Bhansali-Kokoszka (2006) Estimation of the Long-Memory Parameter

Summary

Bhansali and Kokoszka review current methods for estimating the memory parameter $d$ of a long-range-dependent stationary time series and introduce a new estimator based on fitting a fractionally-differenced autoregression of order $p$. Letting $p\to\infty$ jointly with the sample length $n$, the estimators are shown to be consistent both under finite-variance innovations and under infinite-variance stable innovations with index $\alpha\in(1,2)$. Finite-sample behaviour is examined by simulation and by application to ethernet-traffic data.

Key Claims

• Setup. $\{X_t\}$ is long-memory with parameter $d$, $0<d<\tfrac12$, if its spectral density satisfies $f(\lambda)\sim|\lambda|^{-2d}L(1/\lambda)$ with $L$ slowly varying; equivalently its autocovariances are not absolutely summable (vs. absolutely summable for short memory).
• Review. The paper surveys the existing memory-parameter estimators — R/S, log-periodogram (GPH), local Whittle and related frequency-domain and time-domain methods.
• New autoregressive estimator. Estimate $d$ by fitting a fractionally-differenced autoregression of order $p$, a parametric-fit alternative to the semiparametric estimators, with $p\to\infty$ as $n\to\infty$.
• Consistency under heavy tails. The estimators are consistent both when innovations have finite variance and when they follow an infinite-variance stable law with exponent $\alpha\in(1,2)$ — robustness to heavy tails that finite-variance-based methods lack.
• Empirical illustration. A simulation study compares finite-sample behaviour, and the methods are applied to a real ethernet-traffic series (self-similar network traffic).

Concepts Introduced or Extended

• Long Memory and Fractional Integration

Entities Mentioned

• Rajendra J. Bhansali
• Piotr S. Kokoszka

Quotes

"Current methods of estimating the memory parameter, d, of a long-range dependent stationary time series are reviewed, and a new method of estimating d by fitting a fractionally-differenced autoregression of order p is introduced."

My Take

A useful pairing of survey and contribution: it maps the semiparametric estimator landscape (GPH, local Whittle, R/S) and then offers a parametric-fit route via a fractionally-differenced AR of growing order. The headline value is the heavy-tail robustness — consistency under infinite-variance stable innovations puts it alongside the sign-based tests of Delgado-Velasco (2005) as tools for the fat-tailed data (telecom traffic, finance) where Gaussian-calibrated estimators misbehave. The autoregressive framing also ties long-memory estimation back to ordinary AR model-order selection, at the cost of choosing the truncation order $p$ — the familiar bandwidth/lag-length tuning problem in another guise.