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dynare/matlab/getH.m

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function [H, dA, dOm, Hss, info] = getH(A, B, M_,oo_,kronflag,indx,indexo)
% computes derivative of reduced form linear model w.r.t. deep params
%
% Copyright (C) 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/>.
if nargin<3 | isempty(kronflag), kronflag = 0; end
if nargin<4 | isempty(indx), indx = [1:M_.param_nbr];, end,
if nargin<5 | isempty(indexo), indexo = [];, end,
[I,J]=find(M_.lead_lag_incidence');
yy0=oo_.dr.ys(I);
% yy0=[];
% for j=1:size(M_.lead_lag_incidence,1);
% yy0 = [ yy0; oo_.dr.ys(find(M_.lead_lag_incidence(j,:)))];
% end
[df, gp] = feval([M_.fname,'_params_derivs'],yy0, oo_.exo_steady_state', M_.params, 1);
[residual, g1 ] = feval([M_.fname,'_dynamic'],yy0, oo_.exo_steady_state', M_.params,1);
[residual, g1, g2 ] = feval([M_.fname,'_dynamic'],yy0, oo_.exo_steady_state', M_.params,1);
[residual, gg1] = feval([M_.fname,'_static'],oo_.dr.ys, oo_.exo_steady_state', M_.params);
% df = feval([M_.fname,'_model_derivs'],yy0, oo_.exo_steady_state', M_.params, 1);
dyssdtheta = -gg1\df;
Hss = dyssdtheta(oo_.dr.order_var,indx);
dyssdtheta = dyssdtheta(I,:);
[nr, nc]=size(g2);
nc = sqrt(nc);
ns = max(max(M_.lead_lag_incidence));
gp2 = gp*0;
for j=1:nr,
[I J]=ind2sub([nc nc],find(g2(j,:)));
for i=1:nc,
is = find(I==i);
is = is(find(J(is)<=ns));
if ~isempty(is),
g20=full(g2(j,find(g2(j,:))));
gp2(j,i,:)=g20(is)*dyssdtheta(J(is),:);
end
end
end
gp = gp+gp2;
gp = gp(:,:,indx);
param_nbr = length(indx);
% order_var = [oo_.dr.order_var; ...
% [size(oo_dr.ghx,2)+1:size(oo_dr.ghx,2)+size(oo_.dr.transition_auxiliary_variables,1)]' ];
% [A(order_var,order_var),B(order_var,:)]=dynare_resolve;
% [A,B,ys,info]=dynare_resolve;
% if info(1) > 0
% H = [];
% A0 = [];
% B0 = [];
% dA = [];
% dOm = [];
% return
% end
m = size(A,1);
n = size(B,2);
klen = M_.maximum_endo_lag + M_.maximum_endo_lead + 1;
k1 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_endo_lag+1),:);
a = g1(:,nonzeros(k1'));
da = gp(:,nonzeros(k1'),:);
kstate = oo_.dr.kstate;
GAM1 = zeros(M_.endo_nbr,M_.endo_nbr);
Dg1 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr);
% nf = find(M_.lead_lag_incidence(M_.maximum_endo_lag+2,:));
% GAM1(:, nf) = -g1(:,M_.lead_lag_incidence(M_.maximum_endo_lag+2,nf));
k = find(kstate(:,2) == M_.maximum_endo_lag+2 & kstate(:,3));
GAM1(:, kstate(k,1)) = -a(:,kstate(k,3));
Dg1(:, kstate(k,1), :) = -da(:,kstate(k,3),:);
k = find(kstate(:,2) > M_.maximum_endo_lag+2 & kstate(:,3));
kk = find(kstate(:,2) > M_.maximum_endo_lag+2 & ~kstate(:,3));
nauxe = 0;
if ~isempty(k),
ax(:,k)= -a(:,kstate(k,3));
ax(:,kk)= 0;
dax(:,k,:)= -da(:,kstate(k,3),:);
dax(:,kk,:)= 0;
nauxe=size(ax,2);
GAM1 = [ax GAM1];
Dg1 = cat(2,dax,Dg1);
clear ax
end
[junk,cols_b,cols_j] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+1, ...
