135 lines
8.2 KiB
Matlab
135 lines
8.2 KiB
Matlab
function [dLIK,dlik,a,Pstar] = kalman_filter_d(Y, start, last, a, Pinf, Pstar, kalman_tol, diffuse_kalman_tol, riccati_tol, presample, T, R, Q, H, Z, mm, pp, rr)
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% Computes the diffuse likelihood of a state space model.
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%
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% INPUTS
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% Y [double] pp*smpl matrix of (detrended) data, where pp is the number of observed variables.
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% start [integer] scalar, first observation.
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% last [integer] scalar, last observation.
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% a [double] mm*1 vector, levels of the predicted initial state variables (E_{0}(alpha_1)).
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% Pinf [double] mm*mm matrix used to initialize the covariance matrix of the state vector.
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% Pstar [double] mm*mm matrix used to initialize the covariance matrix of the state vector.
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% kalman_tol [double] scalar, tolerance parameter (rcond) of F_star.
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% diffuse_kalman_tol [double] scalar, tolerance parameter (rcond) of Pinf to signify end of diffuse filtering and Finf.
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% riccati_tol [double] scalar, tolerance parameter (riccati iteration);
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% not used in this filter as usually diffuse phase will be left before convergence of filter to steady state.
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% presample [integer] scalar, presampling if strictly positive.
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% T [double] mm*mm matrix, transition matrix in the state equations.
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% R [double] mm*rr matrix relating the structural innovations to the state vector.
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% Q [double] rr*rr covariance matrix of the structural innovations.
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% H [double] pp*pp covariance matrix of the measurement errors (if H is equal to zero (scalar) there is no measurement error).
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% Z [double] pp*mm matrix, selection matrix or pp linear independent combinations of the state vector.
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% mm [integer] scalar, number of state variables.
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% pp [integer] scalar, number of observed variables.
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% rr [integer] scalar, number of structural innovations.
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%
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% OUTPUTS
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% LIK [double] scalar, minus loglikelihood
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% lik [double] smpl*1 vector, log density of each vector of observations.
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% a [double] mm*1 vector, current estimate of the state vector tomorrow
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% (E_{T}(alpha_{T+1})).
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% Pstar [double] mm*mm matrix, covariance matrix of the state vector.
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%
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% REFERENCES
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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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% and
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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 and 7.2
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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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% Get sample size.
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smpl = last-start+1;
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% Initialize some variables.
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dF = 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); % Variance of R times the vector of structural innovations.
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t = start; % Initialization of the time index.
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dlik = zeros(smpl,1); % Initialization of the vector gathering the densities.
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dLIK = Inf; % Default value of the log likelihood.
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oldK = Inf;
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s = 0;
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while rank(Z*Pinf*Z',diffuse_kalman_tol) && (t<=last)
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s = t-start+1;
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v = Y(:,t)-Z*a; %get prediction error v^(0) in (5.13) DK (2012)
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Finf = Z*Pinf*Z'; % (5.7) in DK (2012)
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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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%do case distinction based on whether F_{\infty,t} has full rank or 0 rank
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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 instead.
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return
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else %rank of F_{\infty,t} is 0
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Fstar = Z*Pstar*Z' + H; % (5.7) in DK (2012)
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if rcond(Fstar) < kalman_tol %F_{*} is singular
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% if ~all(abs(Fstar(:))<kalman_tol), then use univariate diffuse filter,
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% otherwise this is a pathological case and the draw is discarded
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return
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else
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iFstar = inv(Fstar);
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dFstar = det(Fstar);
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Kstar = Pstar*Z'*iFstar; %(5.15) of DK (2012) with Kstar=T^{-1}*K^(0)
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dlik(s)= log(dFstar) + v'*iFstar*v; %set w_t to bottom case in bottom equation page 172, DK (2012)
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Pinf = T*Pinf*transpose(T); % (5.16) DK (2012)
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Pstar = T*(Pstar-Pstar*Z'*Kstar')*T'+QQ; % (5.17) DK (2012)
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a = T*(a+Kstar*v); % (5.13) DK (2012)
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end
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end
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else %F_{\infty,t} positive definite
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%To compare to DK (2012), this block makes use of the following transformation
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%Kstar=T^{-1}*K^{(1)}=M_{*}*F^{(1)}+M_{\infty}*F^{(2)}
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% =P_{*}*Z'*F^{(1)}+P_{\infty}*Z'*((-1)*(-F_{\infty}^{-1})*F_{*}*(F_{\infty}^{-1}))
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% =[P_{*}*Z'-Kinf*F_{*})]*F^{(1)}
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%Make use of L^{0}'=(T-K^{(0)}*Z)'=(T-T*M_{\infty}*F^{(1)}*Z)'
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% =(T-T*P_{\infty*Z'*F^{(1)}*Z)'=(T-T*Kinf*Z)'
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% = (T*(I-*Kinf*Z))'=(I-Z'*Kinf')*T'
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%P_{*}=T*P_{\infty}*L^{(1)}+T*P_{*}*L^{(0)}+RQR
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% =T*[(P_{\infty}*(-K^{(1)*Z}))+P_{*}*(I-Z'*Kinf')*T'+RQR]
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dlik(s)= log(det(Finf)); %set w_t to top case in bottom equation page 172, DK (2012)
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iFinf = inv(Finf);
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Kinf = Pinf*Z'*iFinf; %define Kinf=T^{-1}*K_0 with M_{\infty}=Pinf*Z'
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Fstar = Z*Pstar*Z' + H; %(5.7) DK(2012)
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Kstar = (Pstar*Z'-Kinf*Fstar)*iFinf; %(5.12) DK(2012); 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*(Pstar-Pstar*Z'*Kinf'-Pinf*Z'*Kstar')*T'+QQ; %(5.14) DK(2012)
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Pinf = T*(Pinf-Pinf*Z'*Kinf')*T'; %(5.14) DK(2012)
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a = T*(a+Kinf*v); %(5.13) DK(2012)
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end
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t = t+1;
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end
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if t>last
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warning(['kalman_filter_d: There isn''t enough information to estimate the initial conditions of the nonstationary variables. The diffuse Kalman filter never left the diffuse stage.']);
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dLIK = NaN;
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return
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end
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dlik = dlik(1:s);
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dlik = .5*(dlik + pp*log(2*pi));
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dLIK = sum(dlik(1+presample:end));
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