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## Copyright (C) 1996-2013 John W. Eaton
##
## This file is part of Octave.
##
## Octave is 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.
##
## Octave 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.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{beta}, @var{sigma}, @var{r}] =} ols (@var{y}, @var{x})
## Ordinary least squares estimation for the multivariate model
## @tex
## $y = x b + e$
## with
## $\bar{e} = 0$, and cov(vec($e$)) = kron ($s, I$)
## @end tex
## @ifnottex
## @w{@math{y = x*b + e}} with
## @math{mean (e) = 0} and @math{cov (vec (e)) = kron (s, I)}.
## @end ifnottex
## where
## @tex
## $y$ is a $t \times p$ matrix, $x$ is a $t \times k$ matrix,
## $b$ is a $k \times p$ matrix, and $e$ is a $t \times p$ matrix.
## @end tex
## @ifnottex
## @math{y} is a @math{t} by @math{p} matrix, @math{x} is a @math{t} by
## @math{k} matrix, @math{b} is a @math{k} by @math{p} matrix, and
## @math{e} is a @math{t} by @math{p} matrix.
## @end ifnottex
##
## Each row of @var{y} and @var{x} is an observation and each column a
## variable.
##
## The return values @var{beta}, @var{sigma}, and @var{r} are defined as
## follows.
##
## @table @var
## @item beta
## The OLS estimator for @math{b}.
## @tex
## $beta$ is calculated directly via $(x^Tx)^{-1} x^T y$ if the matrix $x^Tx$ is
## of full rank.
## @end tex
## @ifnottex
## @var{beta} is calculated directly via @code{inv (x'*x) * x' * y} if the
## matrix @code{x'*x} is of full rank.
## @end ifnottex
## Otherwise, @code{@var{beta} = pinv (@var{x}) * @var{y}} where
## @code{pinv (@var{x})} denotes the pseudoinverse of @var{x}.
##
## @item sigma
## The OLS estimator for the matrix @var{s},
##
## @example
## @group
## @var{sigma} = (@var{y}-@var{x}*@var{beta})'
## * (@var{y}-@var{x}*@var{beta})
## / (@var{t}-rank(@var{x}))
## @end group
## @end example
##
## @item r
## The matrix of OLS residuals, @code{@var{r} = @var{y} - @var{x}*@var{beta}}.
## @end table
## @seealso{gls, pinv}
## @end deftypefn
## Author: Teresa Twaroch <twaroch@ci.tuwien.ac.at>
## Created: May 1993
## Adapted-By: jwe
function [beta, sigma, r] = ols (y, x)
if (nargin != 2)
print_usage ();
endif
if (! (isnumeric (x) && isnumeric (y)))
error ("ols: X and Y must be numeric matrices or vectors");
endif
if (ndims (x) != 2 || ndims (y) != 2)
error ("ols: X and Y must be 2-D matrices or vectors");
endif
[nr, nc] = size (x);
[ry, cy] = size (y);
if (nr != ry)
error ("ols: number of rows of X and Y must be equal");
endif
if (isinteger (x))
x = double (x);
endif
if (isinteger (y))
y = double (y);
endif
## Start of algorithm
z = x' * x;
[u, p] = chol (z);
if (p)
beta = pinv (x) * y;
else
beta = u \ (u' \ (x' * y));
endif
if (isargout (2) || isargout (3))
r = y - x * beta;
endif
if (isargout (2))
## z is of full rank, avoid the SVD in rnk
if (p == 0)
rnk = columns (z);
else
rnk = rank (z);
endif
sigma = r' * r / (nr - rnk);
endif
endfunction
%!test
%! x = [1:5]';
%! y = 3*x + 2;
%! x = [x, ones(5,1)];
%! assert (ols (y,x), [3; 2], 50*eps)
%!test
%! x = [1, 2; 3, 4];
%! y = [1; 2];
%! [b, s, r] = ols (x, y);
%! assert (b, [1.4, 2], 2*eps);
%! assert (s, [0.2, 0; 0, 0], 2*eps);
%! assert (r, [-0.4, 0; 0.2, 0], 2*eps);
%!test
%! x = [1, 2; 3, 4];
%! y = [1; 2];
%! [b, s] = ols (x, y);
%! assert (b, [1.4, 2], 2*eps);
%! assert (s, [0.2, 0; 0, 0], 2*eps);
%!test
%! x = [1, 2; 3, 4];
%! y = [1; 2];
%! b = ols (x, y);
%! assert (b, [1.4, 2], 2*eps);
%% Test input validation
%!error ols ()
%!error ols (1)
%!error ols (1, 2, 3)
%!error ols ([true, true], [1, 2])
%!error ols ([1, 2], [true, true])
%!error ols (ones (2,2,2), ones (2,2))
%!error ols (ones (2,2), ones (2,2,2))
%!error ols (ones (1,2), ones (2,2))
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