/usr/share/dynare/matlab/get_Hessian.m is in dynare-common 4.4.1-1build1.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 | function [Hess] = get_Hessian(T,R,Q,H,P,Y,DT,DYss,DOm,DH,DP,D2T,D2Yss,D2Om,D2H,D2P,start,mf,kalman_tol,riccati_tol)
% function [Hess] = get_Hessian(T,R,Q,H,P,Y,DT,DYss,DOm,DH,DP,D2T,D2Yss,D2Om,D2H,D2P,start,mf,kalman_tol,riccati_tol)
%
% computes the 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
% NOTE: the derivative matrices (D2T,D2Om ...) are 4-dim. arrays with last
% two dimensions equal to the number of structural parameters
% Copyright (C) 2011 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;
Hess = zeros(k,k); % Initialization of the Hessian
Da = zeros(mm,k); % State vector.
Dv = zeros(length(mf),k);
D2a = zeros(mm,k,k); % State vector.
D2v = zeros(length(mf),k,k);
C = zeros(length(mf),mm);
for ii=1:length(mf); C(ii,mf(ii))=1;end % SELECTION MATRIX IN MEASUREMENT EQ. (FOR WHEN IT IS NOT CONSTANT)
dC = zeros(length(mf),mm,k);
d2C = zeros(length(mf),mm,k,k);
s = zeros(pp,1); % CONSTANT TERM IN MEASUREMENT EQ. (FOR WHEN IT IS NOT CONSTANT)
ds = zeros(pp,1,k);
d2s = zeros(pp,1,k,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;
[DK,DF,DP1] = computeDKalman(T,DT,DOm,P,DP,DH,mf,iF,K);
[D2K,D2F,D2P1] = computeD2Kalman(T,DT,D2T,D2Om,P,DP,D2P,DH,mf,iF,K,DK);
tmp = (a+K*v);
for ii = 1:k
Dv(:,ii) = -Da(mf,ii) - DYss(mf,ii);
% dai = da(:,:,ii);
dKi = DK(:,:,ii);
diFi = -iF*DF(:,:,ii)*iF;
dtmpi = Da(:,ii)+dKi*v+K*Dv(:,ii);
for jj = 1:ii
dFj = DF(:,:,jj);
diFj = -iF*DF(:,:,jj)*iF;
dKj = DK(:,:,jj);
d2Kij = D2K(:,:,jj,ii);
d2Fij = D2F(:,:,jj,ii);
d2iFij = -diFi*dFj*iF -iF*d2Fij*iF -iF*dFj*diFi;
dtmpj = Da(:,jj)+dKj*v+K*Dv(:,jj);
d2vij = -D2Yss(mf,jj,ii) - D2a(mf,jj,ii);
d2tmpij = D2a(:,jj,ii) + d2Kij*v + dKj*Dv(:,ii) + dKi*Dv(:,jj) + K*d2vij;
D2a(:,jj,ii) = D2T(:,:,jj,ii)*tmp + DT(:,:,jj)*dtmpi + DT(:,:,ii)*dtmpj + T*d2tmpij;
Hesst(ii,jj) = getHesst_ij(v,Dv(:,ii),Dv(:,jj),d2vij,iF,diFi,diFj,d2iFij,dFj,d2Fij);
end
Da(:,ii) = DT(:,:,ii)*tmp + T*dtmpi;
end
% vecDPmf = reshape(DP(mf,mf,:),[],k);
% iPmf = inv(P(mf,mf));
if t>=start
Hess = Hess + Hesst;
end
a = T*(a+K*v);
P = T*(P-K*P(mf,:))*transpose(T)+Om;
DP = DP1;
D2P = D2P1;
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);
tmp = (a+K*v);
for ii = 1:k,
Dv(:,ii) = -Da(mf,ii)-DYss(mf,ii);
dKi = DK(:,:,ii);
diFi = -iF*DF(:,:,ii)*iF;
dtmpi = Da(:,ii)+dKi*v+K*Dv(:,ii);
for jj = 1:ii,
dFj = DF(:,:,jj);
diFj = -iF*DF(:,:,jj)*iF;
dKj = DK(:,:,jj);
d2Kij = D2K(:,:,jj,ii);
d2Fij = D2F(:,:,jj,ii);
d2iFij = -diFi*dFj*iF -iF*d2Fij*iF -iF*dFj*diFi;
