Estimators for Long Range Dependence: An Empirical Study

Abstract: 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

• The paper demonstrates that estimator performance varies significantly with series length and Hurst parameter values.
• The study employs extensive simulations using FGNs and Campito Mountain data to evaluate 10 H estimators and 2 d estimators.
• Findings underscore the need to combine multiple estimators and goodness-of-fit tests for robust time series analysis.

Evaluating Estimators for Long Range Dependence in Time Series: A Comprehensive Empirical Analysis

The paper, "Estimators for Long Range Dependence: An Empirical Study," conducted by William Rea, Les Oxley, Marco Reale, and Jennifer Brown, explores the efficacy of various estimators used to measure long-range dependence in time series data. The authors employ a simulation study to critically examine the performance of these estimators across different scenarios, focusing on their ability to accurately estimate the Hurst parameter ($H$) and the fractional integration parameter ($d$).

Background and Methodology

In the domain of time series analysis, long-range dependence is typically characterized by the Hurst parameter $H$ and the fractional integration parameter $d$. These parameters are central to understanding self-similar processes and fractional Brownian motion. The relationship between the two is given by $H = d + 0.5$. Over the years, numerous estimators have been devised to estimate these parameters, each having distinct theoretical and empirical properties. This study extends previous empirical studies by analyzing twelve different estimators with extensive simulation scenarios, including varying series lengths and values of $H$.

The authors use the R statistical software packages fSeries and fracdiff to implement ten $H$ estimators and two $d$ estimators. These estimators are assessed against simulated Fractional Gaussian Noises (FGNs) and are later applied to real-world data from the Campito Mountain series, a recognized example of a long memory time series.

Results

The empirical analysis employs graphical representations instead of tabular ones due to the vast amount of data. Findings reveal that the performance of estimators varies significantly with different series lengths and $H$ values:

• Absolute Value and Aggregated Variance Methods: These exhibit increasing bias as $H$ rises, particularly in short series lengths.
• Periodogram and Boxed Periodogram Methods: The boxed periodogram generally underestimates $H$ while being less effective than its unmodified counterpart.
• Differenced Variance Method: Shows severe bias for short series but improves marginally as the series length increases.
• Higuchi and Peng Estimators: Both show underestimation of $H$ with varying degrees of bias dependent on the series length.
• Wavelet and Whittle Estimators: These offer minimal bias and narrower confidence intervals, with the wavelet estimator showing slight bias for shorter series only.
• GPH and Haslett-Raftery Estimators: Both demonstrate reliability with more consistent MSE values across larger sample sizes.

The study's detailed simulation further examines the implications for datasets like the Campito Mountain data, illustrating the estimators' practical application.

Implications and Future Research

The paper provides clear evidence on the variability in estimator performance and reinforces the notion that the choice and evaluation of an estimator can significantly affect the conclusions drawn from time series data analysis. Practically, the study advises the use of multiple estimators in tandem, coupled with goodness-of-fit tests such as the Beran test, to ensure robust analysis.

Theoretically, the research guides future work towards enhancing estimator robustness. Potential advancements could involve improving existing methods to handle series with shifting means or outlier presence, or developing entirely new estimation techniques. The influence of structural breaks on estimator performance also remains an engaging area for forthcoming studies.

In closing, this paper contributes extensively to the empirical analysis of long-range dependence estimators, setting a benchmark for future innovations and applications in time series analysis.