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function [Bh,e,xtx,xty,phi,y,ncoe,xr] = sye(z,lags)
% Now [Bh,e,xtx,xty,phi,y,ncoe,xr] = sye(z,lags)
% Old: [Bh,e,xtx,xty,phi,y,ncoe,Sigu,xtxinv] = sye(z,lags)
%
% Estimate a system of equations in the form of y(T*nvar) = XB + u,
% X--phi: T*k, B: k*nvar; where T=sp-lags, k=ncoe,
%
% z: (T+lags)-by-(nvar+1) raw data matrix (nvar of variables + constant).
% lags: number of lags
%--------------------
% Bh: k-by-nvar estimated reduced-form parameter; column: nvar;
% row: k=ncoe=[nvar for 1st lag, ..., nvar for last lag, const]
% e: estimated residual e = y -xBh, T-by-nvar
% xtx: X'X: k-by-k
% xty: X'Y: k-by-nvar
% phi: X; T-by-k; column: [nvar for 1st lag, ..., nvar for last lag, const]
% y: Y: T-by-nvar
% ncoe: number of coeffcients per equation: nvar*lags + 1
% xr: the economy size (k-by-k) in qr(phi) so that xr=chol(X'*X)
% Sigu: e'*e: nvar-by-nvar. Note, not divided (undivided) by "nobs"
% xtxinv: inv(X'X): k-by-k
%
% See also syed.m (allowing for more predetermined terms) which has not
% been yet updated as "sye.m".
%
% Note, "lags" is something I changed recently, so it may not be compatible
% with old programs, 10/15/98 by TAZ.
%
% Revised, 5/2/99. Replaced outputs Sigu and xtxinv with xr so that previous
% programs may be incompatible.
%
% 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 **
[sp,nvar] = size(z); % sp: sample period T include lags
nvar = nvar-1; % -1: takes out the counting of constant
ess = sp-lags; % effective sample size
sb = lags+1; % sample beginning
sl = sp; % sample last period
ncoe = nvar*lags + 1; % with constant
% ** construct X for Y = X*B + U where phi = X **
x = z(:,1:nvar);
C = z(:,nvar+1);
phi = zeros(ess,ncoe);
phi(:,ncoe) = C(1:ess);
for k=1:lags, phi(:,nvar*(k-1)+1:nvar*k) = x(sb-k:sl-k,:); end
% row: T-lags; column: [nvar for 1st lag, ..., nvar for last lag, const]
% Thus, # of columns is nvar*lags+1 = ncoef.
% ** estimate: B, XTX, residuals **
y = x(sb:sl,:);
%
%**** The following, though stable, is too slow *****
% [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;
%**** The following, though stable, is too slow *****
%===== (Fast but perhaps less accurate) alternative to the above =========
[xq,xr]=qr(phi,0);
xtx=xr'*xr;
xty=phi'*y;
Bh = xr\(xr'\xty);
e=y-phi*Bh;
%===== (Fast but perhaps less accurate) alternative to the above =========
%* not numerically stable way of computing e'*e
%Sigu = y'*y-xty'*Bh;
%Sigu = y'*(eye(ess)-phi*xtxinv*phi')*y; % probablly better way, commented out
% by TZ, 2/28/98. See following
%Sigu = y'*(eye(ess)-u*u')*y; % I think this is the best, TZ, 2/28/98
% Note, u*u'=x*inv(x'x)*x.
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