Estimators for Long Range Dependence: An Empirical Study
aa r X i v : . [ s t a t . M E ] J a n Electronic Journal of Statistics
Vol. 0 (2009)ISSN: 1935-7524
Estimators for Long Range Dependence:An Empirical Study
William Rea e-mail: [email protected] and
Les Oxley e-mail: [email protected] and
Marco Reale e-mail: [email protected] and
Jennifer Brown e-mail: [email protected]
Abstract:
We present the results of a simulation study into the properties of 12different estimators of the Hurst parameter, H , or the fractional integra-tion parameter, d , in long memory time series. We compare and contrasttheir performance on simulated Fractional Gaussian Noises and fractionallyintegrated 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 applyall 12 estimators to the Campito Mountain data and estimate the accuracyof their estimates using the Beran goodness of fit test for long memory timeseries.MCS code: 37M10 Keywords and phrases:
Strong dependence, Global dependence, Longrange dependence, Hurst parameter estimators.
1. Introduction
The subject of long-memory time series was brought to prominence by Hurst(1951) and has subsequently received extensive attention in the literature. Seethe volumes by Beran (1994), Embrechts and Maejima (2002), and Palma (2007)and the collections of Doukhan et al. (2003) and Robinson (2003) and the ref-erences therein.Of critical importance in analyzing and modeling long memory time series isestimating the strength of the long-range dependence. Two measures are com-monly used. The parameter H , known as the Hurst or self-similarity parameter, ∗ imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators was introduced to applied statistics by Mandelbrot and van Ness (1968) andarises naturally from the study of self-similar processes. The other measure,the fractional integration parameter, d , arises from the generalization of theBox-Jenkins ARIMA(p,d,q) models from integer to non-integer values of the in-tegration parameter d . This generalization was accomplished independently byGranger and Joyeux (1980) and by Hosking (1981). The fractional integrationparameter d is also the discrete time counterpart to the self-similarity parameter H and the two are related by the simple formula H = d + 1 / H and d have been developed. These are usuallyvalidated by an appeal to some aspect of self-similarity, or by an asymptoticanalysis of the distributional properties of the estimator as the length of thetime series converges to infinity.A number of theoretical results on the asymptotic properties of various esti-mators have been obtained. The aggregated variance method was shown to beasymptotically biased of the order 1 / log N , where N is the number of observa-tions by Giraitis et al. (1999) who also showed the GPH (Geweke and Porter-Hudak,1983) estimator was asymptotically normal and unbiased. Robinson (1994) provedthe averaged periodogram estimator was consistent under very mild conditions.Lobato and Robinson (1996) obtained its limiting distribution. The Peng et al.(1994) estimator was proved to be asymptotically unbiased by Taqqu et al.(1995). Some theoretical properties of the R/S estimator have been examinedby Mandelbrot (1975) and Mandelbrot and Taqqu (1979). Mandelbrot (1975)proved that the R/S statistic is robust to the increment process having a long-tailed distribution in the sense that E [ X i ] = ∞ . However, Bhattacharya et al.(1983) proved that the R/S statistic was not robust to departures from sta-tionarity. Thus for a short memory process with slowly decaying deterministictrend the R/S statistic will report an estimate of H which implies the pres-ence of long-memory. An estimator based on wavelets was proved asymptoti-cally unbiased and efficient by Abry et al. (1998). They also showed the tradi-tional variance type estimators were fundamentally flawed and could not leadto good estimators of H . Fox and Taqqu (1986) proved the Whittle estimatorwas consistent and asymptotically normal for Gaussian long range dependentsequences. Dalhaus (1989) proved the estimator of Fox and Taqqu (1986) wasefficient. Further theoretical results on the Whittle estimator can be found inHorvath and Shao (1999).Because the finite sample properties of these estimators can be quite differentfrom their asymptotic properties some previous authors have undertaken em-pirical comparisons of estimators of H and d . Nine estimators were discussed insome detail by Taqqu et al. (1995) who carried out an empirical study of theseestimators for a single series length of 10,000 data points, five values of both H and d , and 50 replications. Teverovsky and Taqqu (1999) showed in a simula-tion study that the differenced variance estimator was unbiased for five valuesof H (0.5, 0.6, 0.7, 0.8, and 0.9) for series with 10,000 observations whereasthe aggregated variance estimator was downwards biased. Jensen (1999) under-took a comparison of two estimators based on wavelets, one proposed by Jensen(1999) and the other proposed by McCoy and Walden (1996), with the GPH imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators estimator for four series lengths (2 , 2 , 2 , 2 observations), five values of d and 1000 replications. They reported the wavelet estimators had lower meansquared errors (MSEs) than the GPH estimator for all d values and series lengthsinvestigated. Jeong et al. (2007) carried out a comparison of six estimators onsimulated fractional Gaussian noises (FGNs) with 32,768 (2 ) observations, fivevalues of H and 100 replications.Several of the above empirical investigations would have been limited bythe then available computer power which has since increased considerably. Wehave extended these studies to a larger number of parameters, higher numberof replications and 12 estimators as detailed in Section (2) below.The remainder of the paper is organized as follows. Section (2) gives detailsof the method. Section (3) presents the results. Section (4) applies the methodsto the Campito Mountain data which is regarded as a standard example of along memory time series. Section (5) contains the discussion and Section (6)gives our conclusions and suggests avenues of future research.
