Two-step estimation of a multivariate L\'evy process

TL;DR

This paper introduces a two-step estimation method for multivariate Lévy processes using Lévy copulas, addressing dependence structure estimation with theoretical guarantees and simulation validation.

Contribution

It proposes a novel two-step estimation procedure for multivariate Lévy processes with Lévy copulas, including asymptotic normality proof and efficiency analysis.

Findings

Asymptotic normality of parameter estimates established

Efficiency loss due to truncation and two-step approach quantified

Simulation confirms theoretical properties and practical performance

Abstract

Based on the concept of a L\'evy copula to describe the dependence structure of a multivariate L\'evy process we present a new estimation procedure. We consider a parametric model for the marginal L\'evy processes as well as for the L\'evy copula and estimate the parameters by a two-step procedure. We first estimate the parameters of the marginal processes, and then estimate in a second step only the dependence structure parameter. For infinite L\'evy measures we truncate the small jumps and base our statistical analysis on the large jumps of the model. Prominent example will be a bivariate stable \lp, which allows for analytic calculations and, hence, for a comparison of different methods. We prove asymptotic normality of the parameter estimates from the two-step procedure and, in particular, we derive the Godambe information matrix, whose inverse is the covariance matrix of the normal limit law. A simulation study investigates the loss of efficiency because of the two-step procedure and the truncation.