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dynare/matlab/dr1.m

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function [dr,info,M_,options_,oo_] = dr1(dr,task,M_,options_,oo_)
% function [dr,info,M_,options_,oo_] = dr1(dr,task,M_,options_,oo_)
% computes the reduced form solution of a rational expectation model (first or second order
% approximation of the stochastic model around the deterministic steady state).
%
% INPUTS
% dr [matlab structure] Decision rules for stochastic simulations.
% task [integer] if task = 0 then dr1 computes decision rules.
% if task = 1 then dr1 computes eigenvalues.
% M_ [matlab structure] Definition of the model.
% options_ [matlab structure] Global options.
% oo_ [matlab structure] Results
%
% OUTPUTS
% dr [matlab structure] Decision rules for stochastic simulations.
% info [integer] info=1: the model doesn't define current variables uniquely
% info=2: problem in mjdgges.dll info(2) contains error code.
% info=3: BK order condition not satisfied info(2) contains "distance"
% absence of stable trajectory.
% info=4: BK order condition not satisfied info(2) contains "distance"
% indeterminacy.
% info=5: BK rank condition not satisfied.
% info=6: The jacobian matrix evaluated at the steady state is complex.
% M_ [matlab structure]
% options_ [matlab structure]
% oo_ [matlab structure]
%
% ALGORITHM
% ...
%
% SPECIAL REQUIREMENTS
% none.
%
% Copyright (C) 1996-2011 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/>.
info = 0;
if M_.maximum_endo_lag == 0 && options_.order > 1
error(['2nd and 3rd order approximation not implemented for purely forward models'])
end
if options_.k_order_solver;
dr = set_state_space(dr,M_);
[dr,info] = k_order_pert(dr,M_,options_,oo_);
oo_.dr = dr;
return;
end
xlen = M_.maximum_endo_lead + M_.maximum_endo_lag + 1;
klen = M_.maximum_endo_lag + M_.maximum_endo_lead + 1;
iyv = M_.lead_lag_incidence';
iyv = iyv(:);
iyr0 = find(iyv) ;
it_ = M_.maximum_lag + 1 ;
if M_.exo_nbr == 0
oo_.exo_steady_state = [] ;
end
klen = M_.maximum_lag + M_.maximum_lead + 1;
iyv = M_.lead_lag_incidence';
iyv = iyv(:);
iyr0 = find(iyv) ;
it_ = M_.maximum_lag + 1 ;
if M_.exo_nbr == 0
oo_.exo_steady_state = [] ;
end
it_ = M_.maximum_lag + 1;
z = repmat(dr.ys,1,klen);
if ~options_.bytecode
z = z(iyr0) ;
end;
exo_ss = [oo_.exo_steady_state' oo_.exo_det_steady_state'];
if options_.order == 1
if (options_.bytecode)
[chck, junk, loc_dr] = bytecode('dynamic','evaluate', z,exo_ss, ...
M_.params, dr.ys, 1);
jacobia_ = [loc_dr.g1 loc_dr.g1_x loc_dr.g1_xd];
else
[junk,jacobia_] = feval([M_.fname '_dynamic'],z,exo_ss, ...
M_.params, dr.ys, 1);
end;
elseif options_.order == 2
if (options_.bytecode)
[chck, junk, loc_dr] = bytecode('dynamic','evaluate', z,exo_ss, ...
M_.params, dr.ys, 1);
jacobia_ = [loc_dr.g1 loc_dr.g1_x];
else
[junk,jacobia_,hessian1] = feval([M_.fname '_dynamic'],z,...
exo_ss, ...
