function [LIK, lik] = kalman_filter(T,R,Q,H,P,Y,start,mf,kalman_tol,riccati_tol)
% Computes the likelihood of a stationnary state space model.
%
% INPUTS
% T [double] mm*mm transition matrix of the state equation.
% R [double] mm*rr matrix, mapping structural innovations to state variables.
% Q [double] rr*rr covariance matrix of the structural innovations.
% H [double] pp*pp (or 1*1 =0 if no measurement error) covariance matrix of the measurement errors.
% P [double] mm*mm variance-covariance matrix with stationary variables
% Y [double] pp*smpl matrix of (detrended) data, where pp is the maximum number of observed variables.
% start [integer] scalar, likelihood evaluation starts at 'start'.
% mf [integer] pp*1 vector of indices.
% kalman_tol [double] scalar, tolerance parameter (rcond).
% riccati_tol [double] scalar, tolerance parameter (riccati iteration).
%
% OUTPUTS
% LIK [double] scalar, MINUS loglikelihood
% lik [double] vector, density of observations in each period.
%
% REFERENCES
% See "Filtering and Smoothing of State Vector for Diffuse State Space
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
% Analysis, vol. 24(1), pp. 85-98).
%
% NOTES
% The vector "lik" is used to evaluate the jacobian of the likelihood.
% Copyright (C) 2004-2010 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see .
smpl = size(Y,2); % Sample size.
mm = size(T,2); % Number of state variables.
pp = size(Y,1); % Maximum number of observed variables.
a = zeros(mm,1); % State vector.
dF = 1; % det(F).
QQ = R*Q*transpose(R); % Variance of R times the vector of structural innovations.
t = 0; % Initialization of the time index.
lik = zeros(smpl,1); % Initialization of the vector gathering the densities.
LIK = Inf; % Default value of the log likelihood.
oldK = Inf;
notsteady = 1; % Steady state flag.
F_singular = 1;
while notsteady & t riccati_tol;
oldK = K(:);
end
end
if F_singular
error('The variance of the forecast error remains singular until the end of the sample')
end
if t < smpl
t0 = t+1;
while t < smpl
t = t+1;
v = Y(:,t)-a(mf);
a = T*(a+K*v);
lik(t) = transpose(v)*iF*v;
end
lik(t0:smpl) = lik(t0:smpl) + log(dF);
end
% adding log-likelihhod constants
lik = (lik + pp*log(2*pi))/2;
LIK = sum(lik(start:end)); % Minus the log-likelihood.