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Extended for second order derivatives and full Hessian.

time-shift
Marco Ratto 2011-11-05 11:04:08 +01:00
parent 8313b158b6
commit db61c7c144
1 changed files with 274 additions and 107 deletions
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381
matlab/getH.m
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@ -1,4 +1,4 @@
function [H, dA, dOm, Hss, gp, info] = getH(A, B, M_,oo_,kronflag,indx,indexo)
function [H, dA, dOm, Hss, gp, d2A, d2Om, H2ss] = getH(A, B, M_,oo_,kronflag,indx,indexo)
% computes derivative of reduced form linear model w.r.t. deep params
%
@ -31,10 +31,29 @@ yy0=oo_.dr.ys(I);
% yy0 = [ yy0; oo_.dr.ys(find(M_.lead_lag_incidence(j,:)))];
% end
dyssdtheta=zeros(length(oo_.dr.ys),M_.param_nbr);
df = feval([M_.fname,'_params_derivs'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1, dyssdtheta);
[residual, gg1] = feval([M_.fname,'_static'],oo_.dr.ys, oo_.exo_steady_state', M_.params);
if nargout>5,
[residual, gg1, gg2] = feval([M_.fname,'_static'],oo_.dr.ys, oo_.exo_steady_state', M_.params);
[df, gp, d2f] = feval([M_.fname,'_params_derivs'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1, dyssdtheta);
d2f = get_all_resid_2nd_derivs(d2f,length(oo_.dr.ys),M_.param_nbr);
d2f = d2f(:,indx,indx);
if isempty(find(gg2)),
for j=1:length(indx),
d2yssdtheta(:,:,j) = -gg1\d2f(:,:,j);
end
else
gam = d2f*0;
for j=1:length(indx),
d2yssdtheta(:,:,j) = -gg1\(d2f(:,:,j)+gam(:,:,j));
end
end
else
[residual, gg1] = feval([M_.fname,'_static'],oo_.dr.ys, oo_.exo_steady_state', M_.params);
df = feval([M_.fname,'_params_derivs'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1, dyssdtheta);
end
dyssdtheta = -gg1\df;
if any(any(isnan(dyssdtheta))),
[U,T] = schur(gg1);
global options_
@ -45,15 +64,29 @@ if any(any(isnan(dyssdtheta))),
[U,T] = ordschur(U,T,e1);
T = T(k+1:end,k+1:end);
dyssdtheta = -U(:,k+1:end)*(T\U(:,k+1:end)')*df;
if nargout>5,
for j=1:length(indx),
d2yssdtheta(:,:,j) = -U(:,k+1:end)*(T\U(:,k+1:end)')*d2f(:,:,j);
end
end
end
if nargout>5,
[df, gp, d2f, gpp] = feval([M_.fname,'_params_derivs'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1, dyssdtheta);
H2ss = d2yssdtheta(oo_.dr.order_var,:,:);
[residual, g1, g2, g3] = feval([M_.fname,'_dynamic'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1);
else
[df, gp] = feval([M_.fname,'_params_derivs'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1, dyssdtheta);
[residual, g1, g2 ] = feval([M_.fname,'_dynamic'],yy0, oo_.exo_steady_state', ...
M_.params, oo_.dr.ys, 1);
end
Hss = dyssdtheta(oo_.dr.order_var,indx);
dyssdtheta = dyssdtheta(I,:);
[nr, nc]=size(g2);
[m, nelem]=size(g1);
nc = sqrt(nc);
ns = max(max(M_.lead_lag_incidence));
gp2 = gp*0;
@ -73,19 +106,9 @@ 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
if nargout>5,
param_nbr_2 = param_nbr*(param_nbr+1)/2;
end
m = size(A,1);
n = size(B,2);
@ -93,9 +116,15 @@ 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'),:);
k11 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_endo_lag+1),:);
a = g1(:,nonzeros(k11'));
da = gp(:,nonzeros(k11'),:);
if nargout > 5,
nelem = size(g1,2);
g22 = get_all_2nd_derivs(gpp,m,nelem,M_.param_nbr);
g22 = g22(:,:,indx,indx);
d2a = g22(:,nonzeros(k11'),:,:);
end
kstate = oo_.dr.kstate;
GAM1 = zeros(M_.endo_nbr,M_.endo_nbr);
@ -103,22 +132,26 @@ 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
k1 = find(kstate(:,2) == M_.maximum_endo_lag+2 & kstate(:,3));
GAM1(:, kstate(k1,1)) = -a(:,kstate(k1,3));
Dg1(:, kstate(k1,1), :) = -da(:,kstate(k1,3),:);
if nargout > 5,
D2g1 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr,param_nbr);
D2g1(:, kstate(k1,1), :, :) = -d2a(:,kstate(k1,3),:,:);
end
% 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, ...
