A frequency domain analysis of the error distribution from noisy high-frequency data

TL;DR

This paper develops a frequency domain method to estimate measurement error distribution and integrated volatility from high-frequency data, achieving optimal convergence rates and validated through simulations and real data.

Contribution

It introduces a deconvolution-based estimator for error density that does not require equally spaced data and a frequency domain estimator for integrated volatility with optimal convergence.

Findings

Estimator is consistent and minimax rate optimal.

Frequency domain estimator for volatility achieves optimal convergence rate.

Method validated through simulations and real data analysis.

Abstract

Data observed at high sampling frequency are typically assumed to be an additive composite of a relatively slow-varying continuous-time component, a latent stochastic process or a smooth random function, and measurement error. Supposing that the latent component is an It\^{o} diffusion process, we propose to estimate the measurement error density function by applying a deconvolution technique with appropriate localization. Our estimator, which does not require equally-spaced observed times, is consistent and minimax rate optimal. We also investigate estimators of the moments of the error distribution and their properties, propose a frequency domain estimator for the integrated volatility of the underlying stochastic process, and show that it achieves the optimal convergence rate. Simulations and a real data analysis validate our analysis.