oo_.dr.order_var));
GAM0 = zeros(M_.endo_nbr,M_.endo_nbr);
Dg0 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr);
GAM0(:,cols_b) = g1(:,cols_j);
Dg0(:,cols_b,:) = gp(:,cols_j,:);
% GAM0 = g1(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,:));
k = find(kstate(:,2) == M_.maximum_endo_lag+1 & kstate(:,4));
GAM2 = zeros(M_.endo_nbr,M_.endo_nbr);
Dg2 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr);
% nb = find(M_.lead_lag_incidence(1,:));
% GAM2(:, nb) = -g1(:,M_.lead_lag_incidence(1,nb));
GAM2(:, kstate(k,1)) = -a(:,kstate(k,4));
Dg2(:, kstate(k,1), :) = -da(:,kstate(k,4),:);
naux = 0;
k = find(kstate(:,2) < M_.maximum_endo_lag+1 & kstate(:,4));
kk = find(kstate(:,2) < M_.maximum_endo_lag+1 );
if ~isempty(k),
ax(:,k)= -a(:,kstate(k,4));
ax = ax(:,kk);
dax(:,k,:)= -da(:,kstate(k,4),:);
dax = dax(:,kk,:);
naux = size(ax,2);
GAM2 = [GAM2 ax];
Dg2 = cat(2,Dg2,dax);
end
GAM0 = blkdiag(GAM0,eye(naux));
Dg0 = cat(1,Dg0,zeros(naux+nauxe,M_.endo_nbr,param_nbr));
Dg0 = cat(2,Dg0,zeros(m+nauxe,naux,param_nbr));
Dg0 = cat(2,zeros(m+nauxe,nauxe,param_nbr),Dg0);
GAM2 = [GAM2 ; A(M_.endo_nbr+1:end,:)];
Dg2 = cat(1,Dg2,zeros(naux+nauxe,m,param_nbr));
Dg2 = cat(2,zeros(m+nauxe,nauxe,param_nbr),Dg2);
GAM2 = [zeros(m+nauxe,nauxe) [GAM2; zeros(nauxe,m)]];
GAM0 = [[zeros(m,nauxe);(eye(nauxe))] [GAM0; zeros(nauxe,m)]];
GAM3 = -g1(:,length(yy0)+1:end);
% GAM3 = -g1(oo_.dr.order_var,length(yy0)+1:end);
GAM3 = [GAM3; zeros(naux+nauxe,size(GAM3,2))];
% Dg3 = -gp(oo_.dr.order_var,length(yy0)+1:end,:);
Dg3 = -gp(:,length(yy0)+1:end,:);
Dg3 = cat(1,Dg3,zeros(naux+nauxe,size(GAM3,2),param_nbr));
auxe = zeros(0,1);
k0 = kstate(find(kstate(:,2) >= M_.maximum_endo_lag+2),:);;
i0 = find(k0(:,2) == M_.maximum_endo_lag+2);
for i=M_.maximum_endo_lag+3:M_.maximum_endo_lag+2+M_.maximum_endo_lead,
i1 = find(k0(:,2) == i);
n1 = size(i1,1);
j = zeros(n1,1);
for j1 = 1:n1
j(j1) = find(k0(i0,1)==k0(i1(j1),1));
end
auxe = [auxe; i0(j)];
i0 = i1;
end
auxe = [(1:size(auxe,1))' auxe(end:-1:1)];
n_ir1 = size(auxe,1);
nr = m + n_ir1;
GAM1 = [[GAM1 zeros(size(GAM1,1),naux)]; zeros(naux+nauxe,m+nauxe)];
GAM1(m+1:end,:) = sparse(auxe(:,1),auxe(:,2),ones(n_ir1,1),n_ir1,nr);
Dg1 = cat(2,Dg1,zeros(M_.endo_nbr,naux,param_nbr));
Dg1 = cat(1,Dg1,zeros(naux+nauxe,m+nauxe,param_nbr));
Ax = A;
A1 = A;
Bx = B;
B1 = B;
for j=1:M_.maximum_endo_lead-1,
A1 = A1*A;
B1 = A*B1;
k = find(kstate(:,2) == M_.maximum_endo_lag+2+j );
Ax = [A1(kstate(k,1),:); Ax];
Bx = [B1(kstate(k,1),:); Bx];
end
Ax = [zeros(m+nauxe,nauxe) Ax];
A0 = A;
A=Ax; clear Ax A1;
B0=B;
B = Bx; clear Bx B1;
m = size(A,1);
n = size(B,2);
% Dg1 = zeros(m,m,param_nbr);
% Dg1(:, nf, :) = -gp(:,M_.lead_lag_incidence(3,nf),:);
% Dg0 = gp(:,M_.lead_lag_incidence(2,:),:);
% Dg2 = zeros(m,m,param_nbr);
% Dg2(:, nb, :) = -gp(:,M_.lead_lag_incidence(1,nb),:);
% Dg3 = -gp(:,length(yy0)+1:end,:);
if kronflag==1, % kronecker products
Dg0=reshape(Dg0,m^2,param_nbr);
Dg1=reshape(Dg1,m^2,param_nbr);
Dg2=reshape(Dg2,m^2,param_nbr);
Dg3=reshape(Dg3,m*n,param_nbr);
Om = B*B';
Im = eye(m);
Dm = duplication(m);
DmPl = inv(Dm'*Dm)*Dm';
Kmm = commutation(m,m);
Kmn = commutation(m,n);
Da = [eye(m^2),zeros(m^2,m*(m+1)/2)];
Dom = [zeros(m*(m+1)/2,m^2),eye(m*(m+1)/2)];
Df1Dtau = ( kron(Im,GAM0) - kron(A',GAM1) - kron(Im,GAM1*A) )*Da;
Df1Dthet = kron(A',Im)*Dg0 - kron( (A')^2,Im)*Dg1 - Dg2;
Df2Dtau = DmPl*( kron(GAM0,GAM0) - kron(GAM0,GAM1*A) - kron(GAM1*A,GAM0) + kron(GAM1*A,GAM1*A) )*Dm*Dom - ...
