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

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.