2. Method
Ten H estimators are implemented in the contributed package fSeries ofWuertz (2005) for the popular statistical software R (R Development Core Team,2005). They are the absolute value, aggregated variance, boxed periodogram, dif-ferenced variance, Higuchi, Peng, periodogram, rescaled range, wavelet, and theWhittle. The wavelet estimator is discussed in some detail by Abry and Veitch(1998), Abry et al. (1998), and Veitch and Abry (1999) and the other nine arediscussed by Taqqu et al. (1995). Further, the GPH (Geweke and Porter-Hudak,1983) and Haslett and Raftery (1989) are implemented as estimators for d inthe contributed package fracdiff of Fraley et al. (2006).Taqqu et al. (1995) simulated FGNs and the corresponding discrete time frac-tionally integrated (FI(d)) series and found that each estimator performed sim-ilarly whether estimating H in simulated FGNs or d in simulated FI(d)s. Forexample, if an estimator was biased when estimating H it was also biased in avery similar manner when estimating d . Thus, with the exception of the GPHand Haslett-Raftery estimators, we only investigated each estimator’s perfor-mance in estimating H for simulated FGNs. FGNs were generated using thefunction fgnSim in fSeries . We ran 1000 replications of simulated FGNs with100 different lengths and eight different H values. The lengths were between100 and 10,000 data points in steps of 100. The H values were between 0.55and 0.90 in steps of 0.05. For each series H was estimated by each of these tenestimators. For each H value and series length we estimated the median, 75%and 95% confidence intervals empirically from the simulated data. The H or d estimates were sorted into ascending order and the median obtained by aver-aging the 500th and 501st values. Similar calculations were done for the upperand lower values of the 75% and 95% confidence intervals.For the GPH and Haslett-Raftery estimators we generated FI(d) series withthe function farimaSim in fSeries over the range 0.05 to 0.40 in steps of 0.05. imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators The other details are the same as above. In the presentation of the resultswe converted the GPH and Haslett-Raftery d estimates to H equivalents tofacilitate comparisons among the estimators.The simulations and estimations were performed on a SunBlade 1000 with a750Mhz UltraSPARC-III CPU with 2Gb of memory and a Sun Ultra 10 with a440Mhz UltraSPARC-IIi CPU and 1Gb of memory.
3. Results
To present the results in tabular form would require a very large amount of space.Thus we present them in graphical form. Figures (1) through (7) present someof the results. Figures (1) through (6) are presented with the vertical axis witha range of 1.2 H units to facilitate comparisons among the estimators’ standarddeviation of their estimates. It should be noted that stationary long memoryoccurs in the range 0 . < H < .
0. Baillie (1996) states that for 1 . ≤ H < . ≤ H ≤ . H was low (0.55 or 0.60) but became progressivelybiased and underestimated H as H increased.The results for the aggregated variance method are presented in Figures (1)(b) and (d). The aggregated variance method exhibited bias and underestimated H in short series when H was low. As H increased the estimator became in-creasingly biased at all series lengths examined. With H = 0 .
