Fractional Gaussian Noise: Spectral Density and Estimation Methods
本文给出了分数高斯噪声谱密度的可计算表达式，并比较了多种估计方法在有限样本下的表现，推荐了兼顾统计与计算效率的方法，对金融波动建模有参考价值。
The fractional Brownian motion (fBm) process, governed by a fractional parameter , is a continuous‐time Gaussian process with its increment being the fractional Gaussian noise (fGn). This article first provides a computationally feasible expression for the spectral density of fGn. This expression enables us to assess the accuracy of a range of approximation methods, including the truncation method, Paxson's approximation, and the Taylor series expansion at the near‐zero frequency. Next, we conduct an extensive Monte Carlo study comparing the finite sample performance and computational cost of alternative estimation methods for under the fGn specification. These methods include two semi‐parametric methods (based on the Taylor series expansion), two versions of the Whittle method (utilising either the computationally feasible expression or Paxson's approximation of the spectral density), a time‐domain maximum likelihood (ML) method (employing a recursive approach for its likelihood calculation), and a change‐of‐frequency method. Special attention is paid to highly anti‐persistent processes with close to zero, which are of empirical relevance to financial volatility modelling. Considering the trade‐off between statistical and computational efficiency, we recommend using either the Whittle ML method based on Paxson's approximation or the time‐domain ML method. We model the log realized volatility dynamics of 40 financial assets in the US market from 2012 to 2019 with fBm. Although all estimation methods suggest rough volatility, the implied degree of roughness varies substantially with the estimation methods, highlighting the importance of understanding the finite sample performance of various estimation methods.