Gaussian quasi-information criteria for ergodic L\'{e}vy driven SDE

TL;DR

This paper introduces Gaussian quasi-information criteria for model selection in ergodic Lévy-driven SDEs, utilizing a two-stage Gaussian quasi-likelihood approach for high-frequency data, with explicit criteria and numerical validation.

Contribution

It develops explicit Gaussian quasi-AIC and BIC criteria for Lévy-driven SDEs, revealing unique mixed-rate asymptotics and providing a stepwise inference procedure for model selection.

Findings

The GQLF exhibits a mixed-rates structure affecting regularization terms.

Explicit GQAIC and GQBIC criteria are proposed for scale and drift coefficients.

Numerical experiments confirm the theoretical properties of the criteria.

Abstract

We consider relative model comparison for the parametric coefficients of a semiparametric ergodic L\'{e}vy driven model observed at high-frequency. Our asymptotics is based on the fully explicit two-stage Gaussian quasi-likelihood function (GQLF) of the Euler-approximation type. For selections of the scale and drift coefficients, we propose explicit Gaussian quasi-AIC (GQAIC) and Gaussian quasi-BIC (GQBIC) statistics through the stepwise inference procedure. In particular, we show that the mixed-rates structure of the joint GQLF, which does not emerge for the case of diffusions, gives rise to the non-standard forms of the regularization terms in the selection of the scale coefficient, quantitatively clarifying the relation between estimation precision and sampling frequency. Numerical experiments are given to illustrate our theoretical findings.