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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, 59 Temple Place - Suite 330, Boston, MA |
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18 ## 02111-1307, USA. |
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19 |
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20 ## @deftypefn {Function File} {[@var{beta}, @var{v}, @var{r}] =} gls (@var{y}, @var{x}, @var{o}) |
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21 ## Generalized least squares estimation for the multivariate model |
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22 ## @iftex |
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23 ## @tex |
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24 ## $y = x b + e$ |
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25 ## with $\bar{e} = 0$ and cov(vec($e$)) = $(s^2)o$, |
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26 ## @end tex |
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27 ## @end iftex |
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28 ## @ifinfo |
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29 ## @code{@var{y} = @var{x} * @var{b} + @var{e}} with @code{mean (@var{e}) = |
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30 ## 0} and @code{cov (vec (@var{e})) = (@var{s}^2)*@var{o}}, |
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31 ## @end ifinfo |
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32 ## where |
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33 ## @iftex |
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34 ## @tex |
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35 ## $y$ is a $t \times p$ matrix, $x$ is a $t \times k$ matrix, $b$ is a $k |
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36 ## \times p$ matrix, $e$ is a $t \times p$ matrix, and $o$ is a $tp \times |
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37 ## tp$ matrix. |
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38 ## @end tex |
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39 ## @end iftex |
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40 ## @ifinfo |
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41 ## @var{Y} is a @var{T} by @var{p} matrix, @var{X} is a @var{T} by @var{k} |
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42 ## matrix, @var{B} is a @var{k} by @var{p} matrix, @var{E} is a @var{T} by |
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43 ## @var{p} matrix, and @var{O} is a @var{T}@var{p} by @var{T}@var{p} |
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44 ## matrix. |
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45 ## @end ifinfo |
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46 ## |
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47 ## @noindent |
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48 ## Each row of Y and X is an observation and each column a variable. |
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49 ## |
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50 ## The return values @var{beta}, @var{v}, 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 GLS estimator for @var{b}. |
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56 ## |
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57 ## @item v |
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58 ## The GLS estimator for @code{@var{s}^2}. |
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59 ## |
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60 ## @item r |
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61 ## The matrix of GLS residuals, @code{@var{r} = @var{y} - @var{x} * |
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62 ## @var{beta}}. |
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63 ## @end table |
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64 ## @end deftypefn |
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65 |
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66 ## Author: Teresa Twaroch <twaroch@ci.tuwien.ac.at> |
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67 ## Created: May 1993 |
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68 ## Adapted-By: jwe |
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69 |
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70 function [BETA, v, R] = gls (Y, X, O) |
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71 |
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72 if (nargin != 3) |
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73 usage ("[BETA, v [, R]] = gls (Y, X, O)"); |
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74 endif |
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75 |
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76 [rx, cx] = size (X); |
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77 [ry, cy] = size (Y); |
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78 if (rx != ry) |
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79 error ("gls: incorrect matrix dimensions"); |
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80 endif |
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81 |
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82 O = O^(-1/2); |
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83 Z = kron (eye (cy), X); |
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84 Z = O * Z; |
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85 Y1 = O * reshape (Y, ry*cy, 1); |
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86 U = Z' * Z; |
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87 r = rank (U); |
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88 |
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89 if (r == cx*cy) |
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90 B = inv (U) * Z' * Y1; |
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91 else |
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92 B = pinv (Z) * Y1; |
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93 endif |
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94 |
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95 BETA = reshape (B, cx, cy); |
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96 R = Y - X * BETA; |
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97 v = (reshape (R, ry*cy, 1))' * (O^2) * reshape (R, ry*cy, 1) / (rx*cy - r); |
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98 |
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99 endfunction |
