720 lines
26 KiB
Matlab
720 lines
26 KiB
Matlab
function [H, dA, dOm, Hss, gp, d2A, d2Om, H2ss] = getH(A, B, estim_params_,M_,oo_,options_,kronflag,indx,indexo,iv)
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% function [H, dA, dOm, Hss, gp, d2A, d2Om, H2ss] = getH(A, B, estim_params_,M_,oo_,options_,kronflag,indx,indexo,iv)
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% computes derivative of reduced form linear model w.r.t. deep params
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%
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% Inputs:
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% A: Transition matrix of lagged states from Kalman filter
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% B: Matrix in state transition equation mapping shocks today to
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% states today
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% M_: structure storing the model information
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% oo_: structure storing the results
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% options_: structure storing the options
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% kronflag: Indicator whether to rely on Kronecker products (1) or
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% not (-1 or -2)
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% indx: Index of estimated parameters in M_.params
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% indexo: Index of estimated standard deviations in M_.exo_names
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% iv: Index of considered variables
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%
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% Outputs:
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% H: dTAU/dTHETA: Jacobian of TAU, vectorized form of
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% linearized reduced form state space model, given ys [steady state],
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% A [transition matrix], B [matrix of shocks], Sigma [covariance of shocks]
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% TAU = [ys; vec(A); dyn_vech(B*Sigma*B')].
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% dA: [endo_nbr by endo_nbr by (indx+indexo)] Jacobian of transition matrix A
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% dOm: [endo_nbr by endo_nbr by (indx+indexo)] Jacobian of Omega = (B*Sigma*B')
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% Hss: [endo_nbr by (indx)] Jacobian of steady state with respect to estimated
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% structural parameters only (indx)
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% gp: Jacobian of linear rational expectation matrices [i.e.
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% Jacobian of dynamic model] with respect to estimated
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% structural parameters only (indx)
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% d2A: Hessian of transition matrix A
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% d2Om: Hessian of Omega
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% H2s: Hessian of steady state with respect to estimated
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% structural parameters only (indx)
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% Copyright (C) 2010-2017 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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if nargin<7 || isempty(kronflag)
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kronflag = 0;
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end
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if nargin<8 || isempty(indx)
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indx = [];
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end
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if nargin<9 || isempty(indexo)
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indexo = [];
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end
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if nargin<10 || isempty(iv)
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iv = (1:length(A))';
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end
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[I,J]=find(M_.lead_lag_incidence');
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yy0=oo_.dr.ys(I);
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param_nbr = length(indx);
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tot_param_nbr = param_nbr + length(indexo);
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if nargout>5
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param_nbr_2 = param_nbr*(param_nbr+1)/2;
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tot_param_nbr_2 = tot_param_nbr*(tot_param_nbr+1)/2;
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end
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m = size(A,1);
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m1=length(iv);
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n = size(B,2);
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if kronflag==-1 % perturbation
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gp=0;
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fun = 'thet2tau';
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params0 = M_.params;
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H = fjaco(fun,[sqrt(diag(M_.Sigma_e(indexo,indexo))); M_.params(indx)], M_, oo_, indx, indexo,0);
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if nargout>1
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dOm = zeros(m1,m1,tot_param_nbr);
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dA=zeros(m1,m1,tot_param_nbr);
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Hss=H(iv,length(indexo)+1:end);
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da = H(m+1:m+m*m,:);
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dom = H(m+m*m+1:end,:);
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for j=1:tot_param_nbr
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tmp = dyn_unvech(dom(:,j));
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dOm(:,:,j) = tmp(iv,iv);
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tmp = reshape(da(:,j),m,m);
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dA(:,:,j) = tmp(iv,iv);
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end
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clear da dom tmp
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end
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if nargout>5
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H2 = hessian_sparse('thet2tau',[sqrt(diag(M_.Sigma_e(indexo,indexo))); M_.params(indx)], ...
