function[LIK, LIKK, a, P] =kalman_filter_fast(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)
%! Computes the likelihood of a stationary state space model, given initial condition for the states (mean and variance).
%! @sp 2
%! @strong{Inputs}
%! @sp 1
%! @table @ @var
%! @item Y
%! Matrix (@var{pp}*T) of doubles, data.
%! @item start
%! Integer scalar, first period.
%! @item last
%! Integer scalar, last period (@var{last}-@var{first} has to be inferior to T).
%! @item a
%! Vector (@var{mm}*1) of doubles, initial mean of the state vector.
%! @item P
%! Matrix (@var{mm}*@var{mm}) of doubles, initial covariance matrix of the state vector.
%! @item kalman_tol
%! Double scalar, tolerance parameter (rcond, inversibility of the covariance matrix of the prediction errors).
%! @item riccati_tol
%! Double scalar, tolerance parameter (iteration over the Riccati equation).
%! @item presample
%! Integer scalar, presampling if strictly positive (number of initial iterations to be discarded when evaluating the likelihood).
%! @item T
%! Matrix (@var{mm}*@var{mm}) of doubles, transition matrix of the state equation.
%! @item Q
%! Matrix (@var{rr}*@var{rr}) of doubles, covariance matrix of the structural innovations (noise in the state equation).
%! @item R
%! Matrix (@var{mm}*@var{rr}) of doubles, second matrix of the state equation relating the structural innovations to the state variables.
%! @item H
%! Matrix (@var{pp}*@var{pp}) of doubles, covariance matrix of the measurement errors (if no measurement errors set H as a zero scalar).
%! @item Z
%! 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}).
%! @item mm
%! Integer scalar, number of state variables.
%! @item pp
%! Integer scalar, number of observed variables.
%! @item rr
%! Integer scalar, number of structural innovations.
%! @item Zflag
%! Integer scalar, equal to 0 if Z is a vector of indices targeting the obseved variables in the state vector, equal to 1 if Z is a @var{pp}*@var{mm} matrix.
%! @item diffuse_periods
%! Integer scalar, number of diffuse filter periods in the initialization step.
%! @end table
%! @sp 2
%! @strong{Outputs}
%! @sp 1
%! @table @ @var
%! @item LIK
%! Double scalar, value of (minus) the likelihood.
%! @item likk
%! Column vector of doubles, values of the density of each observation.
%! @item a
%! Vector (@var{mm}*1) of doubles, mean of the state vector at the end of the (sub)sample.
%! @item P
%! Matrix (@var{mm}*@var{mm}) of doubles, covariance of the state vector at the end of the (sub)sample.
%! @end table
%! @sp 2
%! @strong{This function is called by:}
%! @sp 1
%! @ref{DsgeLikelihood}
%! @sp 2
%! @strong{This function calls:}
%! @sp 1
%! @ref{kalman_filter_ss}
%! @end deftypefn
%@eod:
% Copyright (C) 2004-2013 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/>.
% Set defaults.
ifnargin<17
Zflag=0;
end
ifnargin<18
diffuse_periods=0;
end
ifnargin<19
analytic_derivation=0;
end
ifisempty(Zflag)
Zflag=0;
end
ifisempty(diffuse_periods)
diffuse_periods=0;
end
% Get sample size.
smpl=last-start+1;
% Initialize some variables.
dF=1;
QQ=R*Q*transpose(R);% Variance of R times the vector of structural innovations.
t=start;% Initialization of the time index.
likk=zeros(smpl,1);% Initialization of the vector gathering the densities.
LIK=Inf;% Default value of the log likelihood.
oldK=Inf;
notsteady=1;
F_singular=1;
asy_hess=0;
DLIK=[];
Hess=[];
LIKK=[];
ifZflag
K=T*P*Z';
F=Z*P*Z'+H;
else
K=T*P(:,Z);
F=P(Z,Z)+H;
end
W=K;
iF=inv(F);
Kg=K*iF;
M=-iF;
whilenotsteady&&t<=last
s=t-start+1;
ifZflag
v=Y(:,t)-Z*a;
else
v=Y(:,t)-a(Z);
end
ifrcond(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);
likk(s)=log(dF)+transpose(v)*iF*v;
a=T*a+Kg*v;
ifZflag
ZWM=Z*W*M;
ZWMWp=ZWM*W';
M=M+ZWM'*iF*ZWM;
F=F+ZWMWp*Z';
iF=inv(F);
K=K+T*ZWMWp';
Kg=K*iF;
W=(T-Kg*Z)*W;
else
ZWM=W(Z,:)*M;
ZWMWp=ZWM*W';
M=M+ZWM'*iF*ZWM;
F=F+ZWMWp(:,Z);
iF=inv(F);
K=K+T*ZWMWp';
Kg=K*iF;
W=T*W-Kg*W(Z,:);
end
% notsteady = max(abs(K(:)-oldK))>riccati_tol;
oldK=K(:);
end
t=t+1;
end
ifF_singular
error('The variance of the forecast error remains singular until the end of the sample')
end
% Add observation's densities constants and divide by two.