DmPl*( kron(GAM0*Om,GAM1) + kron(GAM1,GAM0*Om)*Kmm - kron(GAM1*A*Om,GAM1) - kron(GAM1,GAM1*A*Om)*Kmm )*Da;
Df2Dthet = DmPl*( kron(GAM0*Om,Im) + kron(Im,GAM0*Om)*Kmm - kron(Im,GAM1*A*Om)*Kmm - kron(GAM1*A*Om,Im) )*Dg0 - ...
DmPl*( kron(GAM0*Om*A',Im) + kron(Im,GAM0*Om*A')*Kmm - kron(Im,GAM1*A*Om*A')*Kmm - kron(GAM1*A*Om*A',Im) )*Dg1 -...
DmPl*( kron(GAM3,Im) + kron(Im,GAM3)*Kmn )*Dg3;
DfDtau = [Df1Dtau;Df2Dtau];
DfDthet = [Df1Dthet;Df2Dthet];
H = -DfDtau\DfDthet;
x = reshape(H(1:m*m,:),m,m,param_nbr);
y = reshape(Dm*H(m*m+1:end,:),m,m,param_nbr);
x = x(nauxe+1:end,nauxe+1:end,:);
y = y(nauxe+1:end,nauxe+1:end,:);
m = size(y,1);
x = reshape(x,m*m,param_nbr);
Dm = duplication(m);
DmPl = inv(Dm'*Dm)*Dm';
y = DmPl*reshape(y,m*m,param_nbr);
H = [x;y];
H = [ [zeros(M_.endo_nbr,length(indexo)) Hss]; H];
elseif kronflag==-1, % perturbation
fun = 'thet2tau';
params0 = M_.params;
H = fdjac(fun,[sqrt(diag(M_.Sigma_e(indexo,indexo))); M_.params(indx)],indx,indexo);
assignin('base','M_', M_);
assignin('base','oo_', oo_);
else % generalized sylvester equation
% solves a*x+b*x*c=d
a = (GAM0-GAM1*A);
inva = inv(a);
b = -GAM1;
c = A;
elem = zeros(m,m,param_nbr);
d = elem;
for j=1:param_nbr,
elem(:,:,j) = (Dg0(:,:,j)-Dg1(:,:,j)*A);
d(:,:,j) = Dg2(:,:,j)-elem(:,:,j)*A;
end
xx=sylvester3mr(a,b,c,d);
if ~isempty(indexo),
dSig = zeros(M_.exo_nbr,M_.exo_nbr);
for j=1:length(indexo)
dSig(indexo(j),indexo(j)) = 2*sqrt(M_.Sigma_e(indexo(j),indexo(j)));
y = B*dSig*B';
y = y(nauxe+1:end,nauxe+1:end);
H(:,j) = [zeros((m-nauxe)^2,1); vech(y)];
if nargout>1,
dOm(:,:,j) = y;
end
dSig(indexo(j),indexo(j)) = 0;
end
end
for j=1:param_nbr,
x = xx(:,:,j);
y = inva * (Dg3(:,:,j)-(elem(:,:,j)-GAM1*x)*B);
y = y*M_.Sigma_e*B'+B*M_.Sigma_e*y';
x = x(nauxe+1:end,nauxe+1:end);
y = y(nauxe+1:end,nauxe+1:end);
if nargout>1,
dA(:,:,j+length(indexo)) = x;
dOm(:,:,j+length(indexo)) = y;
end
H(:,j+length(indexo)) = [x(:); vech(y)];
end
% for j=1:param_nbr,
% disp(['Derivatives w.r.t. ',M_.param_names(indx(j),:),', ',int2str(j),'/',int2str(param_nbr)])
% elem = (Dg0(:,:,j)-Dg1(:,:,j)*A);
% d = Dg2(:,:,j)-elem*A;
% x=sylvester3(a,b,c,d);
% % x=sylvester3a(x,a,b,c,d);
% y = inva * (Dg3(:,:,j)-(elem-GAM1*x)*B);
% y = y*B'+B*y';
% x = x(nauxe+1:end,nauxe+1:end);
% y = y(nauxe+1:end,nauxe+1:end);
% H(:,j) = [x(:); vech(y)];
% end
H = [[zeros(M_.endo_nbr,length(indexo)) Hss]; H];
end
return
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