M_.params, dr.ys, 1);
end;
if options_.use_dll
% In USE_DLL mode, the hessian is in the 3-column sparse representation
hessian1 = sparse(hessian1(:,1), hessian1(:,2), hessian1(:,3), ...
size(jacobia_, 1), size(jacobia_, 2)*size(jacobia_, 2));
end
end
if options_.debug
save([M_.fname '_debug.mat'],'jacobia_')
end
if ~all(isfinite(jacobia_(:)))
info(1) = 6;
info(2) = 1;
return
elseif ~isreal(jacobia_)
if max(max(abs(imag(jacobia_)))) < 1e-15
jacobia_ = real(jacobia_);
else
info(1) = 6;
info(2) = sum(sum(imag(jacobia_).^2));
return
end
end
dr=set_state_space(dr,M_);
kstate = dr.kstate;
kad = dr.kad;
kae = dr.kae;
nstatic = dr.nstatic;
nfwrd = dr.nfwrd;
npred = dr.npred;
nboth = dr.nboth;
order_var = dr.order_var;
nd = size(kstate,1);
nz = nnz(M_.lead_lag_incidence);
sdyn = M_.endo_nbr - nstatic;
[junk,cols_b,cols_j] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+1, ...
order_var));
b = zeros(M_.endo_nbr,M_.endo_nbr);
b(:,cols_b) = jacobia_(:,cols_j);
if M_.maximum_endo_lead == 0
% backward models: simplified code exist only at order == 1
% If required, use AIM solver if not check only
if options_.order > 1
error(['2nd and 3rd order approximation not implemented for purely ' ...
'backward models'])
end
if (options_.aim_solver == 1) && (task == 0)
if options_.order > 1
error('Option "aim_solver" is incompatible with order >= 2')
end
try
[dr,aimcode]=dynAIMsolver1(jacobia_,M_,dr);
if aimcode ~=1
info(1) = convertAimCodeToInfo(aimcode);
info(2) = 1.0e+8;
return
end
catch
disp(lasterror.message)
error('Problem with AIM solver - Try to remove the "aim_solver" option');
end
else % use original Dynare solver
[k1,junk,k2] = find(kstate(:,4));
dr.ghx(:,k1) = -b\jacobia_(:,k2);
% with simul, the Jacobian doesn't contain derivatives w.r. to shocks
if size(jacobia_,2) > nz
dr.ghu = -b\jacobia_(:,nz+1:end);
end
end % if not use AIM or not...
dr.eigval = eig(transition_matrix(dr));
dr.rank = 0;
if any(abs(dr.eigval) > options_.qz_criterium)
temp = sort(abs(dr.eigval));
nba = nnz(abs(dr.eigval) > options_.qz_criterium);
temp = temp(nd-nba+1:nd)-1-options_.qz_criterium;
info(1) = 3;
info(2) = temp'*temp;
end
if options_.loglinear == 1
klags = find(M_.lead_lag_incidence(1,:));
dr.ghx = repmat(1./dr.ys,1,size(dr.ghx,2)).*dr.ghx.* ...
repmat(dr.ys(klags),size(dr.ghx,1),1);
dr.ghu = repmat(1./dr.ys,1,size(dr.ghu,2)).*dr.ghu;
end
return
end
%forward--looking models
if nstatic > 0
[Q,R] = qr(b(:,1:nstatic));
aa = Q'*jacobia_;
else
aa = jacobia_;
end
% If required, use AIM solver if not check only
if (options_.aim_solver == 1) && (task == 0)
if options_.order > 1
error('Option "aim_solver" is incompatible with order >= 2')
end
try
[dr,aimcode]=dynAIMsolver1(aa,M_,dr);
% reuse some of the bypassed code and tests that may be needed
if aimcode ~=1
info(1) = convertAimCodeToInfo(aimcode);
info(2) = 1.0e+8;
return
end
[A,B] =transition_matrix(dr);
dr.eigval = eig(A);
sdim = sum( abs(dr.eigval) < options_.qz_criterium );
nba = nd-sdim;
nyf = sum(kstate(:,2) > M_.maximum_endo_lag+1);
if nba ~= nyf
temp = sort(abs(dr.eigval));
if nba > nyf
temp = temp(nd-nba+1:nd-nyf)-1-options_.qz_criterium;
info(1) = 3;
elseif nba < nyf;
temp = temp(nd-nyf+1:nd-nba)-1-options_.qz_criterium;
info(1) = 4;
end
info(2) = temp'*temp;
return