90 the true value of H lay above the upper 95% empirical confidence interval for all but the shortestseries lengths.The results for the boxed periodogram method are presented in Figures (2)(a) and (c). The boxed periodogram method was developed specifically to dealwith perceived problems with the periodogram estimator. Comparing the boxedperiodogram with the unmodified periodogram method in Figures (4) (a) and(c) we can see that for FGNs where the series were short and H was high thatthe periodogram method was biased towards over estimating H . The boxedperiodogram was biased towards underestimating H for almost all values of H and series lengths examined.The results for the differenced variance method are presented in Figures (2)(b) and (d). The differenced variance method had one of largest confidenceintervals of the estimators when the series were short but this slowly decreasedas sample size increased. Only the GPH, periodogram and wavelet methods hada similarly wide confidence interval for short series. The differenced varianceestimator exhibited bias towards over estimating H for any series with less than7,000 observations. The bias was very serious in the short series. For series longer imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . (a) H Est and CI Absval H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (b) H Est and CI Aggvar H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (c) H Est and CI Absval H=0.90 Series Length H ActualMedian75% CI95% CI 0 2000 4000 6000 8000 . . . . (d) H Est and CI Aggvar H=0.90 Series Length H ActualMedian75% CI95% CI
Fig 1 . Empirical confidence intervals for the H estimates with H = 0 . and H = 0 . ; (a)and (c) absolute value method, (b) and (d) aggregated variance estimator. than about 9,000 observations the estimator exhibited a small amount of biastowards underestimating H .The results for the Higuchi (1988) estimator are presented in Figures (3) (a)and (c). The Higuchi was biased towards underestimating H but the magnitudeof the bias appeared relatively independent of H . The width of the confidenceinterval of the estimate increased with increasing H .The results for the Peng et al. (1994) estimator are presented in Figures (3)(b) and (d). The Peng estimator was biased toward under estimating H in theseries lengths we investigated. This bias appeared to be independent of H butwas very small though it appeared greater in short series.The results for the periodogram estimator were discussed above in conjunc-tion with the boxed periodogram estimator. imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . (a) H Est and CI BoxPer H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (b) H Est and CI Diffvar H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (c) H Est and CI BoxPer H=0.90 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (d) H Est and CI Diffvar H=0.90 Series Length H ActualMean75% CI95% CI
Fig 2 . Empirical confidence intervals for the H estimates with H = 0 . and H = 0 . ; (a)and (c) boxed periodogram method, (b) and (d) differenced variance estimator. The results for the R/S estimator are presented in Figures (4) (b) and (d). TheR/S estimator is of considerable historical interest because it was first proposedby Hurst and was used extensively in early studies of long-memory processes.However, as can be seen from Figures (4) (b) and (d) the R/S estimator exhibitedthree problems; it was biased upwards when H was low, it was biased downwardswhen H was high, and the confidence interval of the estimate did not decreasewith increasing series length once the series reached about 1000 observations.The results for the Whittle estimator are presented in Figures (5) (a) and(c). Compared to the other nine estimators implemented in fSeries the Whittleestimator was remarkable for its narrow confidence interval. It only displayed asmall amount of downwards bias when the series were short and H was high.There was an implementation issue in the software we used. The Whittle esti- imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . (a) H Est and CI Higuchi H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (b) H Est and CI Peng H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (c) H Est and CI Higuchi H=0.90 Series Length H ActualMedian75% CI95% CI 0 2000 4000 6000 8000 . . . . (d) H Est and CI Peng H=0.90 Series Length H ActualMedian75% CI95% CI
Fig 3 . Empirical confidence intervals for the H estimates with H = 0 . and H = 0 . ; (a)and (c) Higuchi estimator, (b) and (d) Peng estimator. mator would terminate with an error when H was low and the series containedonly a few hundred observations. Thus in Figure (5)(a) there was no data forseries with less than 300 observations in the H = 0 .
