I am studying GARCH processes in Time Series Analysis by Hamilton.

Something that has regularly been used in the book is the assumption that an infinite-order polynomial can be written as the ratio of two finite-order polynomials.

$\pi(L) = \sum_{j=1}^{\infty} \pi_j L^j$

And then

$\pi(L) = \frac{\alpha(L)}{1-\delta(L)} = \frac{\alpha_1 L^1 + \alpha_1 L^2 + ... + \alpha_m L^m}{1 - \delta_1 L^1 - \delta_1 L^2 - ... - \delta_r L^r}$

Followed by "assuming the roots of $1-\delta(L) = 0$ are outside the unit circle".

What is the reasoning behind this transformation? And is the assumption about the roots outside the unit circle required for the infinite order polynomial to have this ratio representation?

clear contextmentioned and AFAIK the coefficients inthat contextwill be in linear recurrecne, so it will be a rational. – user9072 Apr 18 '13 at 13:59