@ -127,84 +160,95 @@ 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,:);
if nargout > 5,
D2g0 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr,param_nbr);
D2g0(:, cols_b, :, :) = g22(:,cols_j,:,:);
end
% GAM0 = g1(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,:));
k = find(kstate(:,2) == M_.maximum_endo_lag+1 & kstate(:,4));
k2 = 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)]];
GAM2(:, kstate(k2,1)) = -a(:,kstate(k2,4));
Dg2(:, kstate(k2,1), :) = -da(:,kstate(k2,4),:);
if nargout > 5,
D2g2 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr,param_nbr);
D2g2(:, kstate(k2,1), :, :) = -d2a(:,kstate(k2,4),:,:);
end% 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))];
% 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;
if nargout>5,
D2g3 = -g22(:,length(yy0)+1:end,:,:);
clear g22 d2a
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));
% Dg3 = cat(1,Dg3,zeros(naux+nauxe,size(GAM3,2),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;
% 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);
@ -312,25 +356,35 @@ else % generalized sylvester equation
d(:,:,j) = Dg2(:,:,j)-elem(:,:,j)*A;
end
xx=sylvester3mr(a,b,c,d);
H=zeros(m*m+m*(m+1)/2,param_nbr+length(indexo));
if nargout>1,
dOm = zeros(m,m,param_nbr+length(indexo));
dA=zeros(m,m,param_nbr+length(indexo));
dB=zeros(m,n,param_nbr);
end
if ~isempty(indexo),
dSig = zeros(M_.exo_nbr,M_.exo_nbr);
dSig = zeros(M_.exo_nbr,M_.exo_nbr,length(indexo));
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); dyn_vech(y)];
dSig(indexo(j),indexo(j),j) = 2*sqrt(M_.Sigma_e(indexo(j),indexo(j)));
y = B*dSig(:,:,j)*B';
% y = y(nauxe+1:end,nauxe+1:end);
% H(:,j) = [zeros((m-nauxe)^2,1); dyn_vech(y)];
H(:,j) = [zeros(m^2,1); dyn_vech(y)];
if nargout>1,
dOm(:,:,j) = y;
end
dSig(indexo(j),indexo(j)) = 0;
% 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);
if nargout>1,
dB(:,:,j) = y;
end
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);
% 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;
@ -353,7 +407,120 @@ else % generalized sylvester equation
end
if nargout > 5,
tot_param_nbr = param_nbr + length(indexo);
tot_param_nbr_2 = tot_param_nbr*(tot_param_nbr+1)/2;
d = zeros(m,m,param_nbr_2);
d2A = zeros(m,m,tot_param_nbr,tot_param_nbr);
d2Om = zeros(m,m,tot_param_nbr,tot_param_nbr);
d2B = zeros(m,n,tot_param_nbr,tot_param_nbr);
cc=triu(ones(param_nbr,param_nbr));
[i2,j2]=find(cc);
cc = blkdiag( zeros(length(indexo),length(indexo)), cc);
[ipar2]=find(cc);
ctot=triu(ones(tot_param_nbr,tot_param_nbr));
ctot(1:length(indexo),1:length(indexo))=eye(length(indexo));
[itot2, jtot2]=find(ctot);
