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function [Bh,e,xtx,phi,y] = syed(z,nn)
% syed: estimate a system of equations: [Bh,e,xtx,phi,y] = syed(z,nn)
% Y((T-lags)*nvar) = XB + u, X: (T-lags)*k, B: k*nvar.
% where z is the T*(nvar+ndt) raw data matrix (nvar of variables +
% ndt -- number of deterministic terms);
% nn is 5 inputs [auindx, ndt, nvar,lags,sample period (total)];
% auindx = 0 (no autoregressive) and 1 (autoregressive);
% total -- including lags, etc.
% Bh: the estimated B; column: nvar; row: [nvar for 1st lag, ...,
% nvar for last lag, deterministic terms (ndt)]
% e: estimated residual e = y -xBh, (T-lags)*nvar
% xtx: X'X
% phi: X; column: [nvar for 1st lag, ...,
% nvar for last lag, deterministic terms (ndt)]
% y: Y
%
% See also "sye.m".
%
% 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/>.
%
% ** setup of orders and lengths **
auindx = nn(1);
ndt = nn(2); % # of deterministic terms including constant
nvar = nn(3); % # of endogenous variables
lags = nn(4);
sp = nn(5); % sample period
ess = sp-lags; % effective sample size
sb = lags+1; % sample beginning
sl = sp; % sample last period
% ** construct X for Y = X*B + U where phi = X **
x = z(:,1:nvar);
C = z(:,nvar+1:nvar+ndt); % C = [] when ndt=0
%
if auindx == 0
ncoe = ndt; % with deterministic terms
phi = zeros(sp,ncoe); % preallocating
y = x;
else
y = x(sb:sl,:);
ncoe = nvar*lags + ndt; % with deterministic terms
phi = zeros(ess,ncoe); % preallocating
for k=1:lags, phi(:,nvar*(k-1)+1:nvar*k) = x(sb-k:sl-k,:); end
end
%
if length(C) == 0
phi(:,ncoe-ndt+1:ncoe) = C;
else
phi(:,ncoe-ndt+1:ncoe) = C(1:sp,:); % perhaps, it should have been be C(sb:sp,:). 2/24/00
end
%
% row: T-lags; column: [nvar for 1st lag, ..., nvar for last lag,
% deterministic terms (ndt)]
% Thus, # of columns is nvar*lags+ndt = ncoe.
% ** estimate: B, XTX, residuals **
[u d v]=svd(phi,0); %trial
%xtx = phi'*phi; % X'X, k*k (ncoe*ncoe)
vd=v.*(ones(size(v,2),1)*diag(d)'); %trial
dinv = 1./diag(d); % inv(diag(d))
vdinv=v.*(ones(size(v,2),1)*dinv'); %trial
xtx=vd*vd';
xtxinv = vdinv*vdinv';
%xty = phi'*y; % X'Y
uy = u'*y; %trial
xty = vd*uy; %trial
%Bh = xtx\xty; %inv(X'X)*(X'Y), k*m (ncoe*nvar).
Bh = xtxinv*xty;
%e = y - phi*Bh; % from Y = XB + U, e: (T-lags)*nvar
e = y - u*uy;
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