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options_.gstep,estim_params_,M_, oo_, indx,indexo,0,[],[],[],iv);
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H2ss = zeros(m1,tot_param_nbr,tot_param_nbr);
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iax=find(triu(rand(tot_param_nbr,tot_param_nbr)));
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H2 = H2(:,iax);
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for j=1:m1
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H2ss(j,:,:)=dyn_unvech(full(H2(j,:)));
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end
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H2ss=H2ss(:,length(indexo)+1:end,length(indexo)+1:end);
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d2A = sparse(m1*m1,tot_param_nbr_2);
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d2Om = sparse(m1*(m1+1)/2,tot_param_nbr_2);
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d2A(:,:) = H2(m1+1:m1+m1*m1,:);
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d2Om(:,:) = H2(m1+m1*m1+1:end,:);
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clear H2
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% tmp0=zeros(m1,m1);
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% tmp0(iv,iv)=1;
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% iax=find(tmp0);
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% d2A=d2a(iax,:);
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% iax=find(dyn_vech(tmp0));
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% d2Om=d2om(iax,:);
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end
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% assignin('base','M_', M_);
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% assignin('base','oo_', oo_);
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return
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end
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if kronflag==-2
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if nargout>5
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[residual, g1, g2 ] = feval([M_.fname,'.dynamic'],yy0, oo_.exo_steady_state', ...
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M_.params, oo_.dr.ys, 1);
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g22 = hessian_sparse('thet2tau',[M_.params(indx)],options_.gstep,estim_params_,M_, oo_, indx,[],-1);
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H2ss=full(g22(1:M_.endo_nbr,:));
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H2ss = reshape(H2ss,[M_.endo_nbr param_nbr param_nbr]);
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for j=1:M_.endo_nbr
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H2ss(j,:,:)=dyn_unvech(dyn_vech(H2ss(j,:,:)));
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end
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g22=g22(M_.endo_nbr+1:end,:);
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inx=find(g22);
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gx22=zeros(length(inx),5);
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for j=1:length(inx)
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[i1, i2] = ind2sub(size(g22),inx(j));
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[ig1, ig2] = ind2sub(size(g1),i1);
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[ip1, ip2] = ind2sub([param_nbr param_nbr],i2);
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gx22(j,:) = [ig1 ig2 ip1 ip2 g22(inx(j))];
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end
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g22 = gx22;
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clear gx22;
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else
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[residual, g1 ] = feval([M_.fname,'.dynamic'],yy0, oo_.exo_steady_state', ...
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M_.params, oo_.dr.ys, 1);
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end
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gp = fjaco('thet2tau',[M_.params(indx)],estim_params_,M_, oo_, indx,[],-1);
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Hss=gp(1:M_.endo_nbr,:);
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gp=gp(M_.endo_nbr+1:end,:);
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gp = reshape(gp,[size(g1) param_nbr]);
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else
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dyssdtheta=zeros(length(oo_.dr.ys),M_.param_nbr);
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d2yssdtheta=zeros(length(oo_.dr.ys),M_.param_nbr,M_.param_nbr);
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[residual, gg1] = feval([M_.fname,'.static'],oo_.dr.ys, oo_.exo_steady_state', M_.params);
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df = feval([M_.fname,'.static_params_derivs'],oo_.dr.ys, repmat(oo_.exo_steady_state',[M_.maximum_exo_lag+M_.maximum_exo_lead+1]), ...
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M_.params);
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dyssdtheta = -gg1\df;
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if nargout>5
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[residual, gg1, gg2] = feval([M_.fname,'.static'],oo_.dr.ys, oo_.exo_steady_state', M_.params);
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[residual, g1, g2, g3] = feval([M_.fname,'.dynamic'],yy0, oo_.exo_steady_state', ...
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M_.params, oo_.dr.ys, 1);
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[nr, nc]=size(gg2);
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[df, gpx, d2f] = feval([M_.fname,'.static_params_derivs'],oo_.dr.ys, oo_.exo_steady_state', ...
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M_.params);%, oo_.dr.ys, 1, dyssdtheta*0, d2yssdtheta);
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d2f = get_all_resid_2nd_derivs(d2f,length(oo_.dr.ys),M_.param_nbr);
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if isempty(find(gg2))
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for j=1:M_.param_nbr
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d2yssdtheta(:,:,j) = -gg1\d2f(:,:,j);
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end
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else
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gam = d2f*0;
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for j=1:nr
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tmp1 = (squeeze(gpx(j,:,:))'*dyssdtheta);
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gam(j,:,:)=transpose(reshape(gg2(j,:),[nr nr])*dyssdtheta)*dyssdtheta ...
