282 lines
11 KiB
Matlab
282 lines
11 KiB
Matlab
function [LIK, lik,a,P] = univariate_kalman_filter(data_index,number_of_observations,no_more_missing_observations,Y,start,last,a,P,kalman_tol,riccati_tol,presample,T,Q,R,H,Z,mm,pp,rr,Zflag,diffuse_periods,analytic_derivation,DT,DYss,DOm,DH,DP,D2T,D2Yss,D2Om,D2H,D2P)
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% Computes the likelihood of a stationnary state space model (univariate approach).
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%@info:
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%! @deftypefn {Function File} {[@var{LIK},@var{likk},@var{a},@var{P} ] =} univariate_kalman_filter (@var{data_index}, @var{number_of_observations},@var{no_more_missing_observations}, @var{Y}, @var{start}, @var{last}, @var{a}, @var{P}, @var{kalman_tol}, @var{riccati_tol},@var{presample},@var{T},@var{Q},@var{R},@var{H},@var{Z},@var{mm},@var{pp},@var{rr},@var{Zflag},@var{diffuse_periods})
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%! @anchor{univariate_kalman_filter}
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%! @sp 1
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%! Computes the likelihood of a stationary state space model, given initial condition for the states (mean and variance).
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%! @sp 2
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%! @strong{Inputs}
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%! @sp 1
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%! @table @ @var
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%! @item data_index
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%! Matlab's cell, 1*T cell of column vectors of indices (in the vector of observed variables).
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%! @item number_of_observations
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%! Integer scalar, effective number of observations.
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%! @item no_more_missing_observations
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%! Integer scalar, date after which there is no more missing observation (it is then possible to switch to the steady state kalman filter).
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%! @item Y
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%! Matrix (@var{pp}*T) of doubles, data.
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%! @item start
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%! Integer scalar, first period.
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%! @item last
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%! Integer scalar, last period (@var{last}-@var{first} has to be inferior to T).
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%! @item a
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%! Vector (@var{mm}*1) of doubles, initial mean of the state vector.
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%! @item P
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%! Matrix (@var{mm}*@var{mm}) of doubles, initial covariance matrix of the state vector.
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%! @item kalman_tol
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%! Double scalar, tolerance parameter (rcond, inversibility of the covariance matrix of the prediction errors).
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%! @item riccati_tol
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%! Double scalar, tolerance parameter (iteration over the Riccati equation).
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%! @item presample
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%! Integer scalar, presampling if strictly positive (number of initial iterations to be discarded when evaluating the likelihood).
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%! @item T
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%! Matrix (@var{mm}*@var{mm}) of doubles, transition matrix of the state equation.
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%! @item Q
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%! Matrix (@var{rr}*@var{rr}) of doubles, covariance matrix of the structural innovations (noise in the state equation).
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%! @item R
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%! Matrix (@var{mm}*@var{rr}) of doubles,
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%! @item H
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%! Vector (@var{pp}) of doubles, diagonal of covariance matrix of the measurement errors (corelation among measurement errors is handled by a model transformation).
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%! @item Z
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%! Matrix (@var{pp}*@var{mm}) of doubles or vector of integers, matrix relating the states to the observed variables or vector of indices (depending on the value of @var{Zflag}).
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%! @item mm
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%! Integer scalar, number of state variables.
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%! @item pp
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%! Integer scalar, number of observed variables.
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%! @item rr
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%! Integer scalar, number of structural innovations.
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%! @item Zflag
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%! Integer scalar, equal to 0 if Z is a vector of indices targeting the observed variables in the state vector, equal to 1 if Z is a @var{pp}*@var{mm} matrix.
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%! @item diffuse_periods
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%! Integer scalar, number of diffuse filter periods in the initialization step.
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%! @end table
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%! @sp 2
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%! @strong{Outputs}
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%! @sp 1
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%! @table @ @var
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%! @item LIK
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%! Double scalar, value of (minus) the likelihood.
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%! @item likk
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%! Column vector of doubles, values of the density of each observation.
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%! @item a
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%! Vector (@var{mm}*1) of doubles, mean of the state vector at the end of the (sub)sample.
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%! @item P
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%! Matrix (@var{mm}*@var{mm}) of doubles, covariance of the state vector at the end of the (sub)sample.
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%! @end table
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%! @sp 2
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%! @strong{This function is called by:}
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%! @sp 1
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%! @ref{dsge_likelihood}
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%! @sp 2
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%! @strong{This function calls:}
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%! @sp 1
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%! @ref{univariate_kalman_filter_ss}
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%! @end deftypefn
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%@eod:
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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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% 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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% AUTHOR(S) stephane DOT adjemian AT univ DASH lemans DOT fr
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if nargin<20 || isempty(Zflag)% Set default value for Zflag ==> Z is a vector of indices.
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Zflag = 0;
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diffuse_periods = 0;
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end
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if nargin<21
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diffuse_periods = 0;
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end
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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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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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lik = zeros(smpl,pp); % Initialization of the matrix gathering the densities at each time and each observable
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LIK = Inf; % Default value of the log likelihood.
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oldP = Inf;
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l2pi = log(2*pi);
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notsteady = 1;
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oldK = Inf;
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K = NaN(mm,pp);
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asy_hess=0;
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if analytic_derivation == 0
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DLIK=[];
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Hess=[];
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else
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k = size(DT,3); % number of structural parameters
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DLIK = zeros(k,1); % Initialization of the score.
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Da = zeros(mm,k); % Derivative State vector.
