > kalman_tol % step needed whe Finf == 0
L_i=eye(mm) - Kstar(:,i,t)*Z(i,:)/Fstar(i,t);
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
end
end
alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t); % KD (2000), eq. (26)
r(:,t) = r0(:,t);
if isoccbin
if isqvec
QRt = Qvec(:,:,t)*transpose(RR(:,:,t));
else
QRt = Q*transpose(RR(:,:,t));
end
R = RR(:,:,t);
T = TT(:,:,t);
else
if isqvec
QRt = Qvec(:,:,t)*transpose(R);
end
end
etahat(:,t) = QRt*r(:,t); % KD (2000), eq. (27)
if state_uncertainty_flag
if smoother_redux
pstmp = [Pstar(:,:,t) R*Q; (R*Q)' Q];
pitmp = [Pinf(:,:,t) zeros(mm,rr); zeros(rr,mm+rr)];
ntmp0 = [N_0(:,:,t) zeros(mm,rr); zeros(rr,mm+rr)];
ntmp1 = [N_1(:,:,t) zeros(mm,rr); zeros(rr,mm+rr)];
ntmp2 = [N_2(:,:,t) zeros(mm,rr); zeros(rr,mm+rr)];
V(:,:,t) = pstmp - pstmp*ntmp0*pstmp...
-(pitmp*ntmp1*pstmp)'...
- pitmp*ntmp1*pstmp...
- pitmp*ntmp2*pitmp; % DK (2012), eq. 5.30
else
V(:,:,t)=Pstar(:,:,t)-Pstar(:,:,t)*N_0(:,:,t)*Pstar(:,:,t)...
-(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
end
end
if t > 1
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
else
r00 = T'*r0(:,t); % KD (2000), below eq. (25) r_{t-1,p_{t-1}}=T'*r_{t,0}
r10 = T'*r1(:,t); % KD (2000), below eq. (25) r_{t-1,p_{t-1}}=T'*r_{t,0}
if state_uncertainty_flag
N_00= T'*N_0(:,:,t)*T; % KD (2000), below eq. (25) N_{t-1,p_{t-1}}=T'*N_{t,0}*T
N_10= T'*N_1(:,:,t)*T; % KD (2000), below eq. (25) N^1_{t-1,p_{t-1}}=T'*N^1_{t,0}*T
N_20= T'*N_2(:,:,t)*T; % KD (2000), below eq. (25) N^2_{t-1,p_{t-1}}=T'*N^2_{t,0}*T
end
end
end
% get smoothed states in t=0
alphahat0 = ainit + Pstar_init*r00 + Pinf_init*r10; % KD (2000), eq. (26)
if state_uncertainty_flag
if smoother_redux
pstmp = [Pstar_init R*Q; (R*Q)' Q];
pitmp = [Pinf_init zeros(mm,rr); zeros(rr,mm+rr)];
ntmp0 = [N_00 zeros(mm,rr); zeros(rr,mm+rr)];
ntmp1 = [N_10 zeros(mm,rr); zeros(rr,mm+rr)];
ntmp2 = [N_20 zeros(mm,rr); zeros(rr,mm+rr)];
V0 = pstmp - pstmp*ntmp0*pstmp...
-(pitmp*ntmp1*pstmp)'...
- pitmp*ntmp1*pstmp...
- pitmp*ntmp2*pitmp; % DK (2012), eq. 5.30
else
V0=Pstar_init-Pstar_init*N_00*Pstar_init...
-(Pinf_init*N_10*Pstar_init)'...
- Pinf_init*N_10*Pstar_init...
- Pinf_init*N_20*Pinf_init; % DK (2012), eq. 5.30
end
end
end
if decomp_flag
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
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,:)';
end
end
% calculate eta_tm1t
if isoccbin
if isqvec
QRt = Qvec(:,:,t)*transpose(RR(:,:,t));
else
QRt = Q*transpose(RR(:,:,t));
end
R = RR(:,:,t);
T = TT(:,:,t);
else
if isqvec
QRt = Qvec(:,:,t)*transpose(R);
end
end
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;
if (d==smpl)
warning(['missing_DiffuseKalmanSmootherH3_Z:: There isn''t enough information to estimate the initial conditions of the nonstationary variables']);
return
end
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