end
catch
disp(lasterror.message)
error('Problem with AIM solver - Try to remove the "aim_solver" option')
end
else % use original Dynare solver
k1 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_endo_lag+1),:);
a = aa(:,nonzeros(k1'));
b(:,cols_b) = aa(:,cols_j);
b10 = b(1:nstatic,1:nstatic);
b11 = b(1:nstatic,nstatic+1:end);
b2 = b(nstatic+1:end,nstatic+1:end);
% buildind D and E
d = zeros(nd,nd) ;
e = d ;
k = find(kstate(:,2) >= M_.maximum_endo_lag+2 & kstate(:,3));
d(1:sdyn,k) = a(nstatic+1:end,kstate(k,3)) ;
k1 = find(kstate(:,2) == M_.maximum_endo_lag+2);
e(1:sdyn,k1) = -b2(:,kstate(k1,1)-nstatic);
k = find(kstate(:,2) <= M_.maximum_endo_lag+1 & kstate(:,4));
e(1:sdyn,k) = -a(nstatic+1:end,kstate(k,4)) ;
k2 = find(kstate(:,2) == M_.maximum_endo_lag+1);
k2 = k2(~ismember(kstate(k2,1),kstate(k1,1)));
d(1:sdyn,k2) = b2(:,kstate(k2,1)-nstatic);
if ~isempty(kad)
for j = 1:size(kad,1)
d(sdyn+j,kad(j)) = 1 ;
e(sdyn+j,kae(j)) = 1 ;
end
end
% 1) if mjdgges.dll (or .mexw32 or ....) doesn't exit,
% matlab/qz is added to the path. There exists now qz/mjdgges.m that
% contains the calls to the old Sims code
% 2) In global_initialization.m, if mjdgges.m is visible exist(...)==2,
% this means that the DLL isn't avaiable and use_qzdiv is set to 1
if isempty(options_.qz_criterium)
error('I cannot solve the model because qz_criterium option is empty!')
end
[err,ss,tt,w,sdim,dr.eigval,info1] = mjdgges(e,d,options_.qz_criterium);
mexErrCheck('mjdgges', err);
if info1
info(1) = 2;
info(2) = info1;
info(3) = size(e,2);
return
end
nba = nd-sdim;
nyf = sum(kstate(:,2) > M_.maximum_endo_lag+1);
if task == 1
dr.rank = rank(w(1:nyf,nd-nyf+1:end));
% Under Octave, eig(A,B) doesn't exist, and
% lambda = qz(A,B) won't return infinite eigenvalues
if ~exist('OCTAVE_VERSION')
dr.eigval = eig(e,d);
end
for i=1:nd
if abs(ss(i,i)) < 1e-6 && abs(tt(i,i)) < 1e-6
info(1) = 7;
end
end
return
end
if nba ~= nyf
sorted_roots = sort(abs(dr.eigval));
if isfield(options_,'indeterminacy_continuity')
if options_.indeterminacy_msv == 1
[ss,tt,w,q] = qz(e',d');
[ss,tt,w,q] = reorder(ss,tt,w,q);
ss = ss';
tt = tt';
w = w';
nba = nyf;
end
else
if nba > nyf
temp = sorted_roots(nd-nba+1:nd-nyf)-1-options_.qz_criterium;
info(1) = 3;
elseif nba < nyf;
temp = sorted_roots(nd-nyf+1:nd-nba)-1-options_.qz_criterium;
info(1) = 4;
end
info(2) = temp'*temp;
return
end
end
np = nd - nyf;
n2 = np + 1;
n3 = nyf;
n4 = n3 + 1;
% derivatives with respect to dynamic state variables
% forward variables
w1 =w(1:n3,n2:nd);
if ~isscalar(w1) && (condest(w1) > 1e9)
% condest() fails on a scalar under Octave
info(1) = 5;
info(2) = condest(w1);
return;
else
gx = -w1'\w(n4:nd,n2:nd)';
end
% predetermined variables
hx = w(1:n3,1:np)'*gx+w(n4:nd,1:np)';
hx = (tt(1:np,1:np)*hx)\(ss(1:np,1:np)*hx);
k1 = find(kstate(n4:nd,2) == M_.maximum_endo_lag+1);
k2 = find(kstate(1:n3,2) == M_.maximum_endo_lag+2);
dr.ghx = [hx(k1,:); gx(k2(nboth+1:end),:)];
%lead variables actually present in the model
j3 = nonzeros(kstate(:,3));
j4 = find(kstate(:,3));
% derivatives with respect to exogenous variables
if M_.exo_nbr
fu = aa(:,nz+(1:M_.exo_nbr));
a1 = b;
aa1 = [];
if nstatic > 0
aa1 = a1(:,1:nstatic);
end
dr.ghu = -[aa1 a(:,j3)*gx(j4,1:npred)+a1(:,nstatic+1:nstatic+ ...
