2015-04-03 17:48:25 +02:00
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)
2011-01-13 21:50:26 +01:00
2015-04-03 17:48:25 +02:00
% 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)
2016-04-10 19:18:41 +02:00
% Computes the diffuse kalman smoother without measurement error, in the case of a non-singular var-cov matrix.
2011-01-13 21:50:26 +01:00
%
% 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.
2011-03-25 21:32:45 +01:00
% nk number of forecasting periods
% kalman_tol tolerance for reciprocal condition number
2015-04-03 17:48:25 +02:00
% diffuse_kalman_tol tolerance for reciprocal condition number (for Finf) and the rank of Pinf
2011-03-25 21:32:45 +01:00
% decomp_flag if true, compute filter decomposition
2011-01-13 21:50:26 +01:00
%
% 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
%
2016-04-10 19:18:41 +02:00
% 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
%
2011-01-13 21:50:26 +01:00
% 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).
2016-03-11 16:22:42 +01:00
% Copyright (C) 2004-2016 Dynare Team
2011-01-13 21:50:26 +01:00
%
% 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 ) ;
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 ;
steady = smpl ;
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 ) ;
t = 0 ;
2015-04-03 17:48:25 +02:00
while rank ( Pinf ( : , : , t + 1 ) , diffuse_kalman_tol ) && t < smpl
2011-01-13 21:50:26 +01:00
t = t + 1 ;
di = data_index { t } ;
if isempty ( di )
atilde ( : , t ) = a ( : , t ) ;
Linf ( : , : , t ) = T ;
Pstar ( : , : , t + 1 ) = T * Pstar ( : , : , t ) * T ' + QQ ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * T ' ;
else
ZZ = Z ( di , : ) ;
v ( di , t ) = Y ( di , t ) - ZZ * a ( : , t ) ;
Finf = ZZ * Pinf ( : , : , t ) * ZZ ' ;
2015-04-03 17:48:25 +02:00
if rcond ( Finf ) < diffuse_kalman_tol
if ~ all ( abs ( Finf ( : ) ) < diffuse_kalman_tol )
2011-01-13 21:50:26 +01:00
% The univariate diffuse kalman filter should be used.
2011-11-02 14:02:12 +01:00
alphahat = Inf ;
2011-01-13 21:50:26 +01:00
return
else
Fstar ( : , : , t ) = ZZ * Pstar ( : , : , t ) * ZZ ' + H ( di , di ) ;
if rcond ( Fstar ( : , : , t ) ) < kalman_tol
if ~ all ( abs ( Fstar ( : , : , t ) ) < kalman_tol )
% The univariate diffuse kalman filter should be used.
2011-11-02 14:02:12 +01:00
alphahat = Inf ;
2011-01-13 21:50:26 +01:00
return
else
2016-03-06 21:07:50 +01:00
a ( : , t + 1 ) = T * a ( : , t ) ;
2011-01-13 21:50:26 +01:00
Pstar ( : , : , t + 1 ) = T * Pstar ( : , : , t ) * transpose ( T ) + QQ ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * transpose ( T ) ;
end
else
iFstar = inv ( Fstar ( : , : , t ) ) ;
Kstar ( : , : , t ) = Pstar ( : , : , t ) * ZZ ' * iFstar ( : , : , t ) ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * transpose ( T ) ;
Pstar ( : , : , t + 1 ) = T * ( Pstar ( : , : , t ) - Pstar ( : , : , t ) * ZZ ' * Kstar ( : , : , t ) ' ) * T ' + QQ ;
2016-03-06 21:07:50 +01:00
a ( : , t + 1 ) = T * ( a ( : , t ) + Kstar ( : , : , t ) * v ( : , t ) ) ;
2011-01-13 21:50:26 +01:00
end
end
else
iFinf ( di , di , t ) = inv ( Finf ) ;
PZI = Pinf ( : , : , t ) * ZZ ' * iFinf ( di , di , t ) ;
atilde ( : , t ) = a ( : , t ) + PZI * v ( di , t ) ;
Kinf ( : , di , t ) = T * PZI ;
Linf ( : , : , t ) = T - Kinf ( : , di , t ) * ZZ ;
