818 lines
31 KiB
Matlab
818 lines
31 KiB
Matlab
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_)
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% 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_)
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% Computes the diffuse kalman smoother in the case of a singular var-cov matrix.
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% Univariate treatment of multivariate time series.
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%
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% INPUTS
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% a_initial:mm*1 vector of initial states
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% T: mm*mm matrix state transition matrix
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% Z: pp*mm matrix selector matrix for observables in augmented state vector
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% R: mm*rr matrix second matrix of the state equation relating the structural innovations to the state variables
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% Q: rr*rr matrix covariance matrix of structural errors
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% H: pp*1 vector of variance of measurement errors
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% Pinf1: mm*mm diagonal matrix with with q ones and m-q zeros
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% Pstar1: mm*mm variance-covariance matrix with stationary variables
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% Y: pp*1 vector
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% pp: number of observed variables
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% mm: number of state variables
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% smpl: sample size
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% data_index: [cell] 1*smpl cell of column vectors of indices.
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% nk: number of forecasting periods
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% kalman_tol: tolerance for zero divider
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% diffuse_kalman_tol: tolerance for zero divider
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% decomp_flag: if true, compute filter decomposition
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% state_uncertainty_flag: if true, compute uncertainty about smoothed
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% state estimate
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% decomp_flag: if true, compute filter decomposition
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% filter_covariance_flag: if true, compute filter covariance
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% smoother_redux: if true, compute smoother on restricted
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% state space, recover static variables from this
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%
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% OUTPUTS
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% alphahat: smoothed state variables (a_{t|T})
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% epsilonhat: measurement errors
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% etahat: smoothed shocks
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% a: matrix of updated variables (a_{t|t})
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% aK: 3D array of k step ahead filtered state variables (a_{t+k|t})
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% (meaningless for periods 1:d)
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% P1: 3D array of one-step ahead forecast error variance
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% matrices
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% PK: 4D array of k-step ahead forecast error variance
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% matrices (meaningless for periods 1:d)
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% decomp: decomposition of the effect of shocks on filtered values
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% V: 3D array of state uncertainty matrices
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% aalphahat: filtered states in t-1|t
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% eetahat: updated shocks in t|t
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% d: number of diffuse periods
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%
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% Notes:
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% Outputs are stored in decision-rule order, i.e. to get variables in order of declaration
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% as in M_.endo_names, ones needs code along the lines of:
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% variables_declaration_order(dr.order_var,:) = alphahat
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%
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% Algorithm:
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%
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% Uses the univariate filter as described in Durbin/Koopman (2012): "Time
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% Series Analysis by State Space Methods", Oxford University Press,
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% Second Edition, Ch. 6.4 + 7.2.5
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% and
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% Koopman/Durbin (2000): "Fast Filtering and Smoothing for Multivariatze State Space
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% Models", in Journal of Time Series Analysis, vol. 21(3), pp. 281-296.
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%
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% SPECIAL REQUIREMENTS
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% See "Filtering and Smoothing of State Vector for Diffuse State Space
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% Models", S.J. Koopman and J. Durbin (2003), in Journal of Time Series
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% Analysis, vol. 24(1), pp. 85-98.