jcount=0;
for j=1:param_nbr,
for i=j:param_nbr,
elem1 = (D2g0(:,:,j,i)-D2g1(:,:,j,i)*A);
elem1 = D2g2(:,:,j,i)-elem1*A;
elemj0 = Dg0(:,:,j)-Dg1(:,:,j)*A;
elemi0 = Dg0(:,:,i)-Dg1(:,:,i)*A;
elem2 = -elemj0*xx(:,:,i)-elemi0*xx(:,:,j);
elem2 = elem2 + ( Dg1(:,:,j)*xx(:,:,i) + Dg1(:,:,i)*xx(:,:,j) )*A;
elem2 = elem2 + GAM1*( xx(:,:,i)*xx(:,:,j) + xx(:,:,j)*xx(:,:,i));
jcount=jcount+1;
d(:,:,jcount) = elem1+elem2;
end
end
xx2=sylvester3mr(a,b,c,d);
jcount = 0;
for j=1:param_nbr,
for i=j:param_nbr,
jcount=jcount+1;
x = xx2(:,:,jcount);
elem1 = (D2g0(:,:,j,i)-D2g1(:,:,j,i)*A);
elem1 = elem1 -( Dg1(:,:,j)*xx(:,:,i) + Dg1(:,:,i)*xx(:,:,j) );
elemj0 = Dg0(:,:,j)-Dg1(:,:,j)*A-GAM1*xx(:,:,j);
elemi0 = Dg0(:,:,i)-Dg1(:,:,i)*A-GAM1*xx(:,:,i);
elem0 = elemj0*dB(:,:,i)+elemi0*dB(:,:,j);
y = inva * (D2g3(:,:,j,i)-elem0-(elem1-GAM1*x)*B);
d2B(:,:,j+length(indexo),i+length(indexo)) = y;
d2B(:,:,i+length(indexo),j+length(indexo)) = y;
y = y*M_.Sigma_e*B'+B*M_.Sigma_e*y'+ ...
dB(:,:,j)*M_.Sigma_e*dB(:,:,i)'+dB(:,:,i)*M_.Sigma_e*dB(:,:,j)';
% x = x(nauxe+1:end,nauxe+1:end);
% y = y(nauxe+1:end,nauxe+1:end);
d2A(:,:,j+length(indexo),i+length(indexo)) = x;
d2A(:,:,i+length(indexo),j+length(indexo)) = x;
d2Om(:,:,j+length(indexo),i+length(indexo)) = y;
d2Om(:,:,i+length(indexo),j+length(indexo)) = y;
end
end
if ~isempty(indexo),
d2Sig = zeros(M_.exo_nbr,M_.exo_nbr,length(indexo));
for j=1:length(indexo)
d2Sig(indexo(j),indexo(j),j) = 2;
y = B*d2Sig(:,:,j)*B';
d2Om(:,:,j,j) = y;
% y = y(nauxe+1:end,nauxe+1:end);
% H(:,j) = [zeros((m-nauxe)^2,1); dyn_vech(y)];
% H(:,j) = [zeros(m^2,1); dyn_vech(y)];
% dOm(:,:,j) = y;
for i=1:param_nbr,
y = dB(:,:,i)*dSig(:,:,j)*B'+B*dSig(:,:,j)*dB(:,:,i)';
d2Om(:,:,j,i+length(indexo)) = y;
d2Om(:,:,i+length(indexo),j) = y;
end
end
end
end
return
function g22 = get_2nd_deriv(gpp,m,n,i,j),
g22=zeros(m,n);
is=find(gpp(:,3)==i);
is=is(find(gpp(is,4)==j));
if ~isempty(is),
g22(gpp(is,1),gpp(is,2))=gpp(is,5);
end
return
function g22 = get_all_2nd_derivs(gpp,m,n,npar),
g22=zeros(m,n,npar,npar);
% c=ones(npar,npar);
% c=triu(c);
% ic=find(c);
for is=1:length(gpp),
% d=zeros(npar,npar);
% d(gpp(is,3),gpp(is,4))=1;
% indx = find(ic==find(d));
g22(gpp(is,1),gpp(is,2),gpp(is,3),gpp(is,4))=gpp(is,5);
g22(gpp(is,1),gpp(is,2),gpp(is,4),gpp(is,3))=gpp(is,5);
end
return
function r22 = get_all_resid_2nd_derivs(rpp,m,npar),
r22=zeros(m,npar,npar);
% c=ones(npar,npar);
% c=triu(c);
% ic=find(c);
for is=1:length(rpp),
% d=zeros(npar,npar);
% d(rpp(is,2),rpp(is,3))=1;
% indx = find(ic==find(d));
r22(rpp(is,1),rpp(is,2),rpp(is,3))=rpp(is,4);
r22(rpp(is,1),rpp(is,3),rpp(is,2))=rpp(is,4);
end
return