65 results.The results for the wavelet estimator are presented in Figures (5) (b) and(d). The wavelet estimator was unbiased for all H values at series lengths over4,100 data points. The bias present in series shorter than 4,100 data points wasvery small. The availability of a new octave can be seen in Figures (5) (b) and(d) with each doubling of the series length. New octaves resulted in a series ofsteps in the reduction of the confidence interval of the estimate with increasingseries length. The estimator had constant variance when the number of octaves was fixed.The results for the GPH estimator are presented in Figures (6)(a) and (c). imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . (a) H Est and CI Per. H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (b) H Est and CI R/S H=0.60 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (c) H Est and CI Per. H=0.90 Series Length H ActualMedian75% CI95% CI 0 2000 4000 6000 8000 . . . . (d) H Est and CI R/S H=0.90 Series Length H ActualMedian75% CI95% CI
Fig 4 . Empirical confidence intervals for the H estimates with H = 0 . and H = 0 . ; (a)and (c) periodogram estimator, (b) and (d) R/S estimator. The GPH estimator exhibited a very small amount of bias towards overestimat-ing d at all series lengths examined. It had a very wide confidence interval whichnarrowed slowly as the series length increased.The results for the Haslett-Raftery estimator are presented in Figures (6)(b)and (d). The Haslett-Raftery did not report estimates of d less than zero ( H < . d and short series the distribution was truncated on the lowside at d = 0 or H = 0 . H = 0 . d = 0 . imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . (a) H Est and CI Whittle H=0.65 Series Length H ActualMedian75% CI95% CI 0 2000 4000 6000 8000 . . . . (b) H Est and CI Wavelet H=0.65 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (c) H Est and CI Whittle H=0.90 Series Length H ActualMedian75% CI95% CI 0 2000 4000 6000 8000 . . . . (d) H Est and CI Wavelet H=0.90 Series Length H ActualMedian75% CI95% CI
Fig 5 . Empirical confidence intervals for the H estimates with H = 0 . and H = 0 . ; (a)and (c) Whittle estimator, (b) and (d) wavelet estimator. MSEs for some estimators in the short series. The Whittle and Haslett-Rafteryboth had low MSEs in all series greater than 500 data points in length. Thestep reductions in the MSE for the wavelet estimator can be clearly seen eachtime a new octave became available.
4. Application: Campito Mountain Data
The Campito Mountain bristlecone pine data is regarded as a standard exam-ple of a long memory time series. It is a 5405 year series of annual tree ringwidths of bristlecone pines on Campito Mountain, California. It was studiedby Baillie and Chung (2002) who determined that an ARFIMA(0,0.44,0) modelfitted the data best. The lack of additional short term correlation in the data imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . (a) H Est and CI GPH d=0.10 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . . (b) H Est and CI Haslett−Raftery d=0.10 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . (c) H Est and CI GPH d=0.40 Series Length H ActualMean75% CI95% CI 0 2000 4000 6000 8000 . . . (d) H Est and CI Haslett−Raftery d=0.40 Series Length H ActualMean75% CI95% CI
Fig 6 . Empirical confidence intervals for the H estimates with d = 0 . ( H = 0 . ) and d = 0 . ( H = 0 . ); (a) and (c) GPH estimator, (b) and (d) Haslett-Raftery estimator. means it is a good candidate for modeling with an FGN.The Campito Mountain data is available in the R package tseries as the dataset camp . We applied the 12 estimators to this series and estimated the goodnessof fit to an FGN for all estimators where possible, except the Haslett-Raftery,using the test of Beran (1992) as implemented in the R package longmemo ofBeran et al. (2006). The Beran test is more powerful against under estimationof H than over estimation. The Beran test was unable to be used for H valuesexceeding unity. It is important to note that Deo and Chen (2000) showed thatthe asymptotic properties of the test as presented by Beran (1992) were incor-rect. We subjected the Beran test to a simulation study, the results of whichwill be presented at a later data. This study showed that for sample sizes ofthe order studied here the Beran test over rejects the null hypothesis by a small imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators . . . . . . (a) Absolute ValueAggregated VarianceWhittle (b)
