dynare/matlab/kalman/likelihood/missing_observations_kalman_filter.m

function  [LIK, lik] = missing_observations_kalman_filter(T,R,Q,H,P,Y,start,mf,kalman_tol,riccati_tol,data_index,number_of_observations,no_more_missing_observations)
% Computes the likelihood of a state space model in the case with missing observations.
%
% 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 of the initial state vector.
%    Y                            [double]    pp*smpl matrix of detrended data, where pp is the maximum number of observed variables
%    trend                        [double]    pp*smpl matrix
%    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).
%    data_index                   [cell]      1*smpl cell of column vectors of indices.
%    number_of_observations       [integer]   scalar.
%    no_more_missing_observations [integer]   scalar.    
%    
% OUTPUTS
%    LIK        [double]    scalar, likelihood
%    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) 2004-2008 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 <http://www.gnu.org/licenses/>.

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<smpl 
    t  = t+1;
    d_index = data_index{t};
    if isempty(d_index)
        a = T*a;
        P = T*P*transpose(T)+QQ;
    else
        MF = mf(d_index);                 % Set the selection for observed variables. 
        v  = Y(d_index,t)-a(MF);
        if ~isscalar(H)                         % => Errors in the measurement equation.
            F = P(MF,MF) + H(d_index,d_index); 
        else% => 
            % case 1. No errors in the measurement (H=0) and more than one variable is observed in this state space model. 
            % case 2. Errors in the measurement equation, but only one variable is observed in this state-space model.
            F = P(MF,MF)+H;
        end
        if rcond(F) < kalman_tol
            if ~all(abs(F(:))<kalman_tol)
                return
            else
                a = T*a;
                P = T*P*transpose(T)+QQ;
            end
        else
            F_singular = 0;
            dF     = det(F);
            iF     = inv(F);
            lik(t) = log(dF) + transpose(v)*iF*v + length(d_index)*log(2*pi);
            K      = P(:,MF)*iF;
            a      = T*(a+K*v);
            P      = T*(P-K*P(MF,:))*transpose(T)+QQ;
            if t>no_more_missing_observations
                notsteady = max(abs(K(:)-oldK))>riccati_tol;
                oldK = K(:);
            end
        end
    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) = v'*iF*v;
    end
    lik(t0:smpl) = lik(t0:smpl) + log(dF) + pp*log(2*pi);
end

lik = lik/2;

LIK    = sum(lik(start:end));