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function [alphahat,epsilonhat,etahat,atilde,P,aK,PK,decomp,V] = missing_DiffuseKalmanSmootherH1_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,a,aK,PK,decomp] = DiffuseKalmanSmoother1(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,diffuse_kalman_tol,decomp_flag,state_uncertainty_flag)
% Computes the diffuse kalman smoother without measurement error, in the case of a non-singular var-cov matrix.
%
% INPUTS
% T: mm*mm matrix
% Z: pp*mm matrix
% R: mm*rr matrix
% Q: rr*rr matrix
% H: pp*pp matrix 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 reciprocal condition number
% diffuse_kalman_tol tolerance for reciprocal condition number (for Finf) and the rank of Pinf
% decomp_flag if true, compute filter decomposition
% state_uncertainty_flag if true, compute uncertainty about smoothed
% state estimate
%
% OUTPUTS
% alphahat: smoothed variables (a_{t|T})
% epsilonhat:smoothed measurement errors
% etahat: smoothed shocks
% atilde: 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
%
% 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).
% Durbin/Koopman (2012): "Time Series Analysis by State Space Methods", Oxford University Press,
% Second Edition, Ch. 5
% 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 k-step predictions)
% new options_.nk: the max step ahed prediction in aK (default is 4)
% new crit1 value for rank of Pinf
% it is assured that P is symmetric
d = 0;
decomp = [];
spinf = size(Pinf1);
spstar = size(Pstar1);
v = zeros(pp,smpl);
a = zeros(mm,smpl+1);
atilde = zeros(mm,smpl);
aK = zeros(nk,mm,smpl+nk);
PK = zeros(nk,mm,mm,smpl+nk);
iF = zeros(pp,pp,smpl);
Fstar = zeros(pp,pp,smpl);
iFstar = zeros(pp,pp,smpl);
iFinf = zeros(pp,pp,smpl);
K = zeros(mm,pp,smpl);
L = zeros(mm,mm,smpl);
Linf = zeros(mm,mm,smpl);
Lstar = zeros(mm,mm,smpl);
Kstar = zeros(mm,pp,smpl);
Kinf = zeros(mm,pp,smpl);
P = zeros(mm,mm,smpl+1);
Pstar = zeros(spstar(1),spstar(2),smpl+1);
Pstar(:,:,1) = Pstar1;
Pinf = zeros(spinf(1),spinf(2),smpl+1);
Pinf(:,:,1) = Pinf1;
rr = size(Q,1);
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+1);
Finf_singular = zeros(1,smpl);
if state_uncertainty_flag
V = zeros(mm,mm,smpl);
N = zeros(mm,mm,smpl+1);
else
V=[];
end
t = 0;
while rank(Pinf(:,:,t+1),diffuse_kalman_tol) && t<smpl
t = t+1;
di = data_index{t};
if isempty(di)
%no observations, propagate forward without updating based on
%observations
atilde(:,t) = a(:,t);
a(:,t+1) = T*atilde(:,t);
Linf(:,:,t) = T;
Pstar(:,:,t+1) = T*Pstar(:,:,t)*T' + QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*T';
else
ZZ = Z(di,:); %span selector matrix
v(di,t)= Y(di,t) - ZZ*a(:,t); %get prediction error v^(0) in (5.13) DK (2012)
Finf = ZZ*Pinf(:,:,t)*ZZ'; % (5.7) in DK (2012)
if rcond(Finf) < diffuse_kalman_tol %F_{\infty,t} = 0
if ~all(abs(Finf(:)) < diffuse_kalman_tol) %rank-deficient but not rank 0
% The univariate diffuse kalman filter should be used.
alphahat = Inf;
return
else %rank of F_{\infty,t} is 0
Finf_singular(1,t) = 1;
Fstar(di,di,t) = ZZ*Pstar(:,:,t)*ZZ' + H(di,di); % (5.7) in DK (2012)
if rcond(Fstar(di,di,t)) < kalman_tol %F_{*} is singular
if ~all(all(abs(Fstar(di,di,t))<kalman_tol))
% The univariate diffuse kalman filter should be used.