Boxed PeriodogramDifferenced VarianceHiguchi 2000 4000 6000 8000 10000 . . . . . . (c) PeriodogramPengR/S 2000 4000 6000 8000 10000 (d)
WaveletGPHHaslett−Raftery M ean S qua r ed E rr o r Series Length
Fig 7 . Mean squared errors (MSE) as a function of series length for all 12 estimators with d =0.4 for the GPH and Haslett-Raftery and H =0.9 for the other ten. MSEs are reportedstarting at a series of 500 data points. (a) Absolute Value, Aggregated Variance and Whittle.(b) Boxed Periodogram, Difference Variance and Higuchi. (c) Peng, Periodogram and R/S.(d) Wavelet, GPH and Haslett-Raftery. imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators H Est p-value Seconds H p-valueAbsolute Value 0.862 0.435 0.19 0.831 0.70Aggregated Variance 0.889 0.577 0.34 0.821 0 . ∗ Boxed Periodogram 0.914 0.509 0.09 0.849 0 . ∗∗ Differenced Variance 1.089 - 0.21 0.925 0 . ∗∗ GPH 1.037 - 1.43 0.897 0.16Haslett-Raftery 0.947 0.241 0.17 - -Higuchi 0.966 0.102 19.65 0.845 < . ∗∗∗ Peng 0.936 0.344 18.46 0.875 < . ∗∗∗ Periodogram 1.007 - 0.06 0.908 < . ∗∗∗ Rescaled Range 0.892 0.577 0.04 0.816 0.36Wavelet 0.927 0.421 0.07 0.889 0.25Whittle 0.876 0.540 1.05 0.890 0.15
Table 1
The first two columns of results presents the H estimates and p-values returned by theBeran (1992) test for the Campito Mountain data for each of the 10 estimators of H andthe GPH and Haslett-Raftery estimators of d converted to H equivalent. CPU times are inseconds on the SunBlade described in the text. The expected H column is the expected valuethat the estimator would report the if the Campito data was an FGN with H = 0 . . Theempirical p-value column is estimated empirically from the simulated data. amount (e.g. typically 6 percent at the 5 percent level). Thus for pragmatic test-ing of goodness of fit, the Beran test can still be used with appropriate caution,alternatively critical values can be obtained through simulation.The results are presented in Table (1). The Beran test indicated an H valueclose to 0.89 fitted this data best. The maximum p-value was 0.577 for values of H estimated by the aggregated variance and rescaled range estimators. Nine ofthe 12 estimators reported H or d values which lie in an acceptable range on thebasis of the Beran (1992) test assuming we set our level of statistical significanceat 0.05 to reject the null hypothesis of an FGN. The remaining three could notbe tested.Given the results from our simulated FGNs there were some unexpected H estimates for the Campito data. On the basis of the simulations we expected theaggregated variance, absolute value, boxed periodogram, Higuchi, and rescaledrange to return a low estimate for H . None of these estimators did so. As theBeran test reported that H = 0 .
89 yielded the best fit we used the median valuefrom the simulations with series length 5400 and H = 0 .
90, and adjusted for thedifference of 0.01 H units, to estimate the value of H which would be reported byeach estimator if the data was from an FGN. This value is reported in Table (1)as “Expected H”. The sixth column reports the empirically determined p-valuefor the actual estimate again using the simulated data. We do not report valuesfor the Haslett-Raftery estimator as it estimates d not H . It is interesting thatsix of the estimators reported H estimates which are statistically significantlyhigher than their expected values.The estimator which had the least bias and narrowest confidence interval inthe simulations with series length 5400 and H = 0 .
90, namely the Whittle, wasmarginally out performed by the aggregated variance and rescaled range judgedon the basis of the Beran test. imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators The fourth column of Table (1) reports CPU times in seconds. With presentday computer speeds estimation times on the Campito series are not an issue.Only four estimators required more than one second of CPU time on this 5405observation series. It is evident that some estimators which require longer com-pute times, such as the Higuchi and Peng, did not necessarily yield a moreaccurate estimation of H for this data.
