493 lines
22 KiB
Matlab
493 lines
22 KiB
Matlab
function [alphahat,epsilonhat,etahat,atilde,P,aK,PK,decomp,V,aalphahat,eetahat,d,alphahat0,aalphahat0,V0] = missing_DiffuseKalmanSmootherH1_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)
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% function [alphahat,epsilonhat,etahat,atilde,P,aK,PK,decomp,V,aalphahat,eetahat,d] = missing_DiffuseKalmanSmootherH1_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)
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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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%
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% INPUTS
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% a_initial:mm*1 vector of initial (predicted) states
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% T: mm*mm matrix
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% Z: pp*mm matrix
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% R: mm*rr matrix
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% Q: rr*rr matrix
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% H: pp*pp matrix 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 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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% 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 variables (a_{t|T})
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% epsilonhat: smoothed measurement errors
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% etahat: smoothed shocks
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% atilde: matrix of updated variables (a_{t|t})
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% P: 3D array of one-step ahead forecast error variance
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% matrices
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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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% 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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% 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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% Durbin/Koopman (2012): "Time Series Analysis by State Space Methods", Oxford University Press,
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% Second Edition, Ch. 5
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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 k-step predictions)
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% new options_.nk: the max step ahed prediction in aK (default is 4)
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% new crit1 value for rank of Pinf
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% it is assured that P is symmetric
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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+1);
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a(:,1) = a_initial;
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atilde = zeros(mm,smpl);
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aK = zeros(nk,mm,smpl+nk);
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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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iF = zeros(pp,pp,smpl);
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Fstar = zeros(pp,pp,smpl);
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iFstar = zeros(pp,pp,smpl);
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iFinf = zeros(pp,pp,smpl);
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K = zeros(mm,pp,smpl);
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L = zeros(mm,mm,smpl);
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Linf = zeros(mm,mm,smpl);
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Lstar = zeros(mm,mm,smpl);
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Kstar = zeros(mm,pp,smpl);
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Kinf = zeros(mm,pp,smpl);
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P = zeros(mm,mm,smpl+1);
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Pstar = zeros(spstar(1),spstar(2),smpl+1);
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Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl+1);
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Pinf(:,:,1) = Pinf1;
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rr = size(Q,1);
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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+1);
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Finf_singular = zeros(1,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+1);
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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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t = 0;
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if rank(Pinf(:,:,1),diffuse_kalman_tol)
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% this is needed to get smoothed states in period 0 with diffuse steps
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% i.e. period 0 is a filter step without observables
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Pinf_init = Pinf(:,:,1);
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Pstar_init = Pstar(:,:,1);
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a_init = a(:,1);
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a(:,1) = T*a(:,1);
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% only non-stationary part is affected by following line,
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% hence Pstar on EXIT from diffuse step will NOT change.
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Pstar(:,:,1) = T*Pstar(:,:,1)*T' + QQ;
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end
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while rank(Pinf(:,:,t+1),diffuse_kalman_tol) && t<smpl
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t = t+1;
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di = data_index{t};
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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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if isempty(di)
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%no observations, propagate forward without updating based on
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%observations
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atilde(:,t) = a(:,t);
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a(:,t+1) = T*atilde(:,t);
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Linf(:,:,t) = T;
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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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else
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ZZ = Z(di,:); %span selector matrix
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v(di,t)= Y(di,t) - ZZ*a(:,t); %get prediction error v^(0) in (5.13) DK (2012)
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Finf = ZZ*Pinf(:,:,t)*ZZ'; % (5.7) in DK (2012)
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if rcond(Finf) < diffuse_kalman_tol %F_{\infty,t} = 0
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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
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Finf_singular(1,t) = 1;
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Fstar(di,di,t) = ZZ*Pstar(:,:,t)*ZZ' + H(di,di); % (5.7) in DK (2012)