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+ tmp1 + tmp1';
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end
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for j=1:M_.param_nbr
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d2yssdtheta(:,:,j) = -gg1\(d2f(:,:,j)+gam(:,:,j));
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end
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clear tmp1 gpx gam
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end
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end
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if any(any(isnan(dyssdtheta)))
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[U,T] = schur(gg1);
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qz_criterium=options_.qz_criterium;
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e1 = abs(ordeig(T)) < qz_criterium-1;
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k = sum(e1); % Number non stationary variables.
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% n = length(e1)-k; % Number of stationary variables.
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[U,T] = ordschur(U,T,e1);
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T = T(k+1:end,k+1:end);
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dyssdtheta = -U(:,k+1:end)*(T\U(:,k+1:end)')*df;
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if nargout>5
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for j=1:length(indx)
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d2yssdtheta(:,:,j) = -U(:,k+1:end)*(T\U(:,k+1:end)')*d2f(:,:,j);
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end
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end
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end
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if nargout>5
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[df, gp, d2f, gpp, hp] = feval([M_.fname,'.dynamic_params_derivs'],yy0, oo_.exo_steady_state', ...
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M_.params, oo_.dr.ys, 1, dyssdtheta, d2yssdtheta);
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H2ss = d2yssdtheta(oo_.dr.order_var,indx,indx);
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else
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[df, gp] = feval([M_.fname,'.dynamic_params_derivs'],yy0, repmat(oo_.exo_steady_state',[M_.maximum_exo_lag+M_.maximum_exo_lead+1,1]), ...
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M_.params, oo_.dr.ys, 1, dyssdtheta,d2yssdtheta);
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[residual, g1, g2 ] = feval([M_.fname,'.dynamic'],yy0, repmat(oo_.exo_steady_state',[M_.maximum_exo_lag+M_.maximum_exo_lead+1,1]), ...
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M_.params, oo_.dr.ys, 1);
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end
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[nr, nc]=size(g2);
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nc = sqrt(nc);
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Hss = dyssdtheta(oo_.dr.order_var,indx);
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dyssdtheta = dyssdtheta(I,:);
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ns = max(max(M_.lead_lag_incidence)); % retrieve the number of states excluding columns for shocks
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gp2 = gp*0;
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for j=1:nr
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[II JJ]=ind2sub([nc nc],find(g2(j,:)));
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for i=1:nc
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is = find(II==i);
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is = is(find(JJ(is)<=ns));
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if ~isempty(is)
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g20=full(g2(j,find(g2(j,:))));
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gp2(j,i,:)=g20(is)*dyssdtheta(JJ(is),:);
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end
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end
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end
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gp = gp+gp2;
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gp = gp(:,:,indx);
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if nargout>5
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% h22 = get_all_hess_derivs(hp,nr,nc,M_.param_nbr);
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g22 = gpp;
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gp22 = sparse(nr*nc,param_nbr*param_nbr);
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tmp1 = reshape(g3,[nr*nc*nc nc]);
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tmp2=sparse(size(tmp1,1),M_.param_nbr);
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% tmp2=tmp1*[dyssdtheta; zeros(nc-ns,M_.param_nbr)];
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tmpa=[dyssdtheta; zeros(nc-ns,M_.param_nbr)];
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tmpa=sparse(tmpa);
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for j=1:M_.param_nbr
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tmp2(:,j)=tmp1*tmpa(:,j);
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end
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% tmp2=sparse(tmp2);
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% [i1 i2]=ind2sub([nc M_.param_nbr],[1:nc*M_.param_nbr]');