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% Copyright © 2004-2021 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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% Modified by M. Ratto
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% New output argument aK: 1-step to nk-stpe ahed predictions)
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% New input argument nk: max order of predictions in aK
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if size(H,2)>1
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error('missing_DiffuseKalmanSmootherH3_Z:: H is not a vector. This must not happens')
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end
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d = 0;
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decomp = [];
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spinf = size(Pinf1);
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spstar = size(Pstar1);
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v = zeros(pp,smpl);
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a = zeros(mm,smpl);
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a1 = zeros(mm,smpl+1);
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a(:,1) = a_initial;
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a1(:,1) = a_initial;
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aK = zeros(nk,mm,smpl+nk);
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Fstar = zeros(pp,smpl);
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Finf = zeros(pp,smpl);
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Fi = zeros(pp,smpl);
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Ki = zeros(mm,pp,smpl);
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Kstar = zeros(mm,pp,smpl);
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Kinf = zeros(spstar(1),pp,smpl);
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P = zeros(mm,mm,smpl+1);
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P1 = P;
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if filter_covariance_flag
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PK = zeros(nk,mm,mm,smpl+nk);
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else
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PK = [];
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end
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Pstar = zeros(spstar(1),spstar(2),smpl);
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Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl);
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Pinf(:,:,1) = Pinf1;
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Pstar1 = Pstar;
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Pinf1 = Pinf;
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rr = size(Q,1); % number of structural shocks
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isqvec = false;
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if ndim(Q)>2
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Qvec = Q;
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Q=Q(:,:,1);
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isqvec = true;
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end
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QQ = R*Q*transpose(R);
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QRt = Q*transpose(R);
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alphahat = zeros(mm,smpl);
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etahat = zeros(rr,smpl);
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if smoother_redux
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aalphahat = alphahat;
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eetahat = etahat;
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else
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aalphahat = [];
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eetahat = [];
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end
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epsilonhat = zeros(rr,smpl);
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r = zeros(mm,smpl);
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if state_uncertainty_flag
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if smoother_redux
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V = zeros(mm+rr,mm+rr,smpl);
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else
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V = zeros(mm,mm,smpl);
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end
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N = zeros(mm,mm,smpl);
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else
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V=[];
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end
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alphahat0=[];
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aalphahat0=[];
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V0=[];
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if ~occbin_.status
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isoccbin = 0;
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C=0;
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TT=[];
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RR=[];
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CC=[];
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else
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isoccbin = 1;
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Qt = repmat(Q,[1 1 3]);
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options_=occbin_.info{1};
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oo_=occbin_.info{2};
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M_=occbin_.info{3};
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occbin_options=occbin_.info{4};
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opts_regime = occbin_options.opts_regime;
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% first_period_occbin_update = inf;
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if isfield(opts_regime,'regime_history') && ~isempty(opts_regime.regime_history)
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opts_regime.regime_history=[opts_regime.regime_history(1) opts_regime.regime_history];
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else
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opts_regime.binding_indicator=zeros(smpl+2,M_.occbin.constraint_nbr);
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end
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occbin_options.opts_regime = opts_regime;
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[~, ~, ~, regimes_] = occbin.check_regimes([], [], [], opts_regime, M_, oo_, options_);
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if length(occbin_.info)>4
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if length(occbin_.info)==6 && options_.smoother_redux
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TT=repmat(T,1,1,smpl+1);
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RR=repmat(R,1,1,smpl+1);
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CC=repmat(zeros(mm,1),1,smpl+1);
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T0=occbin_.info{5};
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R0=occbin_.info{6};
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else
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TT=occbin_.info{5};
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RR=occbin_.info{6};
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CC=occbin_.info{7};
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% TT = cat(3,TT,T);
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% RR = cat(3,RR,R);
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% CC = cat(2,CC,zeros(mm,1));
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if options_.smoother_redux
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my_order_var = oo_.dr.restrict_var_list;
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CC = CC(my_order_var,:);
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RR = RR(my_order_var,:,:);
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TT = TT(my_order_var,my_order_var,:);
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T0=occbin_.info{8};
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R0=occbin_.info{9};
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end
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if size(TT,3)<(smpl+1)
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TT=repmat(T,1,1,smpl+1);
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RR=repmat(R,1,1,smpl+1);
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CC=repmat(zeros(mm,1),1,smpl+1);
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end
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end
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else
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TT=repmat(T,1,1,smpl+1);
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RR=repmat(R,1,1,smpl+1);
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CC=repmat(zeros(mm,1),1,smpl+1);
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end
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if ~smoother_redux
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T0=T;
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R0=R;
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end
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if ~isinf(occbin_options.first_period_occbin_update)
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% initialize state matrices (otherwise they are set to 0 for
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% t<first_period_occbin_update!)
