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function [g,badg] = fn_a0cfreegrad_tv(b,nvar,nStates,n0cumsum,Ui,Tkave,Del00invcell_ave,dpDelp0cell_ave)
% [g,badg] = fn_a0cfreegrad_tv(b,nvar,nStates,n0cumsum,Ui,Tkave,Del00invcell_ave,dpDelp0cell_ave)
% Analytical gradient for fn_a0cfreegrad_tv.m when using csminwel.m.
% Note: (1) columns correspond to equations; (2) c stands for constant.
% See TBVAR NOTE pp. 61-62, 34a-34c.
%
% b: n0cumsum(end)-by-1 vector of free constant A0 parameters, vectorized from b_ihatcell.
% nvar: Number of endogeous variables.
% nStates: Number of states.
% n0cumsum: [0;cumsum(n0)] where n0 is nvar-by-1 and its ith element represents the number of
% free constant A0 parameters in ith equation for *all states*.
% Ui: nvar-by-1 cell. In each cell, nvar*nStates-by-(qi+si) orthonormal basis for the null of the ith
% equation contemporaneous restriction matrix where qi is the number of free parameters
% within the state and si is the number of free parameters across the states.
% 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.
% 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.61.
% Del00invcell_ave: Quardratic term for b0_j in each cell. See p.61.
% dpDelp0cell_ave: Cross d+_j and b0_j term in each cell. See p.61.
%----------------
% g: n0cumsum(end)-by-1 analytical gradient for fn_a0cfreegrad_tv.m.
% badg: 0, the value that is used in csminwel.m.
%
% Tao Zha, September 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/>.
%
A0_shat=zeros(nvar,nvar,nStates); % Arranged A0 matrix across states.
B_shat=zeros(nvar,nvar,nStates); % inv(A0_shat);
badg = 0;
g = zeros(size(b(:)));
%**** The derivative of the exponential term w.r.t. each free constanta A0 paramater. See pp. 34a and 62.
for kj = 1:nvar
indxn0j = [n0cumsum(kj)+1:n0cumsum(kj+1)]; % Index for the parameters in the ith equation.
bj = b(indxn0j);
g(indxn0j) = Del00invcell_ave{kj}*bj - dpDelp0cell_ave{kj}';
A0_shat(:,kj,:) = reshape(Ui{kj}*bj,nvar,nStates);
end
%for si=1:nStates % See p.34a.
%end
%**** Add the derivative of -Tk_ave*log|A0(k)| w.r.t. each free constanta A0 paramater. See pp. 62 and 34a.
for kj = 1:nvar
indxn0j = [n0cumsum(kj)+1:n0cumsum(kj+1)]; % Index for the parameters in the ith equation.
for si=1:nStates % See p.34a.
B_shat(:,:,si)=inv(A0_shat(:,:,si)); % See p.62.
g(indxn0j) = g(indxn0j) - Tkave(si)*( B_shat(kj,:,si)*Ui{kj}((si-1)*nvar+1:si*nvar,:) )'; % See p.62.
end
end
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