[Paper Review] Estimating long range dependence: finite sample properties and confidence intervals

Rafał Weron|arXiv (Cornell University)|Mar 24, 2001

Complex Systems and Time Series Analysis39 references397 citations

TL;DR

This paper evaluates three estimators of long-range dependence—R/S analysis, Detrended Fluctuation Analysis (DFA), and periodogram regression—using Monte Carlo simulations on Gaussian white noise. DFA consistently outperforms the others in finite samples, and the study derives empirical confidence intervals for all methods, showing significant discrepancies from heuristic R/S values and strong alignment with asymptotic results for periodogram regression.

ABSTRACT

A major issue in financial economics is the behavior of asset returns over long horizons. Various estimators of long range dependence have been proposed. Even though some have known asymptotic properties, it is important to test their accuracy by using simulated series of different lengths. We test R/S analysis, Detrended Fluctuation Analysis and periodogram regression methods on samples drawn from Gaussian white noise. The DFA statistics turns out to be the unanimous winner. Unfortunately, no asymptotic distribution theory has been derived for this statistics so far. We were able, however, to construct empirical (i.e. approximate) confidence intervals for all three methods. The obtained values differ largely from heuristic values proposed by some authors for the R/S statistics and are very close to asymptotic values for the periodogram regression method.

Motivation & Objective

• To assess the finite-sample performance of R/S analysis, DFA, and periodogram regression in estimating long-range dependence.
• To construct empirical confidence intervals for all three estimators, given the lack of asymptotic distribution theory for DFA.
• To compare empirical confidence intervals with heuristic and asymptotic values, especially for R/S and periodogram methods.
• To apply the derived confidence intervals to real financial time series to test for long-range dependence in returns and volatility.
• To resolve controversies in financial econometrics by providing reliable, simulation-based inference for short to moderate sample sizes.

Proposed method

• Conducts Monte Carlo simulations on Gaussian white noise series of varying lengths to evaluate estimator accuracy.
• Employs R/S analysis using the Anis-Lloyd correction (R/S-AL) to adjust for small-sample bias in the rescaled range.
• Applies Detrended Fluctuation Analysis (DFA) by dividing time series into segments, detrending each, and computing the root mean square fluctuation.
• Uses periodogram regression (Geweke-Porter-Hudak, GPH) to estimate the fractional differencing parameter d from spectral density near zero frequency.
• Derives empirical confidence intervals via simulation-based quantiles for each estimator across multiple sample sizes.
• Compares empirical intervals with theoretical (asymptotic) and heuristic values, particularly for R/S statistics.

Experimental results

Research questions

• RQ1 How do R/S analysis, DFA, and periodogram regression perform in estimating the Hurst exponent H on finite samples from Gaussian white noise?
• RQ2 What are the finite-sample properties of the R/S-AL, DFA, and GPH estimators, especially in terms of bias and variance?
• RQ3 How do empirical confidence intervals for these estimators compare to heuristic or asymptotic values, particularly for R/S analysis?
• RQ4 To what extent do the empirical confidence intervals derived from simulations improve upon heuristic or theoretical intervals in practical applications?
• RQ5 What is the empirical significance of long-range dependence in real financial time series (e.g., stock returns, electricity prices) when using simulation-based confidence intervals?

Key findings

• Detrended Fluctuation Analysis (DFA) consistently outperforms R/S analysis and periodogram regression in finite-sample simulations, showing the most accurate and stable estimates of the Hurst exponent H.
• Empirical confidence intervals for the R/S-AL estimator differ significantly from heuristic values proposed by some authors, indicating that heuristic intervals are unreliable for inference.
• The empirical confidence intervals for the periodogram regression (GPH) method are very close to their asymptotic counterparts, validating the use of asymptotic theory for large samples.
• For the GPH method, empirical confidence intervals are well-approximated by the formula: 95% CI = [0.5 − exp(−0.71·N²/³ + 2.04), exp(−0.68·N²/³ + 1.78) + 0.5], where N = log₂L.
• In empirical applications, no significant long-range dependence was found in stock index returns (H ≈ 0.5), but strong evidence of long memory was detected in absolute returns (H > 0.8), indicating volatility clustering.
• Electricity price returns showed mean-reverting behavior (H < 0.5), with GPH estimates significantly below 0.5, consistent with earlier findings on price volatility in power markets.