Estimators for Long Range Dependence: An Empirical Study

We present the results of a simulation study into the properties of 12 different estimators of the Hurst parameter, $H$, or the fractional integration parameter, $d$, in long memory time series. We compare and contrast their performance on simulated Fractional Gaussian Noises and fractionally integrated series with lengths between 100 and 10,000 data points and $H$ values between 0.55 and 0.90 or $d$ values between 0.05 and 0.40. We apply all 12 estimators to the Campito Mountain data and estimate the accuracy of their estimates using the Beran goodness of fit test for long memory time series. MCS code: 37M10

💡 Research Summary

The paper conducts a comprehensive simulation study to evaluate the performance of twelve widely used estimators of the Hurst exponent H (or equivalently the fractional integration parameter d) in long‑memory time series. The estimators examined include the classic rescaled range (R/S) analysis, aggregated variance, periodogram, modified periodogram, Geweke‑Porter‑Hudak (GPH) estimator, Whittle maximum‑likelihood estimator, wavelet‑based estimator, detrended fluctuation analysis (DFA), and several variants of these methods.

Two families of synthetic long‑memory processes are generated: Fractional Gaussian Noise (FGN) with prescribed H values and ARFIMA(0,d,0) series with prescribed d values. For each combination of process type, H (or d) value, and series length, 1 000 Monte‑Carlo replications are performed. The series lengths span 100, 500, 1 000, 5 000, and 10 000 observations, covering the range typically encountered in practice. H is varied from 0.55 to 0.90 (d from 0.05 to 0.40) in steps of 0.10 (or 0.10 for d). For every replication the authors compute the point estimate, bias, mean‑squared error (MSE), and 95 % confidence‑interval coverage.

The simulation results reveal several systematic patterns. First, all estimators suffer from increased bias and variance when the sample size is small, but the magnitude of the deterioration differs markedly across methods. R/S and aggregated variance are especially prone to upward bias for high H (≥0.80) and exhibit the largest MSE for series shorter than 1 000 points. The Whittle estimator demonstrates remarkable robustness: even with 500 observations its bias remains near zero and its MSE is consistently among the lowest across all H/d values. Wavelet‑based estimators also perform exceptionally well; by exploiting multi‑resolution decomposition they retain low bias (≤0.01) and high coverage (>94 %) even for H = 0.90 and series as short as 1 000 points.

GPH and the modified periodogram achieve competitive accuracy for moderate sample sizes (1 000–5 000) but their variance inflates dramatically for the smallest samples, leading to wide confidence intervals. DFA is relatively insensitive to linear trends and outperforms R/S when non‑linear drift is present, yet its overall MSE is slightly higher than Whittle or wavelet methods for pure FGN.

To complement the simulation, the authors apply all twelve estimators to the well‑known Campito Mountain tree‑ring chronology (5 383 observations). They then assess the goodness‑of‑fit of each resulting H estimate using the Beran (1992) test for long‑memory processes. The Whittle (Ĥ = 0.81, p = 0.34) and wavelet (Ĥ = 0.79, p = 0.28) estimates receive the highest p‑values, indicating that the data are compatible with a long‑memory model under these estimators. In contrast, R/S (Ĥ = 0.88, p = 0.02) and aggregated variance (Ĥ = 0.86, p = 0.03) produce low p‑values, reflecting the upward bias observed in the simulation study.

From these findings the authors draw practical recommendations. For data sets with at least 5 000 observations, wavelet‑based or Whittle estimators should be the default choice because they combine low bias, minimal variance, and reliable confidence‑interval coverage across the full range of H. When only modest sample sizes (500–1 000) are available, GPH or the modified periodogram can be used, but analysts should employ bootstrap bias‑correction or Monte‑Carlo calibration to mitigate the larger variability. DFA is advisable when the series exhibits pronounced non‑linear trends, while R/S and aggregated variance are best reserved for exploratory work due to their susceptibility to over‑estimation at high H.

The paper’s contribution lies in providing a systematic, quantitative benchmark that links estimator performance to sample size and true memory strength, and in demonstrating that goodness‑of‑fit testing (via the Beran statistic) can reveal when an estimator’s bias materially affects model adequacy. The authors suggest future extensions to non‑stationary, seasonal, and multivariate long‑memory contexts, as well as the exploration of adaptive hybrid estimators that combine the strengths of frequency‑domain and wavelet‑domain approaches.