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function [dr, info] = stochastic_solvers(dr, task, M_, options_, exo_steady_state, exo_det_steady_state)
%[dr, info] = stochastic_solvers(dr, task, M_, options_, exo_steady_state, exo_det_steady_state)
% Computes the reduced form solution of a rational expectations model (first, second or third
% order approximation of the stochastic model around the deterministic steady state).
%
% INPUTS
% - dr [struct] Decision rules for stochastic simulations.
% - task [integer] scalar, if task = 0 then decision rules are computed and if task = 1 then only eigenvales are computed.
% - M_ [struct] Definition of the model.
% - options_ [struct] Options.
% - exo_steady_state [vector] steady state value for exogenous variables
% - exo_det_steady_state [vector] steady state value for exogenous deterministic variables
%
% OUTPUTS
% - dr [struct] Decision rules for stochastic simulations.
% - info [integer] scalar, error code:
%
% 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.
% info=9 -> k_order_pert was unable to compute the solution
% Copyright © 1996-2023 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 <https://www.gnu.org/licenses/>.
info = 0;
if options_.linear
options_.order = 1;
end
local_order = options_.order;
if local_order~=1 && M_.hessian_eq_zero
local_order = 1;
warning('stochastic_solvers: using order = 1 because Hessian is equal to zero');
end
if options_.order>2 && ~options_.k_order_solver
error('You need to set k_order_solver for order>2')
end
if options_.aim_solver && (local_order > 1)
error('Option "aim_solver" is incompatible with order >= 2')
end
if M_.maximum_endo_lag == 0
if local_order >= 2
fprintf('\nSTOCHASTIC_SOLVER: Dynare does not solve purely forward models at higher order.\n')
fprintf('STOCHASTIC_SOLVER: To circumvent this restriction, you can add a backward-looking dummy equation of the form:\n')
fprintf('STOCHASTIC_SOLVER: junk=0.9*junk(-1);\n')
error(['2nd and 3rd order approximation not implemented for purely ' ...
'forward models'])
end
if M_.exo_det_nbr~=0
fprintf('\nSTOCHASTIC_SOLVER: Dynare does not solve purely forward models with var_exo_det.\n')
fprintf('STOCHASTIC_SOLVER: To circumvent this restriction, you can add a backward-looking dummy equation of the form:\n')
fprintf('STOCHASTIC_SOLVER: junk=0.9*junk(-1);\n')
error('var_exo_det not implemented for purely forward models')
end
end
if M_.maximum_endo_lead==0 && M_.exo_det_nbr~=0
fprintf('\nSTOCHASTIC_SOLVER: Dynare does not solve purely backward models with var_exo_det.\n')
fprintf('STOCHASTIC_SOLVER: To circumvent this restriction, you can add a foward-looking dummy equation of the form:\n')
fprintf('STOCHASTIC_SOLVER: junk=0.9*junk(+1);\n')
error('var_exo_det not implemented for purely backwards models')
end
if options_.k_order_solver
if options_.bytecode
warning('Option "bytecode" is ignored when computing perturbation solution at higher order')
end
orig_order = options_.order;
options_.order = local_order;
dr = set_state_space(dr,M_);
[dr,info] = k_order_pert(dr,M_,options_);
options_.order = orig_order;
return
end
klen = M_.maximum_lag + M_.maximum_lead + 1;
exo_simul = [repmat(exo_steady_state',klen,1) repmat(exo_det_steady_state',klen,1)];
iyv = M_.lead_lag_incidence';
iyv = iyv(:);
iyr0 = find(iyv) ;
it_ = M_.maximum_lag + 1;
z = repmat(dr.ys,1,klen);
if local_order == 1
if (options_.bytecode)
[~, loc_dr] = bytecode('dynamic','evaluate', M_, options_, z, exo_simul, ...
M_.params, dr.ys, 1);
jacobia_ = [loc_dr.g1 loc_dr.g1_x loc_dr.g1_xd];
else
[~,jacobia_] = feval([M_.fname '.dynamic'],z(iyr0),exo_simul, ...
M_.params, dr.ys, it_);
end
elseif local_order == 2
if (options_.bytecode)
warning('Option "bytecode" is ignored when computing perturbation solution at order = 2')
end
[~,jacobia_,hessian1] = feval([M_.fname '.dynamic'],z(iyr0),...
exo_simul, ...