dtmpj = Da(:,jj)+dKj*v+K*Dv(:,jj);
d2vij = -D2Yss(mf,jj,ii) - D2a(mf,jj,ii);
d2tmpij = D2a(:,jj,ii) + d2Kij*v + dKj*Dv(:,ii) + dKi*Dv(:,jj) + K*d2vij;
D2a(:,jj,ii) = D2T(:,:,jj,ii)*tmp + DT(:,:,jj)*dtmpi + DT(:,:,ii)*dtmpj + T*d2tmpij;
Hesst(ii,jj) = getHesst_ij(v,Dv(:,ii),Dv(:,jj),d2vij,iF,diFi,diFj,d2iFij,dFj,d2Fij);
end
Da(:,ii) = DT(:,:,ii)*tmp + T*dtmpi;
end
if t>=start
Hess = Hess + Hesst;
end
a = T*(a+K*v);
end
% Hess = Hess + .5*(smpl+t0-1)*(vecDPmf' * kron(iPmf,iPmf) * vecDPmf);
% 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
Hess = Hess + tril(Hess,-1)';
Hess = -Hess/2;
% end of main function
function Hesst_ij = getHesst_ij(e,dei,dej,d2eij,iS,diSi,diSj,d2iSij,dSj,d2Sij);
% computes (i,j) term in the Hessian
Hesst_ij = trace(diSi*dSj + iS*d2Sij) + e'*d2iSij*e + 2*(dei'*diSj*e + dei'*iS*dej + e'*diSi*dej + e'*iS*d2eij);
% end of getHesst_ij
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
function [d2K,d2S,d2P1] = computeD2Kalman(A,dA,d2A,d2Om,P0,dP0,d2P0,DH,mf,iF,K0,dK0);
% computes the second derivatives of the Kalman matrices
% note: A=T in main func.
k = size(dA,3);
tmp = P0-K0*P0(mf,:);
[ns,no] = size(K0);
% CPC = C*P0*C'; CPC = .5*(CPC+CPC');iF = inv(CPC);
% APC = A*P0*C';
% APA = A*P0*A';
d2K = zeros(ns,no,k,k);
d2S = zeros(no,no,k,k);
d2P1 = zeros(ns,ns,k,k);
for ii = 1:k
dAi = dA(:,:,ii);
dFi = dP0(mf,mf,ii);
d2Omi = d2Om(:,:,ii);
diFi = -iF*dFi*iF;
dKi = dK0(:,:,ii);
for jj = 1:k
dAj = dA(:,:,jj);
dFj = dP0(mf,mf,jj);
d2Omj = d2Om(:,:,jj);
dFj = dP0(mf,mf,jj);
diFj = -iF*dFj*iF;
dKj = dK0(:,:,jj);
d2Aij = d2A(:,:,jj,ii);
d2Pij = d2P0(:,:,jj,ii);
d2Omij = d2Om(:,:,jj,ii);
% second order
d2Fij = d2Pij(mf,mf) ;
% d2APC = d2Aij*P0*C' + A*d2Pij*C' + A*P0*d2Cij' + dAi*dPj*C' + dAj*dPi*C' + A*dPj*dCi' + A*dPi*dCj' + dAi*P0*dCj' + dAj*P0*dCi';
d2APC = d2Pij(:,mf);
d2iF = -diFi*dFj*iF -iF*d2Fij*iF -iF*dFj*diFi;
d2Kij= d2Pij(:,mf)*iF + P0(:,mf)*d2iF + dP0(:,mf,jj)*diFi + dP0(:,mf,ii)*diFj;
d2KCP = d2Kij*P0(mf,:) + K0*d2Pij(mf,:) + dKi*dP0(mf,:,jj) + dKj*dP0(mf,:,ii) ;
dtmpi = dP0(:,:,ii) - dK0(:,:,ii)*P0(mf,:) - K0*dP0(mf,:,ii);
dtmpj = dP0(:,:,jj) - dK0(:,:,jj)*P0(mf,:) - K0*dP0(mf,:,jj);
d2tmp = d2Pij - d2KCP;
d2AtmpA = d2Aij*tmp*A' + A*d2tmp*A' + A*tmp*d2Aij' + dAi*dtmpj*A' + dAj*dtmpi*A' + A*dtmpj*dAi' + A*dtmpi*dAj' + dAi*tmp*dAj' + dAj*tmp*dAi';
d2K(:,:,ii,jj) = d2Kij; %#ok<NASGU>
d2P1(:,:,ii,jj) = d2AtmpA + d2Omij; %#ok<*NASGU>
d2S(:,:,ii,jj) = d2Fij;
% d2iS(:,:,ii,jj) = d2iF;
end
end
% end of computeD2Kalman
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