The effect of round-off error on long memory processes

TL;DR

This paper investigates how round-off errors affect the statistical properties of Gaussian long memory processes, showing that discretization scales the autocovariance and spectral density, and impacts the bias of Hurst exponent estimators.

Contribution

It provides an exact calculation of the scaling factor due to discretization and analyzes the bias and asymptotic properties of Hurst estimators on discretized long memory processes.

Findings

Discretization scales autocovariance and spectral density by a factor less than one.

Both LW and DFA estimators are severely negatively biased in finite samples with round-off errors.

LW estimator remains consistent and asymptotically normal under discretization.

Abstract

We study how the round-off (or discretization) error changes the statistical properties of a Gaussian long memory process. We show that the autocovariance and the spectral density of the discretized process are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we find that the discretized process is also long memory with the same Hurst exponent as the original process. We consider the properties of two estimators of the Hurst exponent, namely the local Whittle (LW) estimator and the Detrended Fluctuation Analysis (DFA). By using analytical considerations and numerical simulations we show that, in presence of round-off error, both estimators are severely negatively biased in finite samples. Under regularity conditions we prove that the LW estimator applied to discretized processes is consistent and asymptotically normal. Moreover, we compute the asymptotic properties of the DFA for a generic (i.e. non Gaussian) long memory process and we apply the result to discretized processes.