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dlik = zeros(smpl,k);
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if Zflag==0
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C = zeros(pp,mm);
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for ii=1:pp, C(ii,Z(ii))=1; end % SELECTION MATRIX IN MEASUREMENT EQ. (FOR WHEN IT IS NOT CONSTANT)
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else
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C=Z;
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end
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dC = zeros(pp,mm,k); % either selection matrix or schur have zero derivatives
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if analytic_derivation==2
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Hess = zeros(k,k); % Initialization of the Hessian
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D2a = zeros(mm,k,k); % State vector.
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d2C = zeros(pp,mm,k,k);
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else
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asy_hess=D2T;
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Hess=[];
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D2a=[];
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D2T=[];
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D2Yss=[];
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end
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if asy_hess
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Hess = zeros(k,k); % Initialization of the Hessian
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end
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LIK={inf,DLIK,Hess};
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end
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while notsteady && t<=last %loop over t
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s = t-start+1;
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d_index = 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 Zflag
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z = Z(d_index,:);
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else
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z = Z(d_index);
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end
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oldP = P(:);
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for i=1:rows(z) %loop over i
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if Zflag
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prediction_error = Y(d_index(i),t) - z(i,:)*a; % nu_{t,i} in 6.13 in DK (2012)
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PZ = P*z(i,:)'; % Z_{t,i}*P_{t,i}*Z_{t,i}'
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Fi = z(i,:)*PZ + H(d_index(i)); % F_{t,i} in 6.13 in DK (2012), relies on H being diagonal
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else
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prediction_error = Y(d_index(i),t) - a(z(i)); % nu_{t,i} in 6.13 in DK (2012)
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PZ = P(:,z(i)); % Z_{t,i}*P_{t,i}*Z_{t,i}'
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Fi = PZ(z(i)) + H(d_index(i)); % F_{t,i} in 6.13 in DK (2012), relies on H being diagonal
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end
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if Fi>kalman_tol
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Ki = PZ/Fi; %K_{t,i} in 6.13 in DK (2012)
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if t>=no_more_missing_observations
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K(:,i) = Ki;
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end
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lik(s,i) = log(Fi) + (prediction_error*prediction_error)/Fi + l2pi; %Top equation p. 175 in DK (2012)
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if analytic_derivation
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if analytic_derivation==2
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[Da,DP,DLIKt,D2a,D2P, Hesst] = univariate_computeDLIK(k,i,z(i,:),Zflag,prediction_error,Ki,PZ,Fi,Da,DYss,DP,DH(d_index(i),:),notsteady,D2a,D2Yss,D2P);
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else
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[Da,DP,DLIKt,Hesst] = univariate_computeDLIK(k,i,z(i,:),Zflag,prediction_error,Ki,PZ,Fi,Da,DYss,DP,DH(d_index(i),:),notsteady);
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end
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if t>presample
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DLIK = DLIK + DLIKt;
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if analytic_derivation==2 || asy_hess
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Hess = Hess + Hesst;
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end
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end
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dlik(s,:)=dlik(s,:)+DLIKt';
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end
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a = a + Ki*prediction_error; %filtering according to (6.13) in DK (2012)
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P = P - PZ*Ki'; %filtering according to (6.13) in DK (2012)
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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 analytic_derivation
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if analytic_derivation==2
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[Da,DP,D2a,D2P] = univariate_computeDstate(k,a,P,T,Da,DP,DT,DOm,notsteady,D2a,D2P,D2T,D2Om);
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else
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[Da,DP] = univariate_computeDstate(k,a,P,T,Da,DP,DT,DOm,notsteady);
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end
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end
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a = T*a; %transition according to (6.14) in DK (2012)
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P = T*P*T' + QQ; %transition according to (6.14) in DK (2012)
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if t>=no_more_missing_observations && ~isqvec
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notsteady = max(abs(K(:)-oldK))>riccati_tol;
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oldK = K(:);
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end
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t = t+1;
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end
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% Divide by two.
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lik(1:s,:) = .5*lik(1:s,:);
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if analytic_derivation
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DLIK = DLIK/2;
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dlik = dlik/2;
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if analytic_derivation==2 || asy_hess
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% Hess = (Hess + Hess')/2;
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Hess = -Hess/2;
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end
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end
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% Call steady state univariate kalman filter if needed.
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if t <= last
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if analytic_derivation
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if analytic_derivation==2
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[tmp, tmp2] = univariate_kalman_filter_ss(Y,t,last,a,P,kalman_tol,T,H,Z,pp,Zflag, ...
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analytic_derivation,Da,DT,DYss,DP,DH,D2a,D2T,D2Yss,D2P);
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else
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[tmp, tmp2] = univariate_kalman_filter_ss(Y,t,last,a,P,kalman_tol,T,H,Z,pp,Zflag, ...
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analytic_derivation,Da,DT,DYss,DP,DH,asy_hess);
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end
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lik(s+1:end,:)=tmp2{1};
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dlik(s+1:end,:)=tmp2{2};
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DLIK = DLIK + tmp{2};
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if analytic_derivation==2 || asy_hess
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Hess = Hess + tmp{3};
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end
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else
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[tmp, lik(s+1:end,:)] = univariate_kalman_filter_ss(Y,t,last,a,P,kalman_tol,T,H,Z,pp,Zflag);
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end
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end
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% Compute minus the log-likelihood.
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if presample > diffuse_periods
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LIK = sum(sum(lik(1+presample-diffuse_periods:end,:)));
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else
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LIK = sum(sum(lik));
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end
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if analytic_derivation
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if analytic_derivation==2 || asy_hess
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LIK={LIK, DLIK, Hess};
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else
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LIK={LIK, DLIK};
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end
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lik={lik, dlik};
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end
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