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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)
% Computes the diffuse kalman smoother in the case of a singular var-cov matrix.
% Univariate treatment of multivariate time series.
%
% INPUTS
% 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
%
% 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
% V: 3D array of state uncertainty matrices
%
% 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 (C) 2004-2018 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 <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
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);
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;
PK = zeros(nk,mm,mm,smpl+nk);
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
QQ = R*Q*transpose(R);
QRt = Q*transpose(R);
alphahat = zeros(mm,smpl);
etahat = zeros(rr,smpl);
epsilonhat = zeros(rr,smpl);
r = zeros(mm,smpl);
if state_uncertainty_flag
V = zeros(mm,mm,smpl);
N = zeros(mm,mm,smpl);
else
V=[];
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
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
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
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
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;
a(:,t) = a1(:,t);
P1(:,:,t) = P(:,:,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)
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
a1(:,t+1) = T*a(:,t); %transition according to (6.14) in DK (2012)
Pf = P(:,:,t);
aK(1,:,t+1) = a1(:,t+1);
for jnk=1:nk
Pf = T*Pf*T' + QQ;
PK(jnk,:,:,t+jnk) = Pf;
if jnk>1
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
end
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);
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,:);
% $$$ v(i,t) = Y(i,t) - Zi*a(:,t);
% $$$ 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);
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}
V(:,:,t) = P1(:,:,t)-P1(:,:,t)*N(:,:,t)*P1(:,:,t); % KD (2000), eq. (7) with N_{t-1} stored in N(:,:,t)
Ni = T'*Ni*T; % KD (2000), eq. (23), equation for N_{t-1,p_{t-1}}
end
end
if d
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);
etahat(:,t) = QRt*r(:,t); % KD (2000), eq. (27)
if state_uncertainty_flag
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
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
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
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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