The Local Fractional Bootstrap

TL;DR

This paper introduces the local fractional bootstrap, a novel resampling method for high-frequency data of Brownian semistationary processes, improving hypothesis testing of path roughness with practical applications in finance and turbulence.

Contribution

It presents the first valid bootstrap procedure for high-frequency statistics of Brownian semistationary processes, enhancing finite-sample performance over existing methods.

Findings

Bootstrap method shows improved finite-sample accuracy.

Effective in testing roughness of asset prices.

Validates Kolmogorov's law in turbulence data.

Abstract

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.