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1 ## Copyright (C) 1995, 1996, 1997 Kurt Hornik |
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2 ## |
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3 ## This program is free software; you can redistribute it and/or modify |
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4 ## it under the terms of the GNU General Public License as published by |
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5 ## the Free Software Foundation; either version 2, or (at your option) |
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6 ## any later version. |
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7 ## |
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8 ## This program is distributed in the hope that it will be useful, but |
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9 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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11 ## General Public License for more details. |
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12 ## |
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13 ## You should have received a copy of the GNU General Public License |
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14 ## along with this file. If not, write to the Free Software Foundation, |
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15 ## 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
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16 |
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17 ## usage: [a, b] = arch_fit (y, X, p [, ITER [, gamma [, a0, b0]]]) |
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18 ## |
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19 ## Fits an ARCH regression model to the time series y using the scoring |
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20 ## algorithm in Engle's original ARCH paper. The model is |
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21 ## y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t), |
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22 ## h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(p+1) * e(t-p)^2, |
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23 ## where e(t) is N(0, h(t)), given y up to time t-1 and X up to t. |
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24 ## |
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25 ## If invoked as arch_fit (y, k, p) with a positive integer k, fit an |
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26 ## ARCH(k,p) process, i.e., do the above with the t-th row of X given by |
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27 ## [1, y(t-1), ..., y(t-k)]. |
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28 ## |
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29 ## Optionally, one can specify the number of iterations ITER, the |
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30 ## updating factor gamma, and initial values a0 and b0 for the scoring |
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31 ## algorithm. |
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32 ## |
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33 ## The input parameters are: |
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34 ## y ... time series (vector) |
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35 ## X ... matrix of (ordinary) regressors or order of |
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36 ## autoregression |
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37 ## p ... order of the regression of the residual variance |
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38 |
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39 ## Author: KH <Kurt.Hornik@ci.tuwien.ac.at> |
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40 ## Description: Fit an ARCH regression model |
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41 |
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42 function [a, b] = arch_fit (y, X, p, ITER, gamma, a0, b0) |
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43 |
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44 if ((nargin < 3) || (nargin == 6) || (nargin > 7)) |
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45 usage ("arch_fit (y, X, p [, ITER [, gamma [, a0, b0]]])"); |
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46 endif |
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47 |
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48 if !(is_vector (y)) |
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49 error ("arch_test: y must be a vector"); |
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50 endif |
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51 |
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52 T = length (y); |
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53 y = reshape (y, T, 1); |
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54 [rx, cx] = size (X); |
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55 if ((rx == 1) && (cx == 1)) |
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56 X = autoreg_matrix (y, X); |
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57 elseif !(rx == T) |
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58 error (["arch_test: ", ... |
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59 "either rows (X) == length (y), or X is a scalar"]); |
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60 endif |
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61 |
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62 [T, k] = size (X); |
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63 |
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64 if (nargin == 7) |
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65 a = a0; |
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66 b = b0; |
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67 e = y - X * b; |
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68 else |
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69 [b, v_b, e] = ols (y, X); |
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70 a = [v_b, zeros (1,p)]'; |
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71 if (nargin < 5) |
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72 gamma = 0.1; |
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73 if (nargin < 4) |
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74 ITER = 50; |
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75 endif |
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76 endif |
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77 endif |
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78 |
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79 esq = e.^2; |
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80 Z = autoreg_matrix (esq, p); |
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81 |
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82 for i = 1 : ITER; |
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83 h = Z * a; |
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84 tmp = esq ./ h.^2 - 1 ./ h; |
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85 s = 1 ./ h(1:T-p); |
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86 for j = 1 : p; |
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87 s = s - a(j+1) * tmp(j+1:T-p+j); |
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88 endfor |
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89 r = 1 ./ h(1:T-p); |
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90 for j=1:p; |
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91 r = r + 2 * h(j+1:T-p+j).^2 .* esq(1:T-p); |
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92 endfor |
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93 r = sqrt (r); |
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94 X_tilde = X(1:T-p, :) .* (r * ones (1,k)); |
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95 e_tilde = e(1:T-p) .*s ./ r; |
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96 delta_b = inv (X_tilde' * X_tilde) * X_tilde' * e_tilde; |
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97 b = b + gamma * delta_b; |
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98 e = y - X * b; |
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99 esq = e .^ 2; |
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100 Z = autoreg_matrix (esq, p); |
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101 h = Z * a; |
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102 f = esq ./ h - ones(T,1); |
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103 Z_tilde = Z ./ (h * ones (1, p+1)); |
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104 delta_a = inv (Z_tilde' * Z_tilde) * Z_tilde' * f; |
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105 a = a + gamma * delta_a; |
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106 endfor |
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107 |
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108 endfunction |
