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1 ## Copyright (C) 1996, 1997 John W. Eaton |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 2, or (at your option) |
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8 ## any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {[@var{beta}, @var{sigma}, @var{r}] =} ols (@var{y}, @var{x}) |
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22 ## Ordinary least squares estimation for the multivariate model |
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23 ## @iftex |
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24 ## @tex |
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25 ## $y = x b + e$ |
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26 ## with |
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27 ## $\bar{e} = 0$, and cov(vec($e$)) = kron ($s, I$) |
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28 ## @end tex |
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29 ## @end iftex |
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30 ## @ifinfo |
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31 ## @math{y = x b + e} with |
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32 ## @math{mean (e) = 0} and @math{cov (vec (e)) = kron (s, I)}. |
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33 ## @end ifinfo |
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34 ## where |
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35 ## @iftex |
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36 ## @tex |
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37 ## $y$ is a $t \times p$ matrix, $x$ is a $t \times k$ matrix, |
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38 ## $b$ is a $k \times p$ matrix, and $e$ is a $t \times p$ matrix. |
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39 ## @end tex |
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40 ## @end iftex |
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41 ## @ifinfo |
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42 ## @math{y} is a @math{t} by @math{p} matrix, @math{x} is a @math{t} by |
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43 ## @math{k} matrix, @math{b} is a @math{k} by @math{p} matrix, and |
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44 ## @math{e} is a @math{t} by @math{p} matrix. |
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45 ## @end ifinfo |
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46 ## |
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47 ## Each row of @var{y} and @var{x} is an observation and each column a |
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48 ## variable. |
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49 ## |
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50 ## The return values @var{beta}, @var{sigma}, and @var{r} are defined as |
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51 ## follows. |
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52 ## |
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53 ## @table @var |
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54 ## @item beta |
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55 ## The OLS estimator for @var{b}, @code{@var{beta} = pinv (@var{x}) * |
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56 ## @var{y}}, where @code{pinv (@var{x})} denotes the pseudoinverse of |
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57 ## @var{x}. |
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58 ## |
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59 ## @item sigma |
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60 ## The OLS estimator for the matrix @var{s}, |
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61 ## |
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62 ## @example |
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63 ## @group |
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64 ## @var{sigma} = (@var{y}-@var{x}*@var{beta})' |
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65 ## * (@var{y}-@var{x}*@var{beta}) |
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66 ## / (@var{t}-rank(@var{x})) |
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67 ## @end group |
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68 ## @end example |
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69 ## |
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70 ## @item r |
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71 ## The matrix of OLS residuals, @code{@var{r} = @var{y} - @var{x} * |
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72 ## @var{beta}}. |
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73 ## @end table |
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74 ## @end deftypefn |
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75 |
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76 ## Author: Teresa Twaroch <twaroch@ci.tuwien.ac.at> |
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77 ## Created: May 1993 |
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78 ## Adapted-By: jwe |
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79 |
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80 function [BETA, SIGMA, R] = ols (Y, X) |
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81 |
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82 if (nargin != 2) |
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83 usage ("[BETA, SIGMA, R] = ols (Y, X)"); |
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84 endif |
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85 |
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86 [nr, nc] = size (X); |
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87 [ry, cy] = size (Y); |
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88 if (nr != ry) |
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89 error ("ols: incorrect matrix dimensions"); |
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90 endif |
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91 |
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92 Z = X' * X; |
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93 r = rank (Z); |
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94 |
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95 if (r == nc) |
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96 BETA = inv (Z) * X' * Y; |
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97 else |
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98 BETA = pinv (X) * Y; |
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99 endif |
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100 |
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101 R = Y - X * BETA; |
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102 SIGMA = R' * R / (nr - r); |
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103 |
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104 endfunction |
