Fit 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)^2in which e(t) is N(0, h(t)), given a time-series vector y up to time t-1 and a matrix of (ordinary) regressors x up to t. The order of the regression of the residual variance is specified by p.
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.