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function [alphahat,epsilonhat,etahat,atilde,P,aK,PK,decomp] = missing_DiffuseKalmanSmootherH1_Z ( T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,diffuse_kalman_tol,decomp_flag)
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% 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)
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% Computes the diffuse kalman smoother without measurement error, in the case of a non-singular var-cov matrix.
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%
% 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.
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% nk number of forecasting periods
% kalman_tol tolerance for reciprocal condition number
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% diffuse_kalman_tol tolerance for reciprocal condition number (for Finf) and the rank of Pinf
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% decomp_flag if true, compute filter decomposition
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%
% 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
%
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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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% 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).
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% Durbin/Koopman (2012): "Time Series Analysis by State Space Methods", Oxford University Press,
% Second Edition, Ch. 5
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% Copyright (C) 2004-2016 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 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 ) ;
iFinf = zeros ( pp , pp , smpl ) ;
K = zeros ( mm , pp , smpl ) ;
L = zeros ( mm , mm , smpl ) ;
Linf = zeros ( mm , mm , smpl ) ;
Kstar = zeros ( mm , pp , smpl ) ;
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Kinf = zeros ( mm , pp , smpl ) ;
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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 ) ;
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epsilonhat = zeros ( rr , smpl ) ;
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r = zeros ( mm , smpl + 1 ) ;
t = 0 ;
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while rank ( Pinf ( : , : , t + 1 ) , diffuse_kalman_tol ) && t < smpl
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t = t + 1 ;
di = data_index { t } ;
if isempty ( di )
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%no observations, propagate forward without updating based on
%observations
atilde ( : , t ) = a ( : , t ) ;
a ( : , t + 1 ) = T * atilde ( : , t ) ;
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Linf ( : , : , t ) = T ;
Pstar ( : , : , t + 1 ) = T * Pstar ( : , : , t ) * T ' + QQ ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * T ' ;
else
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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
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% The univariate diffuse kalman filter should be used.
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alphahat = Inf ;
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return
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else %rank of F_{\infty,t} is 0
Fstar ( : , : , t ) = ZZ * Pstar ( : , : , t ) * ZZ ' + H ( di , di ) ; % (5.7) in DK (2012)
if rcond ( Fstar ( : , : , t ) ) < kalman_tol %F_{*} is singular
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if ~ all ( abs ( Fstar ( : , : , t ) ) < kalman_tol )
% The univariate diffuse kalman filter should be used.
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alphahat = Inf ;
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return
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else %rank 0
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a ( : , t + 1 ) = T * a ( : , t ) ;
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Pstar ( : , : , t + 1 ) = T * Pstar ( : , : , t ) * transpose ( T ) + QQ ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * transpose ( T ) ;
end
else
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iFstar = inv ( Fstar ( : , : , t ) ) ;
Kstar ( : , : , t ) = Pstar ( : , : , t ) * ZZ ' * iFstar ; %(5.15) of DK (2012) with Kstar=T^{-1}*K^(0)
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Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * transpose ( T ) ; % DK (2012), 5.16
Pstar ( : , : , t + 1 ) = T * ( Pstar ( : , : , t ) - Pstar ( : , : , t ) * ZZ ' * Kstar ( : , : , t ) ' ) * T ' + QQ ; % (5.17) DK (2012) with L_0 plugged in
a ( : , t + 1 ) = T * ( a ( : , t ) + Kstar ( : , : , t ) * v ( : , t ) ) ; % (5.13) DK (2012)
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end
end
else
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%see notes in kalman_filter_d.m for details of computations
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iFinf ( di , di , t ) = inv ( Finf ) ;
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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)
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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
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Pstar ( : , : , t + 1 ) = T * ( Pstar ( : , : , t ) - Pstar ( : , : , t ) * ZZ ' * Kinf ( : , di , t ) ' - Pinf ( : , : , t ) * ZZ ' * Kstar ( : , di , t ) ' ) * T ' + QQ ; %(5.14) DK(2012)
Pinf ( : , : , t + 1 ) = T * ( Pinf ( : , : , t ) - Pinf ( : , : , t ) * ZZ ' * Kinf ( : , di , t ) ' ) * T ' ; %(5.14) DK(2012)
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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
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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 ) ) ;
end
end
end
d = t ;
P ( : , : , d + 1 ) = Pstar ( : , : , d + 1 ) ;
iFinf = iFinf ( : , : , 1 : d ) ;
Linf = Linf ( : , : , 1 : d ) ;
Kstar = Kstar ( : , : , 1 : d ) ;
Pstar = Pstar ( : , : , 1 : d ) ;
Pinf = Pinf ( : , : , 1 : d ) ;
notsteady = 1 ;
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while notsteady && t < smpl
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t = t + 1 ;
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P ( : , : , t ) = tril ( P ( : , : , t ) ) + transpose ( tril ( P ( : , : , t ) , - 1 ) ) ; % make sure P is symmetric
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di = data_index { t } ;
if isempty ( di )
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atilde ( : , t ) = a ( : , t ) ;
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L ( : , : , t ) = T ;
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P ( : , : , t + 1 ) = T * P ( : , : , t ) * T ' + QQ ; %p. 111, DK(2012)
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else
ZZ = Z ( di , : ) ;
v ( di , t ) = Y ( di , t ) - ZZ * a ( : , t ) ;
F = ZZ * P ( : , : , t ) * ZZ ' + H ( di , di ) ;
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sig = sqrt ( diag ( F ) ) ;
if any ( diag ( F ) < kalman_tol ) || rcond ( F ./ ( sig * sig ' ) ) < kalman_tol
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alphahat = Inf ;
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return
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end
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iF ( di , di , t ) = inv ( F ./ ( sig * sig ' ) ) ./ ( sig * sig ' ) ;
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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 ;
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P ( : , : , t + 1 ) = T * P ( : , : , t ) * L ( : , : , t ) ' + QQ ;
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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
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% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
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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
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%% backward pass
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t = smpl + 1 ;
while t > d + 1
t = t - 1 ;
di = data_index { t } ;
if isempty ( di )
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% in this case, L is simply T due to Z=0, so that DK (2012), eq. 4.93 obtains
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r ( : , t ) = L ( : , : , t ) ' * r ( : , t + 1 ) ;
else
ZZ = Z ( di , : ) ;
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r ( : , t ) = ZZ ' * iF ( di , di , t ) * v ( di , t ) + L ( : , : , t ) ' * r ( : , t + 1 ) ; %DK (2012), eq. 4.38
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end
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alphahat ( : , t ) = a ( : , t ) + P ( : , : , t ) * r ( : , t ) ; %DK (2012), eq. 4.35
etahat ( : , t ) = QRt * r ( : , t ) ; %DK (2012), eq. 4.63
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end
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if d %diffuse periods
% initialize r_d^(0) and r_d^(1) as below DK (2012), eq. 5.23
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r0 = zeros ( mm , d + 1 ) ;
r0 ( : , d + 1 ) = r ( : , d + 1 ) ;
r1 = zeros ( mm , d + 1 ) ;
for t = d : - 1 : 1
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r0 ( : , t ) = Linf ( : , : , t ) ' * r0 ( : , t + 1 ) ; % DK (2012), eq. 5.21 where L^(0) is named Linf
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di = data_index { t } ;
if isempty ( di )
r1 ( : , t ) = Linf ( : , : , t ) ' * r1 ( : , t + 1 ) ;
else
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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)
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end
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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
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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
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 ;