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function g = fn_tran_f2g(F,Vi,nvar,np)
% g = fn_tran_f2g(F,Vi,nvar,np)
% Transform F (A+) to free parameters g's. Note: columns correspond to equations
% See Waggoner and Zha's ``A Gibbs sampler for structural VARs''
%
% F: ncoef-by-nvar matrix of original lagged parameters A+. Column corresponding to equation.
% Note that ncoef is the number of original lagged variables per equation
% Vi: nvar-by-1 cell. In each cell, k-by-ri orthonormal basis for the null of the ith
% equation lagged restriction matrix where k is a total of exogenous variables and
% ri is the number of free parameters. With this transformation, we have fi = Vi*gi
% or Vi'*fi = gi where fi is a vector of total original parameters and gi is a
% vector of free parameters. There must be at least one free parameter left for
% the ith equation.
% nvar: number of endogeous variables
% np: nvar-element vector, ith element represents the number of free A+ parameters in ith equation
%---------------
% g: sum(np)-by-1 stacked vector of all free lagged parameters A+.
%
% August 2000, Tao Zha
%
% 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/>.
%
np=np(:);
npcum = [0;cumsum(np)];
g = zeros(npcum(end),1);
for kj=1:nvar
g(npcum(kj)+1:npcum(kj+1)) = Vi{kj}'*F(:,kj);
end
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