5. Discussion
It is clear from the simulations that not all estimators are created equal. Longmemory occurs in the range 0 . < H <
1. Thus any estimator used to estimatethe strength of the long memory needs to be both accurate and have a lowvariance.The boxed periodogram method was developed specifically to deal with theproblem of having most of the points used to estimate H on the right-hand sideof the graph. This was believed to, possibly, cause bias in the periodogram esti-mator. Beran (1994, p133) and Taqqu et al. (1995) outline some of the reasonssuch a method could be expected to be biased. In the series we investigated herethe box periodogram estimator is inferior to the periodogram estimator it wasintended to improve upon.The differenced variance method was developed to be robust to trends whichwere known to cause spurious long memory in the R/S estimator (Bhattacharya et al.,1983). We did not test its robustness. Teverovsky and Taqqu (1999) establishedthat the differenced variance method had a higher variance than the aggregatedvariance method, a result supported by our simulation study. In fairness to themethod it must be pointed out that Teverovsky and Taqqu (1999) did not intendfor it to be used alone but rather in conjunction with the aggregated variancemethod to test for the presence of shifting means or deterministic trends.The Higuchi (1988) estimator only indirectly estimates H . It estimates thefractal dimension, D , of a series by estimating its path length. As implementedit then converts the estimate of D to H by the simple relationship H = 2 − D .This should be borne in mind if a researcher wishes to estimate D rather than H as it is a simple matter to recover D from the H estimate report by thisimplementation.Taqqu et al. (1995) give a detailed proof that the method of Peng et al. (1994)is asymptotically unbiased. In the simulations the bias was never large buteven at a sample size of 10,000 observations the estimator cannot be consideredunbiased. However, its MSE approaches that of the Periodogram method as theseries length increases which, in turn, is better than several others.The wavelet estimator is asymptotically unbiased. In the simulations the biaswas always small and was unbiased for series with longer than or equal to 4,100observations. imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators
6. Conclusions and Future Research
Of the twelve estimators examined here the Whittle and Haslett-Raftery esti-mators performed the best on simulated series. If we require an estimator tobe close to unbiased across the full range of H values for which long memoryoccurs and have a 95 percent confidence interval width of less than 0.1 H or d units (that is 20 percent of the range for H or d values in which long mem-ory is observed), then for series with less than 4,000 data points they were theonly two estimators worth considering. It should be noted that these estimatorsdid not meet these criteria until the series lengths exceeded 700 and 1000 datapoints respectively. For series with 4,000 or more data points, the Peng estima-tor gave acceptable performance. For series with more than 7,000 data pointsthe periodogram estimator was a worthwhile choice. For series with more than8,200 data points the wavelet became a viable estimator. The remaining sevenestimators did not give acceptable performance at any series lengths examinedand are not recommended.The Higuchi estimator is useful if the researcher wishes to recover the fractaldimension of the time series. In contrast to the other estimators it providesuseful information about a time series if the series is not an FGN (or FI(d))series. As an estimator of H it is inferior to several others.The boxed periodogram method is clearly inferior to the periodogram methodit was intended to improve upon for FGNs. Further research would be neededto test if it is more robust than the periodogram method in series with depar-tures from a pure FGN. This could be accomplished, for example, by simulatingARFIMA series with non-zero AR and MA components or series with structuralbreaks.The R/S estimator is of considerable historical interest but had a majordeficiency in that its MSE plateaued while all other estimators’ MSEs decreasedwith increasing series length. Against this we must note that it was one of thetwo best performing estimators when applied to the Campito data when judgedby the Beran test.The differenced variance estimator was the worst of the twelve estimators inshort series. For series longer than 6,000 data points its MSE was better thanthe R/S and on a par with the absolute value, aggregated variance and Higuchimethods. As noted above, Teverovsky and Taqqu (1999) do not recommend itsuse in isolation as it is part of a test for shifting means or deterministic trends.Teverovsky and Taqqu (1999) also recommend the aggregated and differencedvariance plots always be examined visually. We agree with these recommenda-tions. We did not test its robustness to shifting means or deterministic trends.Some numerical results on its performance in these two situations can be foundin Teverovsky and Taqqu (1999).The application to the Campito data six of the estimators reported statis-tically significantly different H estimates than expected based on the evidencefrom the simulated series. Although the fit of the Campito data to an FGN isgood (p=0.577), these six estimators do not seem to be robust to whatever spe-cific departures from an FGN that are present in the data. This suggests that imsart-ejs ver. 2008/08/29 file: ejs_2009_353.tex date: May 28, 2018 . Rea at al./Long Memory Estimators a researcher should not rely on a single estimator when estimating H and thatthe Beran (1992) test should always be applied to test the goodness of fit of thedata to an FGN, while being aware that its asymptotic properties are currentlyunknown.Because of the apparent lack of robustness, a useful avenue of future researchwould be to quantify the sensitivity of these estimators to various types ofdepartures from an FGN, e.g. FGN series with a small number of shifts in meanor a small number of outlier data points. References
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