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if rcond(Fstar(di,di,t)) < kalman_tol %F_{*} is singular
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if ~all(all(abs(Fstar(di,di,t))<kalman_tol))
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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 0
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a(:,t+1) = T*a(:,t);
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
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end
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else
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iFstar(di,di,t) = inv(Fstar(di,di,t));
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Kstar(:,di,t) = Pstar(:,:,t)*ZZ'*iFstar(di,di,t); %(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
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Lstar(:,:,t) = T - T*Kstar(:,di,t)*ZZ; %L^(0) in DK (2012), eq. 5.12
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*Lstar(:,:,t)'+QQ; % (5.17) DK (2012)
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a(:,t+1) = T*(a(:,t)+Kstar(:,di,t)*v(di,t)); % (5.13) DK (2012)
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end
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end
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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'
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atilde(:,t) = a(:,t) + Kinf(:,di,t)*v(di,t);
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Linf(:,:,t) = T - T*Kinf(:,di,t)*ZZ; %L^(0) in DK (2012), eq. 5.12
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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)*Linf(:,:,t)'-T*Kinf(:,di,t)*Finf*Kstar(:,di,t)'*T' + QQ; %(5.14) DK(2012)
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*Linf(:,:,t)'; %(5.14) DK(2012)
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end
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a(:,t+1) = T*atilde(:,t);
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aK(1,:,t+1) = a(:,t+1);
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% 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));
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end
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end
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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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iFinf = iFinf(:,:,1:d);
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iFstar= iFstar(:,:,1:d);
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Linf = Linf(:,:,1:d);
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Lstar = Lstar(:,:,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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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};
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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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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
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ZZ = Z(di,:);
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v(di,t) = Y(di,t) - ZZ*a(:,t);
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F = ZZ*P(:,:,t)*ZZ' + H(di,di);
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sig=sqrt(diag(F));
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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);
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atilde(:,t) = a(:,t) + PZI*v(di,t);
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K(:,di,t) = T*PZI;
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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
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if smoother_redux
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ri=zeros(mm,1);
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for st=t:-1:max(d+1,t-1)
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di = data_index{st};
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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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ri = L(:,:,t)'*ri; %compute r_{t-1}, DK (2012), eq. 4.38 with Z=0
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else
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ZZ = Z(di,:);
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ri = ZZ'*iF(di,di,st)*v(di,st) + L(:,:,st)'*ri; %compute r_{t-1}, DK (2012), eq. 4.38
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end
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if st==t-1
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% get states in t-1|t
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aalphahat(:,st) = a(:,st) + P(:,:,st)*ri; %DK (2012), eq. 4.35
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else
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% get shocks in t|t
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eetahat(:,st) = QRt*ri; %DK (2012), eq. 4.63
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end
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end
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if t==1
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ri = T'*ri; %compute r_{t-1}, DK (2012), eq. 4.38 with Z=0
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aalphahat0 = P(:,:,1)*ri; %DK (2012), eq. 4.35
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end
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end
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a(:,t+1) = T*atilde(:,t);
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if filter_covariance_flag
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Pf = P(:,:,t+1);
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end
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aK(1,:,t+1) = a(:,t+1);
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for jnk=1:nk
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if filter_covariance_flag
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if jnk>1
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Pf = T*Pf*T' + QQ;
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end
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PK(jnk,:,:,t+jnk) = Pf;
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end
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if jnk>1
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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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end
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% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
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end
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% $$$ if t<smpl
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% $$$ PZI_s = PZI;
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% $$$ K_s = K(:,:,t);
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% $$$ iF_s = iF(:,:,t);
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% $$$ P_s = P(:,:,t+1);
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% $$$ P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
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% $$$ iF = cat(3,iF(:,:,1:t),repmat(iF_s,[1 1 smpl-t]));
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% $$$ L = cat(3,L(:,:,1:t),repmat(T-K_s*Z,[1 1 smpl-t]));
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% $$$ K = cat(3,K(:,:,1:t),repmat(T*P_s*Z'*iF_s,[1 1 smpl-t]));
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% $$$ end