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for j=1:nr
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tmp0=reshape(g2(j,:),[nc nc]);
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tmp0 = tmp0(:,1:ns)*reshape(d2yssdtheta(I,:,:),[ns,M_.param_nbr*M_.param_nbr]);
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for i=1:nc
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indo = sub2ind([nr nc nc], ones(nc,1)*j ,ones(nc,1)*i, (1:nc)');
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tmpx = (tmp2(indo,:))'*[dyssdtheta; zeros(nc-ns,M_.param_nbr)];
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% gp22(j,i,:,:)=squeeze(tmp1(j,i,:,:))'*[dyssdtheta; zeros(nc-ns,M_.param_nbr)];
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tmpu = (get_hess_deriv(hp,j,i,nc,M_.param_nbr))'*[dyssdtheta; zeros(nc-ns,M_.param_nbr)];
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tmpy = tmpx+tmpu+tmpu'+reshape(tmp0(i,:,:),[M_.param_nbr M_.param_nbr]);
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tmpy = tmpy + get_2nd_deriv_mat(gpp,j,i,M_.param_nbr);
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tmpy = tmpy(indx,indx);
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if any(any(tmpy))
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ina = find(triu(tmpy));
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gp22(sub2ind([nr nc],j,i),ina)=transpose(tmpy(ina));
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% gp22(j,i,:,:)= reshape(tmpy,[1 1 M_.param_nbr M_.param_nbr]);
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end
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end
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% gp22(j,:,:,:)=gp22(j,:,:,:)+reshape(tmp0(:,1:ns)*d2yssdtheta(I,:,:),[1 nc M_.param_nbr M_.param_nbr]);
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end
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% g22 = g22+gp22;
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% g22 = g22(:,:,indx,indx);
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clear tmp0 tmp1 tmp2 tmpu tmpx tmpy
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inx=find(gp22);
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gx22=zeros(length(inx),5);
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for j=1:length(inx)
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[i1, i2] = ind2sub(size(gp22),inx(j));
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[ig1, ig2] = ind2sub(size(g1),i1);
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[ip1, ip2] = ind2sub([param_nbr param_nbr],i2);
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gx22(j,:) = [ig1 ig2 ip1 ip2 gp22(inx(j))];
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end
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g22 = gx22;
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clear gx22 gp22;
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end
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end
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klen = M_.maximum_endo_lag + M_.maximum_endo_lead + 1;
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k11 = M_.lead_lag_incidence(find([1:klen] ~= M_.maximum_endo_lag+1),:);
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a = g1(:,nonzeros(k11'));
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da = gp(:,nonzeros(k11'),:);
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if nargout > 5
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indind = ismember(g22(:,2),nonzeros(k11'));
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tmp = g22(indind,:);
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d2a=tmp;
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for j=1:size(tmp,1)
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inxinx = find(nonzeros(k11')==tmp(j,2));
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d2a(j,2) = inxinx;
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end
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end
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kstate = oo_.dr.kstate;
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GAM1 = zeros(M_.endo_nbr,M_.endo_nbr);
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Dg1 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr);
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k1 = find(kstate(:,2) == M_.maximum_endo_lag+2 & kstate(:,3));
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GAM1(:, kstate(k1,1)) = -a(:,kstate(k1,3));
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Dg1(:, kstate(k1,1), :) = -da(:,kstate(k1,3),:);
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if nargout > 5
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indind = ismember(d2a(:,2),kstate(k1,3));
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tmp = d2a(indind,:);
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tmp(:,end)=-tmp(:,end);
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D2g1 = tmp;
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for j=1:size(tmp,1)
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inxinx = (kstate(k1,3)==tmp(j,2));
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D2g1(j,2) = kstate(k1(inxinx),1);
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end
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end
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[~,cols_b,cols_j] = find(M_.lead_lag_incidence(M_.maximum_endo_lag+1, ...
