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

% ** 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;