% -*- 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