npred) a1(:,nstatic+npred+1:end)]\fu;
else
dr.ghu = [];
end
% static variables
if nstatic > 0
temp = -a(1:nstatic,j3)*gx(j4,:)*hx;
j5 = find(kstate(n4:nd,4));
temp(:,j5) = temp(:,j5)-a(1:nstatic,nonzeros(kstate(:,4)));
temp = b10\(temp-b11*dr.ghx);
dr.ghx = [temp; dr.ghx];
temp = [];
end
end % if not use AIM and ....
% End of if... and if not... main AIM Blocks, continue as per usual...
if options_.loglinear == 1
k = find(dr.kstate(:,2) <= M_.maximum_endo_lag+1);
klag = dr.kstate(k,[1 2]);
k1 = dr.order_var;
dr.ghx = repmat(1./dr.ys(k1),1,size(dr.ghx,2)).*dr.ghx.* ...
repmat(dr.ys(k1(klag(:,1)))',size(dr.ghx,1),1);
dr.ghu = repmat(1./dr.ys(k1),1,size(dr.ghu,2)).*dr.ghu;
end
if options_.aim_solver ~= 1 && options_.use_qzdiv
%% Necessary when using Sims' routines for QZ
gx = real(gx);
hx = real(hx);
dr.ghx = real(dr.ghx);
dr.ghu = real(dr.ghu);
end
%exogenous deterministic variables
if M_.exo_det_nbr > 0
f1 = sparse(jacobia_(:,nonzeros(M_.lead_lag_incidence(M_.maximum_endo_lag+2:end,order_var))));
f0 = sparse(jacobia_(:,nonzeros(M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var))));
fudet = sparse(jacobia_(:,nz+M_.exo_nbr+1:end));
M1 = inv(f0+[zeros(M_.endo_nbr,nstatic) f1*gx zeros(M_.endo_nbr,nyf-nboth)]);
M2 = M1*f1;
dr.ghud = cell(M_.exo_det_length,1);
dr.ghud{1} = -M1*fudet;
for i = 2:M_.exo_det_length
dr.ghud{i} = -M2*dr.ghud{i-1}(end-nyf+1:end,:);
end
end
if options_.order == 1
return
end
% Second order
%tempex = oo_.exo_simul ;
%hessian = real(hessext('ff1_',[z; oo_.exo_steady_state]))' ;
kk = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1));
if M_.maximum_endo_lag > 0
kk = [cumsum(M_.lead_lag_incidence(1:M_.maximum_endo_lag,order_var),1); kk];
end
kk = kk';
kk = find(kk(:));
nk = size(kk,1) + M_.exo_nbr + M_.exo_det_nbr;
k1 = M_.lead_lag_incidence(:,order_var);
k1 = k1';
k1 = k1(:);
k1 = k1(kk);
k2 = find(k1);
kk1(k1(k2)) = k2;
kk1 = [kk1 length(k1)+1:length(k1)+M_.exo_nbr+M_.exo_det_nbr];
kk = reshape([1:nk^2],nk,nk);
kk1 = kk(kk1,kk1);
%[junk,junk,hessian] = feval([M_.fname '_dynamic'],z, oo_.exo_steady_state);
hessian(:,kk1(:)) = hessian1;
clear hessian1
%oo_.exo_simul = tempex ;
%clear tempex
n1 = 0;
n2 = np;
zx = zeros(np,np);
zu=zeros(np,M_.exo_nbr);