alphahat = Inf;
return
else %rank 0
a(:,t+1) = T*a(:,t);
Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
end
else
iFstar(di,di,t) = inv(Fstar(di,di,t));
Kstar(:,di,t) = Pstar(:,:,t)*ZZ'*iFstar(di,di,t); %(5.15) of DK (2012) with Kstar=T^{-1}*K^(0)
Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T); % DK (2012), 5.16
Lstar(:,:,t) = T - T*Kstar(:,di,t)*ZZ; %L^(0) in DK (2012), eq. 5.12
Pstar(:,:,t+1) = T*Pstar(:,:,t)*Lstar(:,:,t)'+QQ; % (5.17) DK (2012)
a(:,t+1) = T*(a(:,t)+Kstar(:,di,t)*v(di,t)); % (5.13) DK (2012)
end
end
else
%see notes in kalman_filter_d.m for details of computations
iFinf(di,di,t) = inv(Finf);
Kinf(:,di,t) = Pinf(:,:,t)*ZZ'*iFinf(di,di,t); %define Kinf=T^{-1}*K_0 with M_{\infty}=Pinf*Z'
atilde(:,t) = a(:,t) + Kinf(:,di,t)*v(di,t);
Linf(:,:,t) = T - T*Kinf(:,di,t)*ZZ; %L^(0) in DK (2012), eq. 5.12
Fstar(di,di,t) = ZZ*Pstar(:,:,t)*ZZ' + H(di,di); %(5.7) DK(2012)
Kstar(:,di,t) = (Pstar(:,:,t)*ZZ'-Kinf(:,di,t)*Fstar(di,di,t))*iFinf(di,di,t); %(5.12) DK(2012) with Kstar=T^{-1}*K^(1); note that there is a typo in DK (2003) with "+ Kinf" instead of "- Kinf", but it is correct in their appendix
Pstar(:,:,t+1) = T*Pstar(:,:,t)*Linf(:,:,t)'-T*Kinf(:,di,t)*Finf*Kstar(:,di,t)'*T' + QQ; %(5.14) DK(2012)
Pinf(:,:,t+1) = T*Pinf(:,:,t)*Linf(:,:,t)'; %(5.14) DK(2012)
end
a(:,t+1) = T*atilde(:,t);
aK(1,:,t+1) = a(:,t+1);
% isn't a meaningless as long as we are in the diffuse part? MJ
for jnk=2:nk
aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
end
end
end
d = t;
P(:,:,d+1) = Pstar(:,:,d+1);
iFinf = iFinf(:,:,1:d);
iFstar= iFstar(:,:,1:d);
Linf = Linf(:,:,1:d);
Lstar = Lstar(:,:,1:d);
Kstar = Kstar(:,:,1:d);
Pstar = Pstar(:,:,1:d);
Pinf = Pinf(:,:,1:d);
notsteady = 1;
while notsteady && t<smpl
t = t+1;
P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1)); % make sure P is symmetric
di = data_index{t};
if isempty(di)
atilde(:,t) = a(:,t);
L(:,:,t) = T;
P(:,:,t+1) = T*P(:,:,t)*T' + QQ; %p. 111, DK(2012)
else
ZZ = Z(di,:);
v(di,t) = Y(di,t) - ZZ*a(:,t);
F = ZZ*P(:,:,t)*ZZ' + H(di,di);
sig=sqrt(diag(F));
if any(diag(F)<kalman_tol) || rcond(F./(sig*sig')) < kalman_tol
alphahat = Inf;
return
end
iF(di,di,t) = inv(F./(sig*sig'))./(sig*sig');
PZI = P(:,:,t)*ZZ'*iF(di,di,t);
atilde(:,t) = a(:,t) + PZI*v(di,t);
K(:,di,t) = T*PZI;
L(:,:,t) = T-K(:,di,t)*ZZ;
P(:,:,t+1) = T*P(:,:,t)*L(:,:,t)' + QQ;
end
a(:,t+1) = T*atilde(:,t);
Pf = P(:,:,t);
aK(1,:,t+1) = a(:,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
% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
end
% $$$ if t<smpl
% $$$ PZI_s = PZI;
% $$$ K_s = K(:,:,t);
% $$$ iF_s = iF(:,:,t);
% $$$ P_s = P(:,:,t+1);
% $$$ P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
% $$$ iF = cat(3,iF(:,:,1:t),repmat(iF_s,[1 1 smpl-t]));
% $$$ L = cat(3,L(:,:,1:t),repmat(T-K_s*Z,[1 1 smpl-t]));
% $$$ K = cat(3,K(:,:,1:t),repmat(T*P_s*Z'*iF_s,[1 1 smpl-t]));
% $$$ end
% $$$ while t<smpl
% $$$ t=t+1;
% $$$ v(:,t) = Y(:,t) - Z*a(:,t);
% $$$ atilde(:,t) = a(:,t) + PZI*v(:,t);
% $$$ a(:,t+1) = T*atilde(:,t);
% $$$ Pf = P(:,:,t);
% $$$ for jnk=1:nk,
% $$$ Pf = T*Pf*T' + QQ;
% $$$ aK(jnk,:,t+jnk) = T^jnk*atilde(:,t);
% $$$ PK(jnk,:,:,t+jnk) = Pf;
% $$$ end
% $$$ end
%% backward pass; r_T and N_T, stored in entry (smpl+1) were initialized at 0
t = smpl+1;
while t>d+1
t = t-1;
di = data_index{t};
if isempty(di)