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oo_.dr.order_var));
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GAM0 = zeros(M_.endo_nbr,M_.endo_nbr);
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Dg0 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr);
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GAM0(:,cols_b) = g1(:,cols_j);
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Dg0(:,cols_b,:) = gp(:,cols_j,:);
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if nargout > 5
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indind = ismember(g22(:,2),cols_j);
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tmp = g22(indind,:);
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D2g0=tmp;
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for j=1:size(tmp,1)
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inxinx = (cols_j==tmp(j,2));
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D2g0(j,2) = cols_b(inxinx);
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end
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end
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k2 = find(kstate(:,2) == M_.maximum_endo_lag+1 & kstate(:,4));
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GAM2 = zeros(M_.endo_nbr,M_.endo_nbr);
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Dg2 = zeros(M_.endo_nbr,M_.endo_nbr,param_nbr);
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GAM2(:, kstate(k2,1)) = -a(:,kstate(k2,4));
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Dg2(:, kstate(k2,1), :) = -da(:,kstate(k2,4),:);
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if nargout > 5
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indind = ismember(d2a(:,2),kstate(k2,4));
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tmp = d2a(indind,:);
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tmp(:,end)=-tmp(:,end);
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D2g2 = tmp;
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for j=1:size(tmp,1)
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inxinx = (kstate(k2,4)==tmp(j,2));
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D2g2(j,2) = kstate(k2(inxinx),1);
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end
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end
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GAM3 = -g1(:,length(yy0)+1:end);
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Dg3 = -gp(:,length(yy0)+1:end,:);
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if nargout>5
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cols_ex = [length(yy0)+1:size(g1,2)];
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indind = ismember(g22(:,2),cols_ex);
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tmp = g22(indind,:);
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tmp(:,end)=-tmp(:,end);
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D2g3=tmp;
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for j=1:size(tmp,1)
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inxinx = find(cols_ex==tmp(j,2));
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D2g3(j,2) = inxinx;
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end
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clear g22 d2a tmp
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end
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clear g1 g2 g3 df d2f gpp hp residual gg1 gg2 gp2 dyssdtheta d2yssdtheta
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if kronflag==1 % kronecker products
