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########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {[@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 @nospell{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
function [a, b] = arch_fit (y, x, p, iter, gamma, a0, b0)
if (nargin < 3 || nargin == 6)
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 -= a(j+1) * tmp(j+1:T-p+j);
endfor
r = 1 ./ h(1:T-p);
for j = 1:p
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 += 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 += gamma * delta_a;
endfor
endfunction
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