update packages

[CreaPhase.git] / octave_packages / m / signal / arch_fit.m

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+## Copyright (C) 1995-2012 Kurt Hornik
+##
+## 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{a}, @var{b}] =} arch_fit (@var{y}, @var{x}, @var{p}, @var{iter}, @var{gamma}, @var{a0}, @var{b0})
+## Fit an ARCH regression model to the time series @var{y} using the
+## scoring algorithm in Engle's original ARCH paper.  The model is
+##
+## @example
+## @group
+## y(t) = b(1) * x(t,1) + @dots{} + b(k) * x(t,k) + e(t),
+## h(t) = a(1) + a(2) * e(t-1)^2 + @dots{} + a(p+1) * e(t-p)^2
+## @end group
+## @end example
+##
+## @noindent
+## in which @math{e(t)} is @math{N(0, h(t))}, given a time-series vector
+## @var{y} up to time @math{t-1} and a matrix of (ordinary) regressors
+## @var{x} up to @math{t}.  The order of the regression of the residual
+## variance is specified by @var{p}.
+##
+## If invoked as @code{arch_fit (@var{y}, @var{k}, @var{p})} with a
+## positive integer @var{k}, fit an ARCH(@var{k}, @var{p}) process,
+## i.e., do the above with the @math{t}-th row of @var{x} given by
+##
+## @example
+## [1, y(t-1), @dots{}, y(t-k)]
+## @end example
+##
+## Optionally, one can specify the number of iterations @var{iter}, the
+## updating factor @var{gamma}, and initial values @math{a0} and
+## @math{b0} for the scoring algorithm.
+## @end deftypefn
+
+## Author: KH <Kurt.Hornik@wu-wien.ac.at>
+## Description: Fit an ARCH regression model
+
+function [a, b] = arch_fit (y, x, p, iter, gamma, a0, b0)
+
+  if ((nargin < 3) || (nargin == 6) || (nargin > 7))
+    print_usage ();
+  endif
+
+  if (! (isvector (y)))
+    error ("arch_fit: Y must be a vector");
+  endif
+
+  T   = length (y);
+  y   = reshape (y, T, 1);
+  [rx, cx] = size (x);
+  if ((rx == 1) && (cx == 1))
+    x = autoreg_matrix (y, x);
+  elseif (! (rx == T))
+    error ("arch_fit: either rows (X) == length (Y), or X is a scalar");
+  endif
+
+  [T, k] = size (x);
+
+  if (nargin == 7)
+    a   = a0;
+    b   = b0;
+    e   = y - x * b;
+  else
+    [b, v_b, e] = ols (y, x);
+    a   = [v_b, (zeros (1, p))]';
+    if (nargin < 5)
+      gamma = 0.1;
+      if (nargin < 4)
+        iter = 50;
+      endif
+    endif
+  endif
+
+  esq = e.^2;
+  Z   = autoreg_matrix (esq, p);
+
+  for i = 1 : iter;
+    h    = Z * a;
+    tmp  = esq ./ h.^2 - 1 ./ h;
+    s    = 1 ./ h(1:T-p);
+    for j = 1 : p;
+      s = s - a(j+1) * tmp(j+1:T-p+j);
+    endfor
+    r    = 1 ./ h(1:T-p);
+    for j = 1:p;
+      r = r + 2 * h(j+1:T-p+j).^2 .* esq(1:T-p);
+    endfor
+    r   = sqrt (r);
+    X_tilde = x(1:T-p, :) .* (r * ones (1,k));
+    e_tilde = e(1:T-p) .*s ./ r;
+    delta_b = inv (X_tilde' * X_tilde) * X_tilde' * e_tilde;
+    b   = b + gamma * delta_b;
+    e   = y - x * b;
+    esq = e .^ 2;
+    Z   = autoreg_matrix (esq, p);
+    h   = Z * a;
+    f   = esq ./ h - ones(T,1);
+    Z_tilde = Z ./ (h * ones (1, p+1));
+    delta_a = inv (Z_tilde' * Z_tilde) * Z_tilde' * f;
+    a   = a + gamma * delta_a;
+  endfor
+
+endfunction