for i=2:M_.maximum_endo_lag+1
k1 = sum(kstate(:,2) == i);
zx(n1+1:n1+k1,n2-k1+1:n2)=eye(k1);
n1 = n1+k1;
n2 = n2-k1;
end
kk = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1));
k0 = [1:M_.endo_nbr];
gx1 = dr.ghx;
hu = dr.ghu(nstatic+[1:npred],:);
zx = [zx; gx1];
zu = [zu; dr.ghu];
for i=1:M_.maximum_endo_lead
k1 = find(kk(i+1,k0) > 0);
zu = [zu; gx1(k1,1:npred)*hu];
gx1 = gx1(k1,:)*hx;
zx = [zx; gx1];
kk = kk(:,k0);
k0 = k1;
end
zx=[zx; zeros(M_.exo_nbr,np);zeros(M_.exo_det_nbr,np)];
zu=[zu; eye(M_.exo_nbr);zeros(M_.exo_det_nbr,M_.exo_nbr)];
[nrzx,nczx] = size(zx);
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zx,options_.threads.kronecker.sparse_hessian_times_B_kronecker_C);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
rhs = -rhs;
%lhs
n = M_.endo_nbr+sum(kstate(:,2) > M_.maximum_endo_lag+1 & kstate(:,2) < M_.maximum_endo_lag+M_.maximum_endo_lead+1);
A = zeros(n,n);
B = zeros(n,n);
A(1:M_.endo_nbr,1:M_.endo_nbr) = jacobia_(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var));
% variables with the highest lead
k1 = find(kstate(:,2) == M_.maximum_endo_lag+M_.maximum_endo_lead+1);
if M_.maximum_endo_lead > 1
k2 = find(kstate(:,2) == M_.maximum_endo_lag+M_.maximum_endo_lead);
[junk,junk,k3] = intersect(kstate(k1,1),kstate(k2,1));
else
k2 = [1:M_.endo_nbr];
k3 = kstate(k1,1);
end
% Jacobian with respect to the variables with the highest lead
B(1:M_.endo_nbr,end-length(k2)+k3) = jacobia_(:,kstate(k1,3)+M_.endo_nbr);
offset = M_.endo_nbr;
k0 = [1:M_.endo_nbr];
gx1 = dr.ghx;
for i=1:M_.maximum_endo_lead-1
k1 = find(kstate(:,2) == M_.maximum_endo_lag+i+1);
[k2,junk,k3] = find(kstate(k1,3));
A(1:M_.endo_nbr,offset+k2) = jacobia_(:,k3+M_.endo_nbr);
n1 = length(k1);
A(offset+[1:n1],nstatic+[1:npred]) = -gx1(kstate(k1,1),1:npred);
gx1 = gx1*hx;
A(offset+[1:n1],offset+[1:n1]) = eye(n1);
n0 = length(k0);
E = eye(n0);
if i == 1
[junk,junk,k4]=intersect(kstate(k1,1),[1:M_.endo_nbr]);
else
[junk,junk,k4]=intersect(kstate(k1,1),kstate(k0,1));
end
i1 = offset-n0+n1;
B(offset+[1:n1],offset-n0+[1:n0]) = -E(k4,:);
k0 = k1;
offset = offset + n1;
end
[junk,k1,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+M_.maximum_endo_lead+1,order_var));
A(1:M_.endo_nbr,nstatic+1:nstatic+npred)=...
