Two‐Step Estimation of a Multi‐Variate Lévy Process

23 Pages Posted: 16 Nov 2013

See all articles by C. Klüppelberg

C. Klüppelberg

Technische Universität München (TUM)

Habib Esmaeili

Technische Universität München (TUM)

Date Written: November 2013


Based on the concept of a Lévy copula to describe the dependence structure of a multi‐variate Lévy process, we present a new estimation procedure. We consider a parametric model for the marginal Lévy processes as well as for the Lévy 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 Lévy 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 Lévy process, 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.

Keywords: Dependence structure, Godambe information matrix, Lévy copula, maximum likelihood estimation, multi‐variate Lévy process, reduced likelihood, two‐step parameter estimation

JEL Classification: IFM, inference functions for margins

Suggested Citation

Kluppelberg, Claudia and Esmaeili, Habib, Two‐Step Estimation of a Multi‐Variate Lévy Process (November 2013). Journal of Time Series Analysis, Vol. 34, Issue 6, pp. 668-690, 2013, Available at SSRN: or

Claudia Kluppelberg

Technische Universität München (TUM) ( email )

Center for Mathematical Sciences
D-80290 Munich

Habib Esmaeili

Technische Universität München (TUM) ( email )

Arcisstrasse 21
Munich, DE 80333

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