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Dg0=reshape(Dg0,m^2,param_nbr);
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Dg1=reshape(Dg1,m^2,param_nbr);
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Dg2=reshape(Dg2,m^2,param_nbr);
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for j=1:param_nbr
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|
Dg3(:,:,j)=Dg3(:,:,j)*M_.Sigma_e;
|
|
end
|
|
Dg3=reshape(Dg3,m*n,param_nbr);
|
|
Om = B*M_.Sigma_e*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);
|
|
dA = x;
|
|
dOm = y;
|
|
% convert to dyn_vech
|
|
tmpH = Dm*H(m*m+1:end,:);
|
|
Index = find(triu(ones(m)));
|
|
H(m*m+1:end,:) = tmpH(Index,:);
|
|
|
|
Hx = [];
|
|
if ~isempty(indexo)
|
|
dSig = zeros(M_.exo_nbr,M_.exo_nbr);
|
|
dOm = cat(3,zeros(size(dOm,1),size(dOm,1),length(indexo)),dOm);
|
|
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);
|
|
Hx(:,j) = [zeros((m-nauxe)^2,1); dyn_vech(y)];
|
|
if nargout>1
|
|
dOm(:,:,j) = y;
|
|
end
|
|
dSig(indexo(j),indexo(j)) = 0;
|
|
end
|
|
end
|
|
H = [ [zeros(M_.endo_nbr,length(indexo)) Hss]; [Hx H]];
|
|
|
|
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=sylvester3(a,b,c,d);
|
|
flag=1;
|
|
icount=0;
|
|
while flag && icount<4
|
|
[xx, flag]=sylvester3a(xx,a,b,c,d);
|
|
icount=icount+1;
|
|
end
|
|
H=zeros(m1*m1+m1*(m1+1)/2,param_nbr+length(indexo));
|
|
if nargout>1
|
|
dOm = zeros(m1,m1,param_nbr+length(indexo));
|
|
dA=zeros(m1,m1,param_nbr+length(indexo));
|
|
dB=zeros(m,n,param_nbr);
|
|
end
|
|
if ~isempty(indexo)
|
|
dSig = zeros(M_.exo_nbr,M_.exo_nbr,length(indexo));
|
|
for j=1:length(indexo)
|
|
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(m1^2,1); dyn_vech(y(iv,iv))];
|
|
if nargout>1
|
|
dOm(:,:,j) = y(iv,iv);
|
|
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);
|
|
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);
|
|
if nargout>1
|
|
dA(:,:,j+length(indexo)) = x(iv,iv);
|
|
dOm(:,:,j+length(indexo)) = y(iv,iv);
|
|
end
|
|
H(:,j+length(indexo)) = [vec(x(iv,iv)); dyn_vech(y(iv,iv))];
|
|
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(:); dyn_vech(y)];
|
|
% end
|
|
Hss = Hss(iv,:);
|
|
H = [[zeros(m1,length(indexo)) Hss]; H];
|
|
|
|
end
|
|
|
|
if nargout > 5
|
|
H2ss = H2ss(iv,:,:);
|
|
d = zeros(m,m,floor(sqrt(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;
|
|
cumjcount=0;
|
|
jinx = [];
|
|
x2x=sparse(m*m,param_nbr_2);
|
|
% x2x=[];
|
|
for i=1:param_nbr
|
|
for j=1:i
|
|
elem1 = (get_2nd_deriv(D2g0,m,m,j,i)-get_2nd_deriv(D2g1,m,m,j,i)*A);
|
|
elem1 = get_2nd_deriv(D2g2,m,m,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;
|
|
jinx = [jinx; [j i]];
|
|
d(:,:,jcount) = elem1+elem2;
|
|
if jcount==floor(sqrt(param_nbr_2)) || (j*i)==param_nbr^2
|
|
if (j*i)==param_nbr^2
|
|
d = d(:,:,1:jcount);
|
|
end
|
|
% d(find(abs(d)<1.e-12))=0;
|
|
xx2=sylvester3(a,b,c,d);
|
|
flag=1;
|
|
icount=0;
|
|
while flag && icount<4
|
|
[xx2, flag]=sylvester3a(xx2,a,b,c,d);
|
|
icount = icount + 1;
|
|
end
|
|
% inx = find(abs(xx2)>1.e-12);
|
|
% xx2(find(abs(xx2)<1.e-12))=0;
|
|
x2x(:,cumjcount+1:cumjcount+jcount)=reshape(xx2,[m*m jcount]);
|
|
cumjcount=cumjcount+jcount;
|
|
% [i1 i2 i3]=ind2sub(size(xx2),inx);
|
|
% x2x = [x2x; [i1 i2 jinx(i3,:) xx2(inx)]];
|
|
jcount = 0;
|
|
jinx = [];
|
|
end
|
|
end
|
|
end
|
|
clear d xx2;
|
|
jcount = 0;
|
|
icount = 0;
|
|
cumjcount = 0;
|
|
MAX_DIM_MAT = 100000000;
|
|
ncol = max(1,floor(MAX_DIM_MAT/(8*m1*(m1+1)/2)));
|
|
ncol = min(ncol, tot_param_nbr_2);
|
|
d2A = sparse(m1*m1,tot_param_nbr_2);
|
|
d2Om = sparse(m1*(m1+1)/2,tot_param_nbr_2);
|
|
d2A_tmp = zeros(m1*m1,ncol);
|
|