A(1:M_.endo_nbr,nstatic+[1:npred])+jacobia_(:,k2)*gx1(k1,1:npred);
C = hx;
D = [rhs; zeros(n-M_.endo_nbr,size(rhs,2))];
[err, dr.ghxx] = gensylv(2,A,B,C,D);
mexErrCheck('gensylv', err);
%ghxu
%rhs
hu = dr.ghu(nstatic+1:nstatic+npred,:);
%kk = reshape([1:np*np],np,np);
%kk = kk(1:npred,1:npred);
%rhs = -hessian*kron(zx,zu)-f1*dr.ghxx(end-nyf+1:end,kk(:))*kron(hx(1:npred,:),hu(1:npred,:));
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zx,zu,options_.threads.kronecker.sparse_hessian_times_B_kronecker_C);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
nyf1 = sum(kstate(:,2) == M_.maximum_endo_lag+2);
hu1 = [hu;zeros(np-npred,M_.exo_nbr)];
%B1 = [B(1:M_.endo_nbr,:);zeros(size(A,1)-M_.endo_nbr,size(B,2))];
[nrhx,nchx] = size(hx);
[nrhu1,nchu1] = size(hu1);
[abcOut,err] = A_times_B_kronecker_C(dr.ghxx,hx,hu1,options_.threads.kronecker.A_times_B_kronecker_C);
mexErrCheck('A_times_B_kronecker_C', err);
B1 = B*abcOut;
rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1;
%lhs
dr.ghxu = A\rhs;
%ghuu
%rhs
kk = reshape([1:np*np],np,np);
kk = kk(1:npred,1:npred);
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zu,options_.threads.kronecker.sparse_hessian_times_B_kronecker_C);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
[B1, err] = A_times_B_kronecker_C(B*dr.ghxx,hu1,options_.threads.kronecker.A_times_B_kronecker_C);
mexErrCheck('A_times_B_kronecker_C', err);
rhs = -[rhs; zeros(n-M_.endo_nbr,size(rhs,2))]-B1;
%lhs
dr.ghuu = A\rhs;
dr.ghxx = dr.ghxx(1:M_.endo_nbr,:);
dr.ghxu = dr.ghxu(1:M_.endo_nbr,:);
dr.ghuu = dr.ghuu(1:M_.endo_nbr,:);
% dr.ghs2
% derivatives of F with respect to forward variables
% reordering predetermined variables in diminishing lag order
O1 = zeros(M_.endo_nbr,nstatic);
O2 = zeros(M_.endo_nbr,M_.endo_nbr-nstatic-npred);
LHS = jacobia_(:,M_.lead_lag_incidence(M_.maximum_endo_lag+1,order_var));
RHS = zeros(M_.endo_nbr,M_.exo_nbr^2);
kk = find(kstate(:,2) == M_.maximum_endo_lag+2);
gu = dr.ghu;
guu = dr.ghuu;
Gu = [dr.ghu(nstatic+[1:npred],:); zeros(np-npred,M_.exo_nbr)];
Guu = [dr.ghuu(nstatic+[1:npred],:); zeros(np-npred,M_.exo_nbr*M_.exo_nbr)];
E = eye(M_.endo_nbr);
M_.lead_lag_incidenceordered = flipud(cumsum(flipud(M_.lead_lag_incidence(M_.maximum_endo_lag+1:end,order_var)),1));
if M_.maximum_endo_lag > 0
M_.lead_lag_incidenceordered = [cumsum(M_.lead_lag_incidence(1:M_.maximum_endo_lag,order_var),1); M_.lead_lag_incidenceordered];
end
M_.lead_lag_incidenceordered = M_.lead_lag_incidenceordered';
M_.lead_lag_incidenceordered = M_.lead_lag_incidenceordered(:);
k1 = find(M_.lead_lag_incidenceordered);
M_.lead_lag_incidenceordered(k1) = [1:length(k1)]';
M_.lead_lag_incidenceordered =reshape(M_.lead_lag_incidenceordered,M_.endo_nbr,M_.maximum_endo_lag+M_.maximum_endo_lead+1)';
kh = reshape([1:nk^2],nk,nk);
kp = sum(kstate(:,2) <= M_.maximum_endo_lag+1);
E1 = [eye(npred); zeros(kp-npred,npred)];
H = E1;
hxx = dr.ghxx(nstatic+[1:npred],:);
for i=1:M_.maximum_endo_lead
for j=i:M_.maximum_endo_lead
[junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+j+1,order_var));
[junk,k3a,k3] = ...
