Moment conditions for random coefficient AR($\infty$) under non-negativity assumptions

TL;DR

This paper develops new combinatorial methods to determine when moments of non-negative random coefficient AR(∞) models are finite, providing both necessary and sufficient conditions and extending previous results.

Contribution

It introduces novel combinatorial techniques for analyzing moment finiteness in non-negative AR(∞) models, including a second moment criterion that complements existing conditions.

Findings

Second moment condition is necessary and sufficient for finiteness of second moments.

Methods recover previous sufficient conditions by Doukhan and Wintenberger.

New criteria are equivalent to classical ones in finite order cases.

Abstract

We consider random coefficient autoregressive models of infinite order (AR($\infty$)) under the assumption of non-negativity of the coefficients. We develop novel methods yielding sufficient or necessary conditions for finiteness of moments, based on combinatorial expressions of first and second moments. The methods based on first moments recover previous sufficient conditions by Doukhan and Wintenberger in our setting. The second moment method provides in particular a necessary and sufficient condition for finiteness of second moments which is different, but shown to be equivalent to the classical criterion of Nicholls and Quinn in the case of finite order. We further illustrate our results through two examples.