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arch_fit.m

## Copyright (C) 1995, 1996, 1997  Kurt Hornik
## 
## This program 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 2, or (at your option)
## any later version.
## 
## This program 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 this file.  If not, write to the Free Software Foundation,
## 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

## usage:  [a, b] = arch_fit (y, X, p [, ITER [, gamma [, a0, b0]]]) 
##
## Fits an ARCH regression model to the time series y using the scoring
## algorithm in Engle's original ARCH paper.  The model is
##    y(t) = b(1) * x(t,1) + ... + b(k) * x(t,k) + e(t),
##    h(t) = a(1) + a(2) * e(t-1)^2 + ... + a(p+1) * e(t-p)^2,
## where e(t) is N(0, h(t)), given y up to time t-1 and X up to t.
##
## If invoked as arch_fit (y, k, p) with a positive integer k, fit an
## ARCH(k,p) process, i.e., do the above with the t-th row of X given by
## [1, y(t-1), ..., y(t-k)].
##
## Optionally, one can specify the number of iterations ITER, the
## updating factor gamma, and initial values a0 and b0 for the scoring
## algorithm.
##
## The input parameters are:
##    y     ... time series (vector)
##    X     ... matrix of (ordinary) regressors or order of
##              autoregression 
##    p     ... order of the regression of the residual variance

## Author:  KH <Kurt.Hornik@ci.tuwien.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))
    usage ("arch_fit (y, X, p [, ITER [, gamma [, a0, b0]]])");
  endif
  
  if !(is_vector (y))
    error ("arch_test:  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_test:  ", ...
          "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

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