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TTT=repmat(T0,1,1,smpl+1);
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RRR=repmat(R0,1,1,smpl+1);
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CCC=repmat(zeros(length(T0),1),1,smpl+1);
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end
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end
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t = 0;
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icc=0;
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if ~isempty(Pinf(:,:,1))
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newRank = rank(Z*Pinf(:,:,1)*Z',diffuse_kalman_tol);
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else
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newRank = rank(Pinf(:,:,1),diffuse_kalman_tol);
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end
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if newRank
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% add this to get smoothed states in period 0
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Pinf_init = Pinf(:,:,1);
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Pstar_init = Pstar(:,:,1);
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Pstar(:,:,1) = T*Pstar(:,:,1)*T' + QQ;
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ainit = a1(:,1);
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end
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while newRank && t < smpl
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t = t+1;
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a(:,t) = a1(:,t);
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Pstar1(:,:,t) = Pstar(:,:,t);
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Pinf1(:,:,t) = Pinf(:,:,t);
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di = data_index{t}';
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for i=di
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Zi = Z(i,:);
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v(i,t) = Y(i,t)-Zi*a(:,t); % nu_{t,i} in 6.13 in DK (2012)
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Fstar(i,t) = Zi*Pstar(:,:,t)*Zi' +H(i); % F_{*,t} in 5.7 in DK (2012), relies on H being diagonal
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Finf(i,t) = Zi*Pinf(:,:,t)*Zi'; % F_{\infty,t} in 5.7 in DK (2012)
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Kstar(:,i,t) = Pstar(:,:,t)*Zi'; % KD (2000), eq. (15)
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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}
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icc=icc+1;
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Kinf(:,i,t) = Pinf(:,:,t)*Zi'; % KD (2000), eq. (15)
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Kinf_Finf = Kinf(:,i,t)/Finf(i,t);
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a(:,t) = a(:,t) + Kinf_Finf*v(i,t); % KD (2000), eq. (16)
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Pstar(:,:,t) = Pstar(:,:,t) + ...
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Kinf(:,i,t)*Kinf_Finf'*(Fstar(i,t)/Finf(i,t)) - ...
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Kstar(:,i,t)*Kinf_Finf' - ...
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Kinf_Finf*Kstar(:,i,t)'; % KD (2000), eq. (16)
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Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*Kinf(:,i,t)'/Finf(i,t); % KD (2000), eq. (16)
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elseif Fstar(i,t) > kalman_tol
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a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t); % KD (2000), eq. (17)
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Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*Kstar(:,i,t)'/Fstar(i,t); % KD (2000), eq. (17)
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% Pinf is passed through unaltered, see eq. (17) of
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% Koopman/Durbin (2000)
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else
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% do nothing as a_{t,i+1}=a_{t,i} and P_{t,i+1}=P_{t,i}, see
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% p. 157, DK (2012)
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end
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end
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if newRank
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if ~isempty(Pinf(:,:,t))
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oldRank = rank(Z*Pinf(:,:,t)*Z',diffuse_kalman_tol);
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else
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oldRank = rank(Pinf(:,:,t),diffuse_kalman_tol);
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end
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else
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oldRank = 0;
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end
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if isoccbin
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TT(:,:,t+1)= T;
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RR(:,:,t+1)= R;
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end
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a1(:,t+1) = T*a(:,t);
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aK(1,:,t+1) = a1(:,t+1);
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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));
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end
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if isqvec
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QQ = R*Qvec(:,:,t+1)*transpose(R);
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end
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*T'+ QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*T';
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if newRank
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if ~isempty(Pinf(:,:,t+1))
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newRank = rank(Z*Pinf(:,:,t+1)*Z',diffuse_kalman_tol);
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else
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newRank = rank(Pinf(:,:,t+1),diffuse_kalman_tol);
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end
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end
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if oldRank ~= newRank
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disp('univariate_diffuse_kalman_filter:: T does influence the rank of Pinf!')
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disp('This may happen for models with order of integration >1.')
