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1 ## Copyright (C) 1995, 1996, 1997, 1998, 1999, 2000, 2002, 2005, 2006, |
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2 ## 2007 Kurt Hornik |
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3 ## |
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4 ## This file is part of Octave. |
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5 ## |
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6 ## Octave is free software; you can redistribute it and/or modify it |
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7 ## under the terms of the GNU General Public License as published by |
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8 ## the Free Software Foundation; either version 3 of the License, or (at |
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9 ## your option) any later version. |
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10 ## |
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11 ## Octave is distributed in the hope that it will be useful, but |
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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14 ## General Public License for more details. |
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15 ## |
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16 ## You should have received a copy of the GNU General Public License |
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17 ## along with Octave; see the file COPYING. If not, see |
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18 ## <http://www.gnu.org/licenses/>. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {[@var{a}, @var{b}] =} arch_fit (@var{y}, @var{x}, @var{p}, @var{iter}, @var{gamma}, @var{a0}, @var{b0}) |
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22 ## Fit an ARCH regression model to the time series @var{y} using the |
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23 ## scoring algorithm in Engle's original ARCH paper. The model is |
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24 ## |
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25 ## @example |
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26 ## y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t), |
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27 ## h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(p+1) * e(t-p)^2 |
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28 ## @end example |
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29 ## |
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30 ## @noindent |
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31 ## in which @math{e(t)} is @math{N(0, h(t))}, given a time-series vector |
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32 ## @var{y} up to time @math{t-1} and a matrix of (ordinary) regressors |
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33 ## @var{x} up to @math{t}. The order of the regression of the residual |
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34 ## variance is specified by @var{p}. |
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35 ## |
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36 ## If invoked as @code{arch_fit (@var{y}, @var{k}, @var{p})} with a |
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37 ## positive integer @var{k}, fit an ARCH(@var{k}, @var{p}) process, |
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38 ## i.e., do the above with the @math{t}-th row of @var{x} given by |
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39 ## |
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40 ## @example |
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41 ## [1, y(t-1), ..., y(t-k)] |
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42 ## @end example |
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43 ## |
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44 ## Optionally, one can specify the number of iterations @var{iter}, the |
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45 ## updating factor @var{gamma}, and initial values @math{a0} and |
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46 ## @math{b0} for the scoring algorithm. |
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47 ## @end deftypefn |
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48 |
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49 ## Author: KH <Kurt.Hornik@wu-wien.ac.at> |
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50 ## Description: Fit an ARCH regression model |
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51 |
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52 function [a, b] = arch_fit (y, X, p, ITER, gamma, a0, b0) |
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53 |
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54 if ((nargin < 3) || (nargin == 6) || (nargin > 7)) |
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55 print_usage (); |
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56 endif |
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57 |
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58 if (! (isvector (y))) |
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59 error ("arch_test: y must be a vector"); |
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60 endif |
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61 |
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62 T = length (y); |
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63 y = reshape (y, T, 1); |
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64 [rx, cx] = size (X); |
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65 if ((rx == 1) && (cx == 1)) |
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66 X = autoreg_matrix (y, X); |
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67 elseif (! (rx == T)) |
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68 error ("arch_test: either rows (X) == length (y), or X is a scalar"); |
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69 endif |
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70 |
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71 [T, k] = size (X); |
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72 |
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73 if (nargin == 7) |
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74 a = a0; |
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75 b = b0; |
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76 e = y - X * b; |
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77 else |
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78 [b, v_b, e] = ols (y, X); |
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79 a = [v_b, (zeros (1, p))]'; |
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80 if (nargin < 5) |
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81 gamma = 0.1; |
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82 if (nargin < 4) |
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83 ITER = 50; |
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84 endif |
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85 endif |
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86 endif |
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87 |
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88 esq = e.^2; |
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89 Z = autoreg_matrix (esq, p); |
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90 |
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91 for i = 1 : ITER; |
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92 h = Z * a; |
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93 tmp = esq ./ h.^2 - 1 ./ h; |
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94 s = 1 ./ h(1:T-p); |
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95 for j = 1 : p; |
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96 s = s - a(j+1) * tmp(j+1:T-p+j); |
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97 endfor |
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98 r = 1 ./ h(1:T-p); |
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99 for j = 1:p; |
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100 r = r + 2 * h(j+1:T-p+j).^2 .* esq(1:T-p); |
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101 endfor |
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102 r = sqrt (r); |
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103 X_tilde = X(1:T-p, :) .* (r * ones (1,k)); |
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104 e_tilde = e(1:T-p) .*s ./ r; |
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105 delta_b = inv (X_tilde' * X_tilde) * X_tilde' * e_tilde; |
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106 b = b + gamma * delta_b; |
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107 e = y - X * b; |
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108 esq = e .^ 2; |
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109 Z = autoreg_matrix (esq, p); |
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110 h = Z * a; |
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111 f = esq ./ h - ones(T,1); |
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112 Z_tilde = Z ./ (h * ones (1, p+1)); |
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113 delta_a = inv (Z_tilde' * Z_tilde) * Z_tilde' * f; |
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114 a = a + gamma * delta_a; |
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115 endfor |
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116 |
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117 endfunction |
