On the spectral density of fractional Ornstein–Uhlenbeck processes

Abstract

This paper introduces a novel and easy-to-implement method for accurately approximating the spectral density of discretely sampled fractional Ornstein–Uhlenbeck (fOU) processes. The method offers a substantial reduction in approximation error, particularly within the rough region of the fractional parameter $H\in (0,0.5)$. This approximate spectral density has the potential to enhance the performance of estimation methods and hypothesis testing that make use of spectral densities. We introduce the approximate Whittle maximum likelihood (AWML) method for discretely sampled fOU processes, utilizing the approximate spectral density, and demonstrate that the AWML estimator exhibits properties of consistency and asymptotic normality when $H\in (0,1)$, akin to the conventional Whittle maximum likelihood method. Through extensive simulation studies, we show that AWML outperforms existing methods in terms of estimation accuracy in finite samples. We then apply the AWML method to the trading volume of 40 financial assets. Our empirical findings reveal that the estimated Hurst parameters for these assets fall within the range of 0.10 to 0.21, indicating a rough dynamic.