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% $$$ while t<smpl
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% $$$ t=t+1;
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% $$$ v(:,t) = Y(:,t) - Z*a(:,t);
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% $$$ atilde(:,t) = a(:,t) + PZI*v(:,t);
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% $$$ a(:,t+1) = T*atilde(:,t);
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% $$$ Pf = P(:,:,t);
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% $$$ for jnk=1:nk,
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% $$$ Pf = T*Pf*T' + QQ;
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% $$$ aK(jnk,:,t+jnk) = T^jnk*atilde(:,t);
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% $$$ PK(jnk,:,:,t+jnk) = Pf;
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% $$$ end
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% $$$ end
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%% backward pass; r_T and N_T, stored in entry (smpl+1) were initialized at 0
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t = smpl+1;
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while t>d+1
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t = t-1;
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di = data_index{t};
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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); %compute r_{t-1}, DK (2012), eq. 4.38 with Z=0
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if state_uncertainty_flag
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N(:,:,t)=L(:,:,t)'*N(:,:,t+1)*L(:,:,t); %compute N_{t-1}, DK (2012), eq. 4.42 with Z=0
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end
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else
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ZZ = Z(di,:);
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r(:,t) = ZZ'*iF(di,di,t)*v(di,t) + L(:,:,t)'*r(:,t+1); %compute r_{t-1}, DK (2012), eq. 4.38
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if state_uncertainty_flag
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N(:,:,t)=ZZ'*iF(di,di,t)*ZZ+L(:,:,t)'*N(:,:,t+1)*L(:,:,t); %compute N_{t-1}, DK (2012), eq. 4.42
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end
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end
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alphahat(:,t) = a(:,t) + P(:,:,t)*r(:,t); %DK (2012), eq. 4.35
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if isqvec
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QRt = Qvec(:,:,t)*transpose(R);
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end
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etahat(:,t) = QRt*r(:,t); %DK (2012), eq. 4.63
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if state_uncertainty_flag
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if smoother_redux
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ptmp = [P(:,:,t) R*Q; (R*Q)' Q];
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ntmp = [N(:,:,t) zeros(mm,rr); zeros(rr,mm+rr)];
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V(:,:,t) = ptmp - ptmp*ntmp*ptmp;
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else
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V(:,:,t) = P(:,:,t)-P(:,:,t)*N(:,:,t)*P(:,:,t); %DK (2012), eq. 4.43
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end
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end
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end
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if d==0 % get smoother in period t=0
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a0 = a(:,1);
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P0 = P(:,:,1);
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L0=T;
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r0 = L0'*r(:,1); %compute r_{t-1}, DK (2012), eq. 4.38 with Z=0
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alphahat0 = a0 + P0*r0; %DK (2012), eq. 4.35
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if state_uncertainty_flag
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N0=L0'*N(:,:,1)*L0; %compute N_{t-1}, DK (2012), eq. 4.42 with Z=0
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if smoother_redux
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ptmp = [P0 R*Q; (R*Q)' Q];
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ntmp = [N0 zeros(mm,rr); zeros(rr,mm+rr)];
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V0 = ptmp - ptmp*ntmp*ptmp;
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else
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V0 = P0-P0*N0*P0; %DK (2012), eq. 4.43
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|
end
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|
end
|
|
|
|
else %diffuse periods
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% initialize r_d^(0) and r_d^(1) as below DK (2012), eq. 5.23
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r0 = zeros(mm,d+1);
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r0(:,d+1) = r(:,d+1); %set r0_{d}, i.e. shifted by one period
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r1 = zeros(mm,d+1); %set r1_{d}, i.e. shifted by one period
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if state_uncertainty_flag
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|
%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
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N_2=zeros(mm,mm,d+1); %set N_2_{d}=0, i.e. shifted by one period, below DK (2012), eq. 5.26
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end
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|
for t = d:-1:1
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di = data_index{t};
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|
if isempty(di)
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|
r1(:,t) = Linf(:,:,t)'*r1(:,t+1);
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|
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)) ...
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|
+ 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
|
|
if isqvec
|
|
QRt = Qvec(:,:,t)*transpose(R);
|
|
end
|
|
etahat(:,t) = QRt*r0(:,t); % DK (2012), p. 135
|
|
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)];
|
|
ntmp = [N(:,:,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*ntmp*pstmp...
|
|
-(pitmp*ntmp1*pstmp)'...
|
|
- pitmp*ntmp1*pstmp...
|
|
- pitmp*ntmp2*pitmp; % DK (2012), eq. 5.30
|
|
|
|
else
|
|
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
|
|
% compute states and covarinace in period t=0
|
|
r10 = T'*r1(:,1);
|
|
r00 = T'*r0(:,1);
|
|
alphahat0 = a_init + Pstar_init*r00 + Pinf_init*r10; % DK (2012), eq. 5.23
|
|
if state_uncertainty_flag
|
|
N0=T'*N(:,:,1)*T; % DK (2012), eq. 5.19, noting that L^(0) is named Linf
|
|
N_10=T'*N_1(:,:,1)*T; % DK (2012), eq. 5.19, noting that L^(0) is named Linf
|
|
N_20=T'*N_2(:,:,1)*T; % DK (2012), eq. 5.19, noting that L^(0) is named Linf
|
|
if smoother_redux
|
|
pstmp = [Pstar_init R*Q; (R*Q)' Q];
|
|
pitmp = [Pinf_init zeros(mm,rr); zeros(rr,mm+rr)];
|
|
ntmp = [N0 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*ntmp*pstmp...
|
|
-(pitmp*ntmp1*pstmp)'...
|
|
- pitmp*ntmp1*pstmp...
|
|
- pitmp*ntmp2*pitmp; % DK (2012), eq. 5.30
|
|
|
|
else
|
|
V0 = Pstar_init-Pstar_init*N0*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
|
|
di = data_index{t};
|
|
% calculate eta_tm1t
|
|
if isqvec
|
|
QRt = Qvec(:,:,t)*transpose(R);
|
|
end
|
|
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;
|