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function [AHess, DLIK, LIK] = AHessian(T,R,Q,H,P,Y,DT,DYss,DOm,DH,DP,start,mf,kalman_tol,riccati_tol)
% function [AHess, DLIK, LIK] = AHessian(T,R,Q,H,P,Y,DT,DYss,DOm,DH,DP,start,mf,kalman_tol,riccati_tol)
%
% computes the asymptotic hessian matrix of the log-likelihood function of
% a state space model (notation as in kalman_filter.m in DYNARE
% Thanks to Nikolai Iskrev
%
% NOTE: the derivative matrices (DT,DR ...) are 3-dim. arrays with last
% dimension equal to the number of structural parameters
% Copyright (C) 2011-2017 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/>.
k = size(DT,3); % number of structural parameters
smpl = size(Y,2); % Sample size.
pp = size(Y,1); % Maximum number of observed variables.
mm = size(T,2); % Number of state variables.
a = zeros(mm,1); % State vector.
Om = R*Q*transpose(R); % Variance of R times the vector of structural innovations.
t = 0; % Initialization of the time index.
oldK = 0;
notsteady = 1; % Steady state flag.
F_singular = 1;
lik = zeros(smpl,1); % Initialization of the vector gathering the densities.
LIK = Inf; % Default value of the log likelihood.
if nargout > 1
DLIK = zeros(k,1); % Initialization of the score.
end
AHess = zeros(k,k); % Initialization of the Hessian
Da = zeros(mm,k); % State vector.
Dv = zeros(length(mf),k);
% for ii = 1:k
% DOm = DR(:,:,ii)*Q*transpose(R) + R*DQ(:,:,ii)*transpose(R) + R*Q*transpose(DR(:,:,ii));
% end
while notsteady && t<smpl
t = t+1;
v = Y(:,t)-a(mf);
F = P(mf,mf) + H;
if rcond(F) < kalman_tol
if ~all(abs(F(:))<kalman_tol)
return
else
a = T*a;
P = T*P*transpose(T)+Om;
end
else
F_singular = 0;
iF = inv(F);
K = P(:,mf)*iF;
lik(t) = log(det(F))+transpose(v)*iF*v;
[DK,DF,DP1] = computeDKalman(T,DT,DOm,P,DP,DH,mf,iF,K);
for ii = 1:k
Dv(:,ii) = -Da(mf,ii) - DYss(mf,ii);
Da(:,ii) = DT(:,:,ii)*(a+K*v) + T*(Da(:,ii)+DK(:,:,ii)*v + K*Dv(:,ii));
if t>=start && nargout > 1
DLIK(ii,1) = DLIK(ii,1) + trace( iF*DF(:,:,ii) ) + 2*Dv(:,ii)'*iF*v - v'*(iF*DF(:,:,ii)*iF)*v;
end
end
vecDPmf = reshape(DP(mf,mf,:),[],k);
% iPmf = inv(P(mf,mf));
if t>=start
AHess = AHess + Dv'*iF*Dv + .5*(vecDPmf' * kron(iF,iF) * vecDPmf);
end
a = T*(a+K*v);
P = T*(P-K*P(mf,:))*transpose(T)+Om;
DP = DP1;
end
notsteady = max(max(abs(K-oldK))) > riccati_tol;
oldK = K;
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);
for ii = 1:k
Dv(:,ii) = -Da(mf,ii)-DYss(mf,ii);
Da(:,ii) = DT(:,:,ii)*(a+K*v) + T*(Da(:,ii)+DK(:,:,ii)*v + K*Dv(:,ii));
if t>=start && nargout >1
DLIK(ii,1) = DLIK(ii,1) + trace( iF*DF(:,:,ii) ) + 2*Dv(:,ii)'*iF*v - v'*(iF*DF(:,:,ii)*iF)*v;
end
end
if t>=start
AHess = AHess + Dv'*iF*Dv;
end
a = T*(a+K*v);
lik(t) = transpose(v)*iF*v;
end
AHess = AHess + .5*(smpl-t0+1)*(vecDPmf' * kron(iF,iF) * vecDPmf);
if nargout > 1
for ii = 1:k
% DLIK(ii,1) = DLIK(ii,1) + (smpl-t0+1)*trace( iF*DF(:,:,ii) );
end
end
lik(t0:smpl) = lik(t0:smpl) + log(det(F));
% for ii = 1:k;
% for jj = 1:ii
% H(ii,jj) = trace(iPmf*(.5*DP(mf,mf,ii)*iPmf*DP(mf,mf,jj) + Dv(:,ii)*Dv(:,jj)'));
% end
% end
end
AHess = -AHess;
if nargout > 1
DLIK = DLIK/2;
end
% adding log-likelihhod constants
lik = (lik + pp*log(2*pi))/2;
LIK = sum(lik(start:end)); % Minus the log-likelihood.
% end of main function
function [DK,DF,DP1] = computeDKalman(T,DT,DOm,P,DP,DH,mf,iF,K)
k = size(DT,3);
tmp = P-K*P(mf,:);
for ii = 1:k
DF(:,:,ii) = DP(mf,mf,ii) + DH(:,:,ii);
DiF(:,:,ii) = -iF*DF(:,:,ii)*iF;
DK(:,:,ii) = DP(:,mf,ii)*iF + P(:,mf)*DiF(:,:,ii);
Dtmp = DP(:,:,ii) - DK(:,:,ii)*P(mf,:) - K*DP(mf,:,ii);
DP1(:,:,ii) = DT(:,:,ii)*tmp*T' + T*Dtmp*T' + T*tmp*DT(:,:,ii)' + DOm(:,:,ii);
end
% end of computeDKalman
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