d2Om_tmp = zeros(m1*(m1+1)/2,ncol);
|
|
tmpDir = CheckPath('tmp_derivs',M_.dname);
|
|
offset = length(indexo);
|
|
% d2B = zeros(m,n,tot_param_nbr,tot_param_nbr);
|
|
d2Sig = zeros(M_.exo_nbr,M_.exo_nbr,length(indexo));
|
|
for j=1:tot_param_nbr
|
|
for i=1:j
|
|
jcount=jcount+1;
|
|
if j<=offset
|
|
if i==j
|
|
d2Sig(indexo(j),indexo(j),j) = 2;
|
|
y = B*d2Sig(:,:,j)*B';
|
|
% y(abs(y)<1.e-8)=0;
|
|
d2Om_tmp(:,jcount) = dyn_vech(y(iv,iv));
|
|
end
|
|
else
|
|
jind = j-offset;
|
|
iind = i-offset;
|
|
if i<=offset
|
|
y = dB(:,:,jind)*dSig(:,:,i)*B'+B*dSig(:,:,i)*dB(:,:,jind)';
|
|
% y(abs(y)<1.e-8)=0;
|
|
d2Om_tmp(:,jcount) = dyn_vech(y(iv,iv));
|
|
else
|
|
icount=icount+1;
|
|
x = reshape(x2x(:,icount),[m m]);
|
|
% x = get_2nd_deriv(x2x,m,m,iind,jind);%xx2(:,:,jcount);
|
|
elem1 = (get_2nd_deriv(D2g0,m,m,iind,jind)-get_2nd_deriv(D2g1,m,m,iind,jind)*A);
|
|
elem1 = elem1 -( Dg1(:,:,jind)*xx(:,:,iind) + Dg1(:,:,iind)*xx(:,:,jind) );
|
|
elemj0 = Dg0(:,:,jind)-Dg1(:,:,jind)*A-GAM1*xx(:,:,jind);
|
|
elemi0 = Dg0(:,:,iind)-Dg1(:,:,iind)*A-GAM1*xx(:,:,iind);
|
|
elem0 = elemj0*dB(:,:,iind)+elemi0*dB(:,:,jind);
|
|
y = inva * (get_2nd_deriv(D2g3,m,n,iind,jind)-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(:,:,jind)*M_.Sigma_e*dB(:,:,iind)'+dB(:,:,iind)*M_.Sigma_e*dB(:,:,jind)';
|
|
% x(abs(x)<1.e-8)=0;
|
|
d2A_tmp(:,jcount) = vec(x(iv,iv));
|
|
% y(abs(y)<1.e-8)=0;
|
|
d2Om_tmp(:,jcount) = dyn_vech(y(iv,iv));
|
|
end
|
|
end
|
|
if jcount==ncol || i*j==tot_param_nbr^2
|
|
d2A(:,cumjcount+1:cumjcount+jcount) = d2A_tmp(:,1:jcount);
|
|
% d2A(:,:,j+length(indexo),i+length(indexo)) = x;
|
|
% d2A(:,:,i+length(indexo),j+length(indexo)) = x;
|
|
d2Om(:,cumjcount+1:cumjcount+jcount) = d2Om_tmp(:,1:jcount);
|
|
% d2Om(:,:,j+length(indexo),i+length(indexo)) = y;
|
|
% d2Om(:,:,i+length(indexo),j+length(indexo)) = y;
|
|
save([tmpDir filesep 'd2A_' int2str(cumjcount+1) '_' int2str(cumjcount+jcount) '.mat'],'d2A')
|
|
save([tmpDir filesep 'd2Om_' int2str(cumjcount+1) '_' int2str(cumjcount+jcount) '.mat'],'d2Om')
|
|
cumjcount = cumjcount+jcount;
|
|
jcount=0;
|
|
% d2A = sparse(m1*m1,tot_param_nbr*(tot_param_nbr+1)/2);
|
|
% d2Om = sparse(m1*(m1+1)/2,tot_param_nbr*(tot_param_nbr+1)/2);
|
|
d2A_tmp = zeros(m1*m1,ncol);
|
|
d2Om_tmp = zeros(m1*(m1+1)/2,ncol);
|
|
|
|
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(sub2ind([m,n],gpp(is,1),gpp(is,2)))=gpp(is,5)';
|
|
end
|
|
return
|
|
|
|
function g22 = get_2nd_deriv_mat(gpp,i,j,n)
|
|
|
|
g22=zeros(n,n);
|
|
is=find(gpp(:,1)==i);
|
|
is=is(find(gpp(is,2)==j));
|
|
|
|
if ~isempty(is)
|
|
g22(sub2ind([n,n],gpp(is,3),gpp(is,4)))=gpp(is,5)';
|
|
end
|
|
return
|
|
|
|
function g22 = get_all_2nd_derivs(gpp,m,n,npar,fsparse)
|
|
|
|
if nargin==4 || isempty(fsparse)
|
|
fsparse=0;
|
|
end
|
|
if fsparse
|
|
g22=sparse(m*n,npar*npar);
|
|
else
|
|
g22=zeros(m,n,npar,npar);
|
|
end
|
|
% 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));
|
|
if fsparse
|
|
g22(sub2ind([m,n],gpp(is,1),gpp(is,2)),sub2ind([npar,npar],gpp(is,3),gpp(is,4)))=gpp(is,5);
|
|
else
|
|
g22(gpp(is,1),gpp(is,2),gpp(is,3),gpp(is,4))=gpp(is,5);
|
|
end
|
|
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);
|
|
end
|
|
|
|
return
|
|
|
|
function h2 = get_all_hess_derivs(hp,r,m,npar)
|
|
|
|
h2=zeros(r,m,m,npar);
|
|
|
|
for is=1:length(hp)
|
|
h2(hp(is,1),hp(is,2),hp(is,3),hp(is,4))=hp(is,5);
|
|
end
|
|
|
|
return
|
|
|
|
function h2 = get_hess_deriv(hp,i,j,m,npar)
|
|
|
|
h2=zeros(m,npar);
|
|
is1=find(hp(:,1)==i);
|
|
is=is1(find(hp(is1,2)==j));
|
|
|
|
if ~isempty(is)
|
|
h2(sub2ind([m,npar],hp(is,3),hp(is,4)))=hp(is,5)';
|
|
end
|
|
|
|
return |