Fractional Gaussian Noise: Spectral Density and Estimation Methods

Abstract

The fractional Brownian motion (fBm) process, governed by a fractional parameter $H\in (0,1)$, 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 $H$ 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 $H$ 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.