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function A0gbs = fn_gibbsrvar(A0gbs,UT,nvar,fss,n0,Indxcol)
% A0gbs = fn_gibbsrvar(A0gbs,UT,nvar,fss,n0,Indxcol)
% One-step Gibbs sampler for restricted VARs in the structural form
% Ref.: D.F. Waggoner and T. Zha: ``A Gibbs sampler for structural VARs''
% See Note Forecast (2) pp. 44-51 and Theorem 1 and Section 3 in the WZ paper.
%
% A0gbs: the last draw of A0 matrix
% UT: cell(nvar,1) -- U_i*T_i in the proof of Theorem 1 where
% (1) a_i = U_i*b_i with b_i being a vector of free parameters
% (2) T_i (q_i-by-q_i) is from T_i*T_i'=H0inv{i}/T. Note that H0inv is the inverse of
% the covariance martrix NOT divided by fss. See Theorem 1.
% nvar: rank of A0 or # of variables
% fss: effective sample size == nSample (T)-lags+# of dummy observations
% n0: nvar-by-1, ith element represents the number of free A0 parameters in ith equation
% Indxcol: a row vector indicating random columns this Gibbs draws.
% When this input is not supplied, the Gibbs draws all columns
%------------------
% A0bgs: new draw of A0 matrix in this Gibbs step
%
% Written by Tao Zha, August 2000
%
%
% Copyright (C) 1997-2012 Daniel Waggoner and 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/>.
%
if (nargin==5), Indxcol=[1:nvar]; end
%---------------- Local loop for Gibbs given last A0gbs ----------
%
w = zeros(nvar,1);
for ki=Indxcol % given last A0gbs and generate new A0bgs
X = A0gbs; % WZ's Section 4.3
X(:,ki) = 0; % want to find non-zero sw s.t., X'*w=0
%*** Solving for w and getting an orthonormal basis. See Theorem 1 and also p.48 in Forecast II
[jL,Ux] = lu(X');
jIx0 = min(find(abs(diag(Ux))<eps)); % if isempty(jIx0), then something is wrong here
w(jIx0+1:end) = 0; % if jIx0+1>end, no effect on w0
w(jIx0) = 1;
jA = Ux(1:jIx0-1,1:jIx0-1);
jb = Ux(1:jIx0-1,jIx0);
jy = -jA\jb;
w(1:jIx0-1) = jy;
% Note: if jIx0=1 (which almost never happens for numerical stability reasons), no effect on w.
%*** Constructing orthonormal basis w_1, ..., w_qi at each Gibbs step
w0 = UT{ki}'*w;
w1 = w0/sqrt(sum(w0.^2));
[W,jnk] = qr(w1); % columns of W form an orthonormal basis w_1,...,w_qi in Section 4.2 in the WZ paper
%*** Draw beta's in Theorem 1
gkbeta = zeros(n0(ki),1); % qi-by-1: greak beta's
jstd = sqrt(1/fss);
gkbeta(2:end) = jstd*randn(n0(ki)-1,1); % for beta_2, ..., beta_qi
%--- Unnormalized (i.e., not normalized) gamma or 1-d Wishart draw of beta_1 in Theorem 1.
%* gamma or 1-d Wishart draw of beta_1
jr = jstd*randn(fss+1,1);
if rand(1)<0.5
gkbeta(1) = sqrt(jr'*jr);
else
gkbeta(1) = -sqrt(jr'*jr);
end
%*** Getting a new ki_th column in A0
A0gbs(:,ki) = UT{ki}*(W*gkbeta); % n-by-1
end
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