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end
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end
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if isoccbin
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first_period_occbin_update = occbin_options.first_period_occbin_update;
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if d>0
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first_period_occbin_update = max(t+2,occbin_options.first_period_occbin_update);
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% kalman update is not yet robust to accommodate diffuse steps
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end
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if occbin_options.opts_regime.waitbar
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hh = dyn_waitbar(0,'Occbin: Piecewise Kalman Filter');
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set(hh,'Name','Occbin: Piecewise Kalman Filter.');
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waitbar_indicator=1;
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else
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waitbar_indicator=0;
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end
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else
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first_period_occbin_update = inf;
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waitbar_indicator=0;
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end
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d = t;
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P(:,:,d+1) = Pstar(:,:,d+1);
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Fstar = Fstar(:,1:d);
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Finf = Finf(:,1:d);
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Kstar = Kstar(:,:,1:d);
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Pstar = Pstar(:,:,1:d);
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Pinf = Pinf(:,:,1:d);
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Pstar1 = Pstar1(:,:,1:d);
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Pinf1 = Pinf1(:,:,1:d);
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notsteady = 1;
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while notsteady && t<smpl
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t = t+1;
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if t==1
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Pinit = P(:,:,1);
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ainit = a1(:,1);
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end
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a(:,t) = a1(:,t);
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P1(:,:,t) = P(:,:,t);
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di = data_index{t}';
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if t>=first_period_occbin_update
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if waitbar_indicator
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dyn_waitbar(t/smpl, hh, sprintf('Period %u of %u', t,smpl));
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end
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occbin_options.opts_regime.waitbar=0;
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if t==1
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if isqvec
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Qt = cat(3,Q,Qvec(:,:,t:t+1));
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end
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a0 = a(:,1);
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a10 = [a0 a(:,1)];
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P0 = P(:,:,1);
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P10 = P1(:,:,[1 1]);
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data_index0{1}=[];
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data_index0(2)=data_index(1);
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v0(:,2)=v(:,1);
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Y0(:,2)=Y(:,1);
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Y0(:,1)=nan;
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Fi0 = Fi(:,1);
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Ki0 = Ki(:,:,1);
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TT01 = cat(3,T,TT(:,:,1));
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RR01 = cat(3,R,RR(:,:,1));
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CC01 = zeros(size(CC,1),2);
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CC01(:,2) = CC(:,1);
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[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_,oo_,options_,occbin_options,kalman_tol,nk);
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else
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if isqvec
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Qt = Qvec(:,:,t-1:t+1);
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end
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[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_,oo_,options_,occbin_options,kalman_tol,nk);
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end
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if ~error_flag
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regimes_(t:t+2)=tmp;
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else
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varargout{1} = [];
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varargout{2} = [];
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varargout{3} = [];
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varargout{4} = [];
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varargout{5} = [];
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varargout{6} = [];
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varargout{7} = [];
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return
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end
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if smoother_redux && t>1
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aalphahat(:,t-1) = aha(:,1);
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end
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eetahat(:,t) = etaha(:,2);
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a(:,t) = ax(:,1);
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a1(:,t) = a1x(:,2);
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a1(:,t+1) = ax(:,2);
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v(di,t) = vx(di,2);
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Fi(di,t) = Fix(di,2);
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Ki(:,di,t) = Kix(:,di,2);
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TT(:,:,t:t+1) = Tx(:,:,1:2);
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RR(:,:,t:t+1) = Rx(:,:,1:2);
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CC(:,t:t+1) = Cx(:,1:2);
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TTT(:,:,t)=TTx;
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|
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(oo_.dr.restrict_var_list)=a(:,t);
|
|
opts_simul.endo_init = tmp(oo_.dr.inv_order_var);
|
|
else
|
|
opts_simul.endo_init = a(oo_.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_,oo_,options_);
|
|
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,oo_.dr.order_var(oo_.dr.restrict_var_list)) - out.ys(oo_.dr.order_var(oo_.dr.restrict_var_list))';
|
|
else
|
|
aK(jnk,oo_.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);
|
|
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)'...
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|
- 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
|