Detection of Long Range Dependence in the Time Domain for (In)Finite-Variance Time Series

TL;DR

This paper improves the time domain detection of long-range dependence in both finite and infinite-variance time series by providing rigorous conditions for the variance plot estimator's consistency, supported by simulations.

Contribution

It offers theoretically grounded asymptotic conditions for the variance plot estimator to reliably detect LRD, including for infinite-variance series, which is a novel extension.

Findings

Variance plot estimator outperforms GPH in simulations

Conditions for estimator consistency are rigorously established

Detects LRD in infinite-variance time series after transformation

Abstract

Empirical detection of long range dependence (LRD) of a time series often consists of deciding whether an estimate of the memory parameter $d$ corresponds to LRD. Surprisingly, the literature offers numerous spectral domain estimators for $d$ but there are only a few estimators in the time domain. Moreover, the latter estimators are criticized for relying on visual inspection to determine an observation window $[n_1, n_2]$ for a linear regression to run on. Theoretically motivated choices of $n_1$ and $n_2$ are often missing for many time series models. In this paper, we take the well-known variance plot estimator and provide rigorous asymptotic conditions on $[n_1, n_2]$ to ensure the estimator's consistency under LRD. We establish these conditions for a large class of square-integrable time series models. This large class enables one to use the variance plot estimator to detect LRD for infinite-variance time series (after suitable transformation). Thus, detection of LRD for infinite-variance time series is another novelty of our paper. A simulation study indicates that the variance plot estimator can detect LRD better than the popular spectral domain GPH estimator.