Successive determination of the coefficients of $AR\left(p\right)$ time series %
using the Cholesky decomposition
Daniel Ciuiu
Technical University of Civil Engineering Bucharest and Romanian Institute for Economic Research, Romanian Academy, Romania
Abstract:
In this paper we will determine successively the $AR\left(p\right)$ coefficients
of a time series starting with the $AR\left(1\right)$ model,
continuing with $AR\left(2\right)$, $AR\left(3\right)$, $\ldots$ and finishing with
$AR\left(p\right)$ model.
We use the essential fact that the matrix of linear system of the Yule--Walker
algorithm is symmetric and positive defined. That's why we solve the linear system by Cholesky method.
We consider all $AR\left(\widetilde{p}\right)$ with $\widetilde{p}$<$p$ having
$p$ coefficients, last $p-\widetilde{p}$ of them being zero. We use also
the linearity of the solutions of linear systems with the same matrix. More exactly, we go from $AR\left(\widetilde{p}\right)$ to
$AR\left(\widetilde{p}+1\right)$ coefficients by determining the difference between coefficients.