Data driven time scale in Gaussian quasi-likelihood inference

TL;DR

This paper develops a method for estimating parameters and unknown sampling stepsize in ergodic diffusion models from high-frequency data, proving asymptotic normality and consistency of model selection criteria.

Contribution

It introduces explicit estimators for both model parameters and sampling stepsize, addressing the challenge of unknown sampling intervals in diffusion process inference.

Findings

Estimators are asymptotically normal and jointly distributed.

The proposed model selection criterion is consistent.

Numerical experiments support theoretical results.

Abstract

We study parametric estimation of ergodic diffusions observed at high frequency. Different from the previous studies, we suppose that sampling stepsize is unknown, thereby making the conventional Gaussian quasi-likelihood not directly applicable. In this situation, we construct estimators of both model parameters and sampling stepsize in a fully explicit way, and prove that they are jointly asymptotically normally distributed. The $L^{q}$-boundedness of the obtained estimator is also derived. Further, we propose the Schwarz (BIC) type statistics for model selection and show its model-selection consistency. We conducted some numerical experiments and found that the observed finite-sample performance well supports our theoretical findings. Also provided is a real data example.