M_.params, dr.ys, it_);
[infrow, ~] = find(isinf(hessian1));
if options_.debug
if ~isempty(infrow)
fprintf('\nSTOCHASTIC_SOLVER: The Hessian of the dynamic model contains Inf.\n')
fprintf('STOCHASTIC_SOLVER: Try running model_diagnostics to find the source of the problem.\n')
save([M_.dname filesep 'Output' filesep M_.fname '_debug.mat'],'hessian1')
end
end
if ~isempty(infrow)
info(1)=11;
return
end
[nanrow, ~] = find(isnan(hessian1));
if options_.debug
if ~isempty(nanrow)
fprintf('\nSTOCHASTIC_SOLVER: The Hessian of the dynamic model contains NaN.\n')
fprintf('STOCHASTIC_SOLVER: Try running model_diagnostics to find the source of the problem.\n')
save([M_.dname filesep 'Output' filesep M_.fname '_debug.mat'],'hessian1')
end
end
if ~isempty(nanrow)
info(1)=12;
return
end
end
[infrow,infcol]=find(isinf(jacobia_));
if options_.debug
if ~isempty(infrow)
fprintf('\nSTOCHASTIC_SOLVER: The Jacobian of the dynamic model contains Inf. The problem is associated with:\n\n')
display_problematic_vars_Jacobian(infrow,infcol,M_,dr.ys,'dynamic','STOCHASTIC_SOLVER: ')
save([M_.dname filesep 'Output' filesep M_.fname '_debug.mat'],'jacobia_')
end
end
if ~isempty(infrow)
info(1)=10;
return
end
if ~isreal(jacobia_)
if max(max(abs(imag(jacobia_)))) < 1e-15
jacobia_ = real(jacobia_);
else
if options_.debug
[imagrow,imagcol]=find(abs(imag(jacobia_))>1e-15);
fprintf('\nMODEL_DIAGNOSTICS: The Jacobian of the dynamic model contains imaginary parts. The problem arises from: \n\n')
display_problematic_vars_Jacobian(imagrow,imagcol,M_,dr.ys,'dynamic','STOCHASTIC_SOLVER: ')
end
info(1) = 6;
info(2) = sum(sum(imag(jacobia_).^2));
return
end
end
[nanrow,nancol]=find(isnan(jacobia_));
if options_.debug
if ~isempty(nanrow)
fprintf('\nSTOCHASTIC_SOLVER: The Jacobian of the dynamic model contains NaN. The problem is associated with:\n\n')
display_problematic_vars_Jacobian(nanrow,nancol,M_,dr.ys,'dynamic','STOCHASTIC_SOLVER: ')
save([M_.dname filesep 'Output' filesep M_.fname '_debug.mat'],'jacobia_')
end
end
if ~isempty(nanrow)
info(1) = 8;
NaN_params=find(isnan(M_.params));
info(2:length(NaN_params)+1) = NaN_params;
return
end
kstate = dr.kstate;
nstatic = M_.nstatic;
nfwrd = M_.nfwrd;
nspred = M_.nspred;
nboth = M_.nboth;
nsfwrd = M_.nsfwrd;
order_var = dr.order_var;
nd = size(kstate,1);
nz = nnz(M_.lead_lag_incidence);
sdyn = M_.endo_nbr - nstatic;
[~,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 local_order == 1
[k1,~,k2] = find(kstate(:,4));
if M_.maximum_endo_lag
dr.state_var = find(M_.lead_lag_incidence(1,:));
else
dr.state_var = [];
end
dr.ghx(:,k1) = -b\jacobia_(:,k2);
if M_.exo_nbr
dr.ghu = -b\jacobia_(:,nz+1:end);
end
dr.eigval = eig(kalman_transition_matrix(dr,nstatic+(1:nspred),1:nspred));
dr.full_rank = 1;
dr.edim = nnz(abs(dr.eigval) > options_.qz_criterium);
dr.sdim = nd-dr.edim;
if dr.edim
temp = sort(abs(dr.eigval));
temp = temp(dr.sdim+1:nd)-1-options_.qz_criterium;
info(1) = 3;
info(2) = temp'*temp;
end
else
fprintf('\nSTOCHASTIC_SOLVER: Dynare does not solve purely backward models at higher order.\n')
fprintf('STOCHASTIC_SOLVER: To circumvent this restriction, you can add a forward-looking dummy equation of the form:\n')
fprintf('STOCHASTIC_SOLVER: junk=0.9*junk(+1);\n')
error(['2nd and 3rd order approximation not implemented for purely ' ...
