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function [g,badg] = fn_a0sfreegrad(b,Uistar,Uibar,nvar,nStates,n0,Tkave,Sgm0tldinvave)
% [g,badg] = fn_a0sfreegrad(b,Uistar,Uibar,nvar,nStates,n0,Tkave,Sgm0tldinvave)
% Analytical gradient for fn_a0sfreefun.m when using csminwel.m.
% The case of no asymmetric prior and no lag restrictions.
% Note: (1) columns correspond to equations; s stands for state.
% See TBVAR NOTE p.34a.
%
% b: sum(n0)*nStates-by-1 vector of free A0 parameters, vectorized from the sum(n0)-by-nStates matrix.
% Uistar: cell(nvar,1). In each cell, nvar*nStates-by-qi orthonormal basis for the null of the ith
% equation contemporaneous restriction matrix where qi is the number of free parameters.
% With this transformation, we have ai = Ui*bi or Ui'*ai = bi where ai is a vector
% of total original parameters and bi is a vector of free parameters. When no
% restrictions are imposed, we have Ui = I. There must be at least one free
% parameter left for the ith equation. See p.33.
% Uibar: cell(nvar,1). In each cell, we have nvar*nStates-by-qi*nStates, rearranged
% from Uistar or Ui. See p.33.
% nvar: Number of endogeous variables.
% nStates: NUmber of states.
% n0: nvar-by-1. n0(i)=qi where ith element represents the number of free A0 parameters in ith equation for each state.
% Tkave: nStates-by-1 of sample sizes (excluding lags but including dummies if provided) for different states k,
% averaged (ave) over E-step draws. For T_k. See p.38.
% Sgm0tldinvave: nvar*nStates-by-nvar*nStates. The matrix inv(\Sigma~_0) averaged (ave)
% over E-step draws. Same for all equations because of no asymmetric prior and no lag
% restrictions. Resembles old SpH in the exponent term in posterior of A0,
% but NOT divided by fss (T) yet. See p.38.
%----------------
% g: sum(n0)*nStates-by-1 analytical gradient for fn_a0sfreefun.m.
% Vectorized frmo the sum(n0)-by-nStates matrix.
% badg: 0, the value that is used in csminwel.m.
%
% Tao Zha, March 2001
%
% Copyright (C) 1997-2012 Tao Zha
%
% This 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.
%
% It 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.
%
% If you did not received a copy of the GNU General Public License
% with this software, see <http://www.gnu.org/licenses/>.
%
n0=n0(:);
A0=zeros(nvar,nvar,nStates);
n0cum = [0;cumsum(n0)];
b=reshape(b,n0cum(end),nStates);
g = zeros(n0cum(end),nStates); % which will be vectorized later.
badg = 0;
%*** The derivative of the exponential term w.r.t. each free paramater
for kj = 1:nvar
lenbjs=length(n0cum(kj)+1:n0cum(kj+1));
bj = zeros(nStates*lenbjs,1);
for si=1:nStates
bj((si-1)*lenbjs+1:si*lenbjs) = b(n0cum(kj)+1:n0cum(kj+1),si); % bj(si). See p.34a.
A0(:,kj,si) = Uistar{kj}((si-1)*nvar+1:si*nvar,:)*b(n0cum(kj)+1:n0cum(kj+1),si);
end
gj = (Uibar{kj}'*Sgm0tldinvave*Uibar{kj})*bj;
for si=1:nStates
g(n0cum(kj)+1:n0cum(kj+1),si) = gj((si-1)*lenbjs+1:si*lenbjs);
end
end
%*** Add the derivative of -T_klog|A0(k)| w.r.t. each free paramater
for si=1:nStates % See p.34a.
B=inv(A0(:,:,si)');
for ki = 1:n0cum(end) % from 1 to sum(q_i)
n = max(find( (ki-n0cum)>0 )); % note, 1<=n<=nvar equations.
g(ki,si) = g(ki,si) - Tkave(si)*B(:,n)'*Uistar{n}((si-1)*nvar+1:si*nvar,ki-n0cum(n)); % See p.34a.
end
end
g = g(:); % vectorized the same way as b is vectorized.
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