Statistical inference for continuous-time locally stationary processes using stationary approximations

TL;DR

This paper develops asymptotic theory for $M$-estimators in continuous-time locally stationary processes, demonstrating their consistency and normality using stationary approximations and applying these results to L\'evy-driven models.

Contribution

It introduces a framework for establishing the asymptotic properties of $M$-estimators in continuous-time locally stationary processes using stationary approximations and dependence conditions.

Findings

Proves consistency and asymptotic normality of localized least squares estimators for L\'evy-driven Ornstein-Uhlenbeck processes.

Shows consistency of localized Whittle and quasi maximum likelihood estimators for time-varying state space models.

Simulation studies confirm the practical applicability of the proposed estimation procedures.

Abstract

We establish asymptotic properties of $M$-estimators, defined in terms of a contrast function and observations from a continuous-time locally stationary process. Using the stationary approximation of the sequence, $\theta$-weak dependence, and hereditary properties, we give sufficient conditions on the contrast function that ensure consistency and asymptotic normality of the $M$-estimator. As an example, we obtain consistency and asymptotic normality of a localized least squares estimator for observations from a sequence of time-varying L\'evy-driven Ornstein-Uhlenbeck processes. Furthermore, for a sequence of time-varying L\'evy-driven state space models, we show consistency of a localized Whittle estimator and an $M$-estimator that is based on a quasi maximum likelihood contrast. Simulation studies show the applicability of the estimation procedures.