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function [Bh,e,xtx,xty] = fn_ols(y,phi)
% [Bh,e,xtx,xty] = fn_ols(y,phi)
% ols: estimate a system of equations: Y(T*nvar) = XB + u, X: T*k, B: k*nvar.
%
% y: Y: T-by-nvar
% phi: X; T-by-k; column: number of r.h.s. variables (including
%------------
% Bh: the estimated B; column: nvar; row: number of r.h.s. variables.
% e: estimated residual e = y -xBh, T*nvar
% xtx: X'X: k-by-k
% xty: X'Y: k-by-nvar
% deterministic terms)%
% See also sye.m and syed.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 **
[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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