find(M_.lead_lag_incidenceordered(M_.maximum_endo_lag+j+1,:));
nk3a = length(k3a);
[B1, err] = sparse_hessian_times_B_kronecker_C(hessian(:,kh(k3,k3)),gu(k3a,:),options_.threads.kronecker.sparse_hessian_times_B_kronecker_C);
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
RHS = RHS + jacobia_(:,k2)*guu(k2a,:)+B1;
end
% LHS
[junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+i+1,order_var));
LHS = LHS + jacobia_(:,k2)*(E(k2a,:)+[O1(k2a,:) dr.ghx(k2a,:)*H O2(k2a,:)]);
if i == M_.maximum_endo_lead
break
end
kk = find(kstate(:,2) == M_.maximum_endo_lag+i+1);
gu = dr.ghx*Gu;
[nrGu,ncGu] = size(Gu);
[G1, err] = A_times_B_kronecker_C(dr.ghxx,Gu,options_.threads.kronecker.A_times_B_kronecker_C);
mexErrCheck('A_times_B_kronecker_C', err);
[G2, err] = A_times_B_kronecker_C(hxx,Gu,options_.threads.kronecker.A_times_B_kronecker_C);
mexErrCheck('A_times_B_kronecker_C', err);
guu = dr.ghx*Guu+G1;
Gu = hx*Gu;
Guu = hx*Guu;
Guu(end-npred+1:end,:) = Guu(end-npred+1:end,:) + G2;
H = E1 + hx*H;
end
RHS = RHS*M_.Sigma_e(:);
dr.fuu = RHS;
%RHS = -RHS-dr.fbias;
RHS = -RHS;
dr.ghs2 = LHS\RHS;
% deterministic exogenous variables
if M_.exo_det_nbr > 0
hud = dr.ghud{1}(nstatic+1:nstatic+npred,:);
zud=[zeros(np,M_.exo_det_nbr);dr.ghud{1};gx(:,1:npred)*hud;zeros(M_.exo_nbr,M_.exo_det_nbr);eye(M_.exo_det_nbr)];
R1 = hessian*kron(zx,zud);
dr.ghxud = cell(M_.exo_det_length,1);
kf = [M_.endo_nbr-nyf+1:M_.endo_nbr];
kp = nstatic+[1:npred];
dr.ghxud{1} = -M1*(R1+f1*dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{1}(kp,:)));
Eud = eye(M_.exo_det_nbr);
for i = 2:M_.exo_det_length
hudi = dr.ghud{i}(kp,:);
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zx,zudi);
dr.ghxud{i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(hx,Eud)+dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{i}(kp,:)))-M1*R2;
end
R1 = hessian*kron(zu,zud);
dr.ghudud = cell(M_.exo_det_length,1);
kf = [M_.endo_nbr-nyf+1:M_.endo_nbr];
dr.ghuud{1} = -M1*(R1+f1*dr.ghxx(kf,:)*kron(dr.ghu(kp,:),dr.ghud{1}(kp,:)));
Eud = eye(M_.exo_det_nbr);
for i = 2:M_.exo_det_length
hudi = dr.ghud{i}(kp,:);
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zu,zudi);
dr.ghuud{i} = -M2*dr.ghxud{i-1}(kf,:)*kron(hu,Eud)-M1*R2;
end
R1 = hessian*kron(zud,zud);
dr.ghudud = cell(M_.exo_det_length,M_.exo_det_length);
dr.ghudud{1,1} = -M1*R1-M2*dr.ghxx(kf,:)*kron(hud,hud);
for i = 2:M_.exo_det_length
hudi = dr.ghud{i}(nstatic+1:nstatic+npred,:);
zudi=[zeros(np,M_.exo_det_nbr);dr.ghud{i};gx(:,1:npred)*hudi+dr.ghud{i-1}(kf,:);zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zudi,zudi);
dr.ghudud{i,i} = -M2*(dr.ghudud{i-1,i-1}(kf,:)+...
2*dr.ghxud{i-1}(kf,:)*kron(hudi,Eud) ...
+dr.ghxx(kf,:)*kron(hudi,hudi))-M1*R2;
R2 = hessian*kron(zud,zudi);
dr.ghudud{1,i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(hud,Eud)+...
dr.ghxx(kf,:)*kron(hud,hudi))...
-M1*R2;
for j=2:i-1
hudj = dr.ghud{j}(kp,:);
zudj=[zeros(np,M_.exo_det_nbr);dr.ghud{j};gx(:,1:npred)*hudj;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian*kron(zudj,zudi);
dr.ghudud{j,i} = -M2*(dr.ghudud{j-1,i-1}(kf,:)+dr.ghxud{j-1}(kf,:)* ...
kron(hudi,Eud)+dr.ghxud{i-1}(kf,:)* ...
kron(hudj,Eud)+dr.ghxx(kf,:)*kron(hudj,hudi))-M1*R2;
end
end
end
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