248 lines
9.3 KiB
Matlab
248 lines
9.3 KiB
Matlab
function dr = dyn_second_order_solver(jacobia,hessian,dr,M_,threads_ABC,threads_BC)
|
|
|
|
%@info:
|
|
%! @deftypefn {Function File} {@var{dr} =} dyn_second_order_solver (@var{jacobia},@var{hessian},@var{dr},@var{M_},@var{threads_ABC},@var{threads_BC})
|
|
%! @anchor{dyn_first_order_solver}
|
|
%! @sp 1
|
|
%! Computes the first order reduced form of the DSGE model
|
|
%! @sp 2
|
|
%! @strong{Inputs}
|
|
%! @sp 1
|
|
%! @table @ @var
|
|
%! @item jacobia
|
|
%! Matrix containing the Jacobian of the model
|
|
%! @item hessian
|
|
%! Matrix containing the second order derivatives of the model
|
|
%! @item dr
|
|
%! Matlab's structure describing the reduced form solution of the model.
|
|
%! @item M_
|
|
%! Matlab's structure describing the model (initialized by @code{dynare}).
|
|
%! @item threads_ABC
|
|
%! Integer controlling number of threads in A_times_B_kronecker_C
|
|
%! @item threads_BC
|
|
%! Integer controlling number of threads in sparse_hessian_times_B_kronecker_C
|
|
%! @end table
|
|
%! @sp 2
|
|
%! @strong{Outputs}
|
|
%! @sp 1
|
|
%! @table @ @var
|
|
%! @item dr
|
|
%! Matlab's structure describing the reduced form solution of the model.
|
|
%! @end table
|
|
%! @end deftypefn
|
|
%@eod:
|
|
|
|
% Copyright (C) 2001-2011 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/>.
|
|
|
|
dr.ghxx = [];
|
|
dr.ghuu = [];
|
|
dr.ghxu = [];
|
|
dr.ghs2 = [];
|
|
Gy = dr.Gy;
|
|
|
|
kstate = dr.kstate;
|
|
kad = dr.kad;
|
|
kae = dr.kae;
|
|
nstatic = dr.nstatic;
|
|
nfwrd = dr.nfwrd;
|
|
npred = dr.npred;
|
|
nboth = dr.nboth;
|
|
nyf = nfwrd+nboth;
|
|
order_var = dr.order_var;
|
|
nd = size(kstate,1);
|
|
lead_lag_incidence = M_.lead_lag_incidence;
|
|
|
|
np = nd - nyf;
|
|
n2 = np + 1;
|
|
n3 = nyf;
|
|
n4 = n3 + 1;
|
|
|
|
k1 = nonzeros(lead_lag_incidence(:,order_var)');
|
|
kk = [k1; length(k1)+(1:M_.exo_nbr+M_.exo_det_nbr)'];
|
|
nk = size(kk,1);
|
|
kk1 = reshape([1:nk^2],nk,nk);
|
|
kk1 = kk1(kk,kk);
|
|
% reordering second order derivatives
|
|
hessian = hessian(:,kk1(:));
|
|
|
|
zx = zeros(np,np);
|
|
zu=zeros(np,M_.exo_nbr);
|
|
zx(1:np,:)=eye(np);
|
|
k0 = [1:M_.endo_nbr];
|
|
gx1 = dr.ghx;
|
|
hu = dr.ghu(nstatic+[1:npred],:);
|
|
k0 = find(lead_lag_incidence(M_.maximum_endo_lag+1,order_var)');
|
|
zx = [zx; gx1(k0,:)];
|
|
zu = [zu; dr.ghu(k0,:)];
|
|
k1 = find(lead_lag_incidence(M_.maximum_endo_lag+2,order_var)');
|
|
zu = [zu; gx1(k1,:)*hu];
|
|
zx = [zx; gx1(k1,:)*Gy];
|
|
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,threads_BC);
|
|
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(M_.endo_nbr,M_.endo_nbr);
|
|
B = zeros(M_.endo_nbr,M_.endo_nbr);
|
|
A(:,k0) = jacobia(:,nonzeros(lead_lag_incidence(M_.maximum_endo_lag+1,order_var)));
|
|
% variables with the highest lead
|
|
k1 = find(kstate(:,2) == M_.maximum_endo_lag+2);
|
|
% Jacobian with respect to the variables with the highest lead
|
|
fyp = jacobia(:,kstate(k1,3)+nnz(M_.lead_lag_incidence(M_.maximum_endo_lag+1,:)));
|
|
B(:,nstatic+npred-dr.nboth+1:end) = fyp;
|
|
offset = M_.endo_nbr;
|
|
gx1 = dr.ghx;
|
|
[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])+fyp*gx1(k1,1:npred);
|
|
C = Gy;
|
|
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,:);
|
|
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zx,zu,threads_BC);
|
|
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
|
|
|
|
hu1 = [hu;zeros(np-npred,M_.exo_nbr)];
|
|
[nrhx,nchx] = size(Gy);
|
|
[nrhu1,nchu1] = size(hu1);
|
|
|
|
[abcOut,err] = A_times_B_kronecker_C(dr.ghxx,Gy,hu1,threads_ABC);
|
|
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
|
|
[rhs, err] = sparse_hessian_times_B_kronecker_C(hessian,zu,threads_BC);
|
|
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
|
|
|
|
[B1, err] = A_times_B_kronecker_C(B*dr.ghxx,hu1,threads_ABC);
|
|
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,:);
|
|
rdr.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 = zeros(M_.endo_nbr,M_.endo_nbr);
|
|
LHS(:,k0) = jacobia(:,nonzeros(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);
|
|
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],:);
|
|
[junk,k2a,k2] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+2,order_var));
|
|
k3 = nnz(M_.lead_lag_incidence(1:M_.maximum_endo_lag+1,:))+(1:dr.nsfwrd)';
|
|
[B1, err] = sparse_hessian_times_B_kronecker_C(hessian(:,kh(k3,k3)),gu(k2a,:),threads_BC);
|
|
mexErrCheck('sparse_hessian_times_B_kronecker_C', err);
|
|
RHS = RHS + jacobia(:,k2)*guu(k2a,:)+B1;
|
|
|
|
% LHS
|
|
LHS = LHS + jacobia(:,k2)*(E(k2a,:)+[O1(k2a,:) dr.ghx(k2a,:)*H O2(k2a,:)]);
|
|
|
|
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(Gy,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
|