'backward models'])
end
else
% If required, use AIM solver if not check only
if options_.aim_solver && (task == 0)
[dr,info] = AIM_first_order_solver(jacobia_,M_,dr,options_.qz_criterium);
else % use original Dynare solver
[dr,info] = dyn_first_order_solver(jacobia_,M_,dr,options_,task);
if info(1) || task
return
end
end
if local_order > 1
% Second order
dr = dyn_second_order_solver(jacobia_,hessian1,dr,M_,...
options_.threads.kronecker.sparse_hessian_times_B_kronecker_C);
% reordering second order derivatives, used for deterministic
% variables below
k1 = nonzeros(M_.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);
hessian1 = hessian1(:,kk1(:));
end
end
%exogenous deterministic variables
if M_.exo_det_nbr > 0
gx = dr.gx;
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,nsfwrd-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-nsfwrd+1:end,:);
end
if local_order > 1
lead_lag_incidence = M_.lead_lag_incidence;
k0 = find(lead_lag_incidence(M_.maximum_endo_lag+1,order_var)');
k1 = find(lead_lag_incidence(M_.maximum_endo_lag+2,order_var)');
hu = dr.ghu(nstatic+[1:nspred],:);
hud = dr.ghud{1}(nstatic+1:nstatic+nspred,:);
zx = [eye(nspred);dr.ghx(k0,:);gx*dr.Gy;zeros(M_.exo_nbr+M_.exo_det_nbr, ...
nspred)];
zu = [zeros(nspred,M_.exo_nbr); dr.ghu(k0,:); gx*hu; zeros(M_.exo_nbr+M_.exo_det_nbr, ...
M_.exo_nbr)];
zud=[zeros(nspred,M_.exo_det_nbr);dr.ghud{1};gx(:,1:nspred)*hud;zeros(M_.exo_nbr,M_.exo_det_nbr);eye(M_.exo_det_nbr)];
R1 = hessian1*kron(zx,zud);
dr.ghxud = cell(M_.exo_det_length,1);
kf = M_.endo_nbr-nfwrd-nboth+1:M_.endo_nbr;
kp = nstatic+[1:nspred];
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(nspred,M_.exo_det_nbr);dr.ghud{i};gx(:,1:nspred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian1*kron(zx,zudi);
dr.ghxud{i} = -M2*(dr.ghxud{i-1}(kf,:)*kron(dr.Gy,Eud)+dr.ghxx(kf,:)*kron(dr.ghx(kp,:),dr.ghud{i}(kp,:)))-M1*R2;
end
R1 = hessian1*kron(zu,zud);
dr.ghudud = cell(M_.exo_det_length,1);
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(nspred,M_.exo_det_nbr);dr.ghud{i};gx(:,1:nspred)*hudi;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian1*kron(zu,zudi);
dr.ghuud{i} = -M2*dr.ghxud{i-1}(kf,:)*kron(hu,Eud)-M1*R2;
end
R1 = hessian1*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+nspred,:);
zudi=[zeros(nspred,M_.exo_det_nbr);dr.ghud{i};gx(:,1:nspred)*hudi+dr.ghud{i-1}(kf,:);zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian1*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 = hessian1*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(nspred,M_.exo_det_nbr);dr.ghud{j};gx(:,1:nspred)*hudj;zeros(M_.exo_nbr+M_.exo_det_nbr,M_.exo_det_nbr)];
R2 = hessian1*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
end
if options_.loglinear
% this needs to be extended for order=2,3
[il,~,ik,k1] = indices_lagged_leaded_exogenous_variables(dr.order_var,M_);
[illag,~,iklag,klag1] = indices_lagged_leaded_exogenous_variables(dr.order_var(M_.nstatic+(1:M_.nspred)),M_);
if ~isempty(ik)
if M_.nspred > 0
dr.ghx(ik,iklag) = repmat(1./dr.ys(k1),1,length(klag1)).*dr.ghx(ik,iklag).* ...
repmat(dr.ys(klag1)',length(ik),1);
dr.ghx(ik,illag) = repmat(1./dr.ys(k1),1,length(illag)).*dr.ghx(ik,illag);
end
if M_.exo_nbr > 0
dr.ghu(ik,:) = repmat(1./dr.ys(k1),1,M_.exo_nbr).*dr.ghu(ik,:);
end
end
if ~isempty(il) && M_.nspred > 0
dr.ghx(il,iklag) = dr.ghx(il,iklag).*repmat(dr.ys(klag1)', ...
length(il),1);
end
if local_order > 1
error('Loglinear options currently only works at order 1')
end
end
end
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