% in this case, L is simply T due to Z=0, so that DK (2012), eq. 4.93 obtains
r(:,t) = L(:,:,t)'*r(:,t+1); %compute r_{t-1}, DK (2012), eq. 4.38 with Z=0
if state_uncertainty_flag
N(:,:,t)=L(:,:,t)'*N(:,:,t+1)*L(:,:,t); %compute N_{t-1}, DK (2012), eq. 4.42 with Z=0
end
else
ZZ = Z(di,:);
r(:,t) = ZZ'*iF(di,di,t)*v(di,t) + L(:,:,t)'*r(:,t+1); %compute r_{t-1}, DK (2012), eq. 4.38
if state_uncertainty_flag
N(:,:,t)=ZZ'*iF(di,di,t)*ZZ+L(:,:,t)'*N(:,:,t+1)*L(:,:,t); %compute N_{t-1}, DK (2012), eq. 4.42
end
end
alphahat(:,t) = a(:,t) + P(:,:,t)*r(:,t); %DK (2012), eq. 4.35
etahat(:,t) = QRt*r(:,t); %DK (2012), eq. 4.63
if state_uncertainty_flag
V(:,:,t) = P(:,:,t)-P(:,:,t)*N(:,:,t)*P(:,:,t); %DK (2012), eq. 4.43
end
end
if d %diffuse periods
% initialize r_d^(0) and r_d^(1) as below DK (2012), eq. 5.23
r0 = zeros(mm,d+1);
r0(:,d+1) = r(:,d+1); %set r0_{d}, i.e. shifted by one period
r1 = zeros(mm,d+1); %set r1_{d}, i.e. shifted by one period
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_1=zeros(mm,mm,d+1); %set N_1_{d}=0, i.e. shifted by one period, below DK (2012), eq. 5.26
N_2=zeros(mm,mm,d+1); %set N_2_{d}=0, i.e. shifted by one period, below DK (2012), eq. 5.26
end
for t = d:-1:1
di = data_index{t};
if isempty(di)
r1(:,t) = Linf(:,:,t)'*r1(:,t+1);
else
if ~Finf_singular(1,t)
r0(:,t) = Linf(:,:,t)'*r0(:,t+1); % DK (2012), eq. 5.21 where L^(0) is named Linf
r1(:,t) = Z(di,:)'*(iFinf(di,di,t)*v(di,t)-Kstar(:,di,t)'*T'*r0(:,t+1)) ...
+ Linf(:,:,t)'*r1(:,t+1); % DK (2012), eq. 5.21, noting that i) F^(1)=(F^Inf)^(-1)(see 5.10), ii) where L^(0) is named Linf, and iii) Kstar=T^{-1}*K^(1)
if state_uncertainty_flag
L_1=(-T*Kstar(:,di,t)*Z(di,:)); % noting that Kstar=T^{-1}*K^(1)
N(:,:,t)=Linf(:,:,t)'*N(:,:,t+1)*Linf(:,:,t); % DK (2012), eq. 5.19, noting that L^(0) is named Linf
N_1(:,:,t)=Z(di,:)'*iFinf(di,di,t)*Z(di,:)+Linf(:,:,t)'*N_1(:,:,t+1)*Linf(:,:,t)...
+L_1'*N(:,:,t+1)*Linf(:,:,t); % 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_2(:,:,t)=Z(di,:)'*(-iFinf(di,di,t)*Fstar(di,di,t)*iFinf(di,di,t))*Z(di,:) ...
+ Linf(:,:,t)'*N_2(:,:,t+1)*Linf(:,:,t)...
+ Linf(:,:,t)'*N_1(:,:,t+1)*L_1...
+ L_1'*N_1(:,:,t+1)'*Linf(:,:,t)...
+ L_1'*N(:,:,t+1)*L_1; % DK (2012), eq. 5.29
end
else
r0(:,t) = Z(di,:)'*iFstar(di,di,t)*v(di,t)-Lstar(:,:,t)'*r0(:,t+1); % DK (2003), eq. (14)
r1(:,t) = T'*r1(:,t+1); % DK (2003), eq. (14)
if state_uncertainty_flag
N(:,:,t)=Z(di,:)'*iFstar(di,di,t)*Z(di,:)...
+Lstar(:,:,t)'*N(:,:,t+1)*Lstar(:,:,t); % DK (2003), eq. (14)
N_1(:,:,t)=T'*N_1(:,:,t+1)*Lstar(:,:,t); % DK (2003), eq. (14)
N_2(:,:,t)=T'*N_2(:,:,t+1)*T'; % DK (2003), eq. (14)
end
end
end
alphahat(:,t) = a(:,t) + Pstar(:,:,t)*r0(:,t) + Pinf(:,:,t)*r1(:,t); % DK (2012), eq. 5.23
etahat(:,t) = QRt*r0(:,t); % DK (2012), p. 135
if state_uncertainty_flag
V(:,:,t)=Pstar(:,:,t)-Pstar(:,:,t)*N(:,:,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
end
if decomp_flag
decomp = zeros(nk,mm,rr,smpl+nk);
ZRQinv = inv(Z*QQ*Z');
for t = max(d,1):smpl
di = data_index{t};
% calculate eta_tm1t
eta_tm1t = QRt*Z(di,:)'*iF(di,di,t)*v(di,t);
AAA = P(:,:,t)*Z(di,:)'*ZRQinv(di,di)*bsxfun(@times,Z(di,:)*R,eta_tm1t');
% calculate decomposition
decomp(1,:,:,t+1) = AAA;
for h = 2:nk
AAA = T*AAA;
decomp(h,:,:,t+h) = AAA;
end
end
end
epsilonhat = Y-Z*alphahat;
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