Fstar ( di , di , t ) = ZZ * Pstar ( : , : , t ) * ZZ ' + H ( di , di ) ;
Kstar ( : , di , t ) = ( T * Pstar ( : , : , t ) * ZZ ' - Kinf ( : , di , t ) * Fstar ( di , di , t ) ) * iFinf ( di , di , t ) ;
Pstar ( : , : , t + 1 ) = T * Pstar ( : , : , t ) * T ' - T * Pstar ( : , : , t ) * ZZ ' * Kinf ( : , di , t ) ' - T * Pinf ( : , : , t ) * ZZ ' * Kstar ( : , di , t ) ' + QQ ;
Pinf ( : , : , t + 1 ) = T * Pinf ( : , : , t ) * T ' - T * Pinf ( : , : , t ) * ZZ ' * Kinf ( : , di , t ) ' ;
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 ) ;
Linf = Linf ( : , : , 1 : d ) ;
Fstar = Fstar ( : , : , 1 : d ) ;
Kstar = Kstar ( : , : , 1 : d ) ;
Pstar = Pstar ( : , : , 1 : d ) ;
Pinf = Pinf ( : , : , 1 : d ) ;
notsteady = 1 ;
2011-02-10 15:54:23 +01:00
while notsteady && t < smpl
2011-01-13 21:50:26 +01:00
t = t + 1 ;
P ( : , : , t ) = tril ( P ( : , : , t ) ) + transpose ( tril ( P ( : , : , t ) , - 1 ) ) ;
di = data_index { t } ;
if isempty ( di )
atilde ( : , t ) = a ( : , t ) ;
L ( : , : , t ) = T ;
P ( : , : , t + 1 ) = T * P ( : , : , t ) * T ' + QQ ;
else
ZZ = Z ( di , : ) ;
v ( di , t ) = Y ( di , t ) - ZZ * a ( : , t ) ;
F = ZZ * P ( : , : , t ) * ZZ ' + H ( di , di ) ;
2015-10-13 17:15:01 +02:00
sig = sqrt ( diag ( F ) ) ;
if any ( diag ( F ) < kalman_tol ) || rcond ( F ./ ( sig * sig ' ) ) < kalman_tol
2011-11-02 14:02:12 +01:00
alphahat = Inf ;
2015-04-03 17:48:25 +02:00
return
2011-01-13 21:50:26 +01:00
end
2015-10-13 17:15:01 +02:00
iF ( di , di , t ) = inv ( F ./ ( sig * sig ' ) ) ./ ( sig * sig ' ) ;
2011-01-13 21:50:26 +01:00
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 ;
2014-01-14 17:32:45 +01:00
P ( : , : , t + 1 ) = T * P ( : , : , t ) * L ( : , : , t ) ' + QQ ;
2011-01-13 21:50:26 +01:00
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
2011-03-25 21:32:45 +01:00
% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
2011-01-13 21:50:26 +01:00
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
t = smpl + 1 ;
while t > d + 1
t = t - 1 ;
di = data_index { t } ;
if isempty ( di )
r ( : , t ) = L ( : , : , t ) ' * r ( : , t + 1 ) ;
else
ZZ = Z ( di , : ) ;
r ( : , t ) = ZZ ' * iF ( di , di , t ) * v ( di , t ) + L ( : , : , t ) ' * r ( : , t + 1 ) ;
end
alphahat ( : , t ) = a ( : , t ) + P ( : , : , t ) * r ( : , t ) ;
etahat ( : , t ) = QRt * r ( : , t ) ;
end
if d
r0 = zeros ( mm , d + 1 ) ;
r0 ( : , d + 1 ) = r ( : , d + 1 ) ;
r1 = zeros ( mm , d + 1 ) ;
for t = d : - 1 : 1
r0 ( : , t ) = Linf ( : , : , t ) ' * r0 ( : , t + 1 ) ;
di = data_index { t } ;
if isempty ( di )
r1 ( : , t ) = Linf ( : , : , t ) ' * r1 ( : , t + 1 ) ;
else
r1 ( : , t ) = Z ( di , : ) ' * ( iFinf ( di , di , t ) * v ( di , t ) - Kstar ( : , di , t ) ' * r0 ( : , t + 1 ) ) ...
+ Linf ( : , : , t ) ' * r1 ( : , t + 1 ) ;
end
alphahat ( : , t ) = a ( : , t ) + Pstar ( : , : , t ) * r0 ( : , t ) + Pinf ( : , : , t ) * r1 ( : , t ) ;
etahat ( : , t ) = QRt * r0 ( : , t ) ;
end
end
2011-03-25 21:32:45 +01:00
if decomp_flag
2011-01-13 21:50:26 +01:00
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
Ttok = eye ( mm , mm ) ;
decomp ( 1 , : , : , t + 1 ) = AAA ;
for h = 2 : nk
AAA = T * AAA ;
decomp ( h , : , : , t + h ) = AAA ;
end
end
end
epsilonhat = Y - Z * alphahat ;