Multi-dimensional parameter estimation of heavy-tailed moving averages

TL;DR

This paper introduces a new parametric estimation method for multi-parameter heavy-tailed Lévy-driven moving averages, extending existing techniques to a multi-parametric framework with proven consistency and finite sample performance.

Contribution

It extends marginal empirical characteristic function methods to a multi-parameter setting, including models like fractional stable motion and Ornstein-Uhlenbeck processes.

Findings

Proves consistency and central limit theorem for the estimator.

Demonstrates numerical performance with finite sample analysis.

Includes applications to various Lévy-driven models.

Abstract

In this paper we present a parametric estimation method for certain multi-parameter heavy-tailed L\'evy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained in [3] via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to the papers [14, 15], which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving L\'evy motion. We extend their work to allow for a multi-parametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.