Finite Sample Comparison of Alternative Estimators for Fractional Gaussian Noise

The fractional Brownian motion (fBm) process is a continuous-time Gaussian process with its increment being the fractional Gaussian noise (fGn). It has enjoyed widespread empirical applications across many fields, from science to economics and finance. The dynamics of fBm and fGn are governed by a fractional parameter H ∈ (0, 1). This paper first derives an analytical expression for the spectral density of fGn and investigates the accuracy of various approximation methods for the spectral density. 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 the log periodogram regression method, the local Whittle method, the time-domain maximum likelihood (ML) method, the Whittle ML method, and the change-of-frequency method. We implement two versions of the Whittle method, one based on the analytical expression for the spectral density and the other based on Paxson’s approximation. 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.