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Estimation methods for the LRD parameter under a change in the mean

  • Aeneas Rooch
  • Ieva Zelo
  • Roland Fried
Regular Article
  • 127 Downloads

Abstract

When analyzing time series which are supposed to exhibit long-range dependence (LRD), a basic issue is the estimation of the LRD parameter, for example the Hurst parameter \(H \in (1/2, 1)\). Conventional estimators of H easily lead to spurious detection of long memory if the time series includes a shift in the mean. This defect has fatal consequences in change-point problems: Tests for a level shift rely on H, which needs to be estimated before, but this estimation is distorted by the level shift. We investigate two blocks approaches to adapt estimators of H to the case that the time series includes a jump and compare them with other natural techniques as well as with estimators based on the trimming idea via simulations. These techniques improve the estimation of H if there is indeed a change in the mean. In the absence of such a change, the methods little affect the usual estimation. As adaption, we recommend an overlapping blocks approach: If one uses a consistent estimator, the adaption will preserve this property and it performs well in simulations.

Keywords

Hurst parameter Estimation Jump Long-range dependence Long memory Change-point problems 

Mathematics Subject Classification

62M10 

Notes

Acknowledgments

The financial support of the Deutsche Forschungsgemeinschaft (SFB 823, “Statistical modelling of nonlinear dynamic processes”) is gratefully acknowledged. We would like to thank two referees for their constructive comments which improved the presentation of our work considerably.

References

  1. Baillie RT (1996) Long memory processes and fractional integration in econometrics. J Econom 73(1):5–59MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barkoulas JT, Baum CF, Travlos N (2000) Long memory in the Greek stock market. Appl Financial Econom 10(2):177–184CrossRefGoogle Scholar
  3. Beran J (1994) Statistics for long-memory processes. Chapman & Hall/CRC, Boca Raton ISBN 0-412-04901-5zbMATHGoogle Scholar
  4. Beran J, Feng Y, Ghosh S, Kulik R (2013) Long-memory processes: probabilistic properties and statistical methods. Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. Berkes I, Horváth L, Kokoszka P, Shao Q-M (2006) On discriminating between long-range dependence and changes in the mean. Ann Stat 34:1140–1165MathSciNetCrossRefzbMATHGoogle Scholar
  6. Breidt FJ, Crato N, de Lima P (1998) The detection and estimation of long memory in stochastic volatility. J Econom 83(1–2):325–348MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cheung Y-W, Lai KS (1995) A search for long memory in international stock market returns. J Int Money Finance 14(4):597–615CrossRefGoogle Scholar
  8. Csörgő M, Horváth L (1997) Limit theorems in change-point analysis. Wiley, Chichester ISBN 0-471-95522-1zbMATHGoogle Scholar
  9. Cutland NJ, Kopp PE, Willinger W (1995) Stock price returns and the Joseph effect: a fractional version of the Black–Scholes model. In: Bolthausen E, Dozzi M, Russo F (eds) Seminar on stochastic analysis, random fields and applications. Birkhäuser, Boston, pp 327–351CrossRefGoogle Scholar
  10. Dehling H, Rooch A, Taqqu MS (2013) Nonparametric change-point tests for long-range dependent data. Scand J Stat 40:153–173MathSciNetCrossRefzbMATHGoogle Scholar
  11. Deo RS, Hurvich CM (2013) Linear trend with fractionally integrated errors. J Time Ser Anal 19(4):379–397MathSciNetCrossRefzbMATHGoogle Scholar
  12. Diebold FX, Inoue A (2001) Long memory and regime switching. J Econom 105(1):131–159MathSciNetCrossRefzbMATHGoogle Scholar
  13. Erramilli A, Narayan O, Willinger W (1996) Experimental queueing analysis with long-range dependent packet traffic. IEEE/ACM Trans Netw 4(2):209–223CrossRefGoogle Scholar
  14. Geweke J, Porter-Hudak S (1983) The estimation and application of long-memory times series models. J Time Ser Anal 4(4):221–238MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gil-Alana LA (2005) Statistical modeling of the temperatures in the Northern Hemisphere using fractional integration techniques. J Clim 18(24):5357–5369CrossRefGoogle Scholar
  16. Giraitis L, Leipus R, Surgailis D (1996) The change-point problem for dependent observations. J Stat Plan Inference 53:297–310MathSciNetCrossRefzbMATHGoogle Scholar
  17. Granger CW, Hyung N (2004) Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. J Empir Finance 11(3):399–421CrossRefGoogle Scholar
  18. Hassler U, Olivares M (2008) Long memory and structural change: new evidence from German stock market returns. Goethe University Frankfurt Discussion PaperGoogle Scholar
  19. Hassler U, Scheithauer J (2009) Detecting changes from short to long memory. Stat Papers 52:847–870MathSciNetCrossRefzbMATHGoogle Scholar
  20. Horváth L, Kokoszka P (1997) The effect of long-range dependence on change-point estimators. J Stat Plan Inference 64:57–81MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hsu C-C (2005) Long memory or structural changes: an empirical examination on inflation rates. Econ Lett 88(2):289–294MathSciNetCrossRefzbMATHGoogle Scholar
  22. Hurvich CM, Beltrao KI (1993) Asymptotics for the low frequency ordinates of the periodogram of a long-memory time series. J Time Ser Anal 14(5):455–472MathSciNetCrossRefzbMATHGoogle Scholar
  23. Hurvich CM, Deo R, Brodsky J (1998) The mean squared error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time series. J Time Ser Anal 19(1):19–46MathSciNetCrossRefzbMATHGoogle Scholar
  24. Iacone F (2010) Local Whittle estimation of the memory parameter in presence of deterministic components. J Time Ser Anal 31(1):37–49MathSciNetCrossRefzbMATHGoogle Scholar
  25. Jandhyala V, Fotopoulos S, MacNeill I, Liu P (2013) Inference for single and multiple change-points in time series. J Time Ser Anal. doi: 10.1111/jtsa12035
  26. Jaruskova D (1997) Some problems with application of change-point detection methods to environmental data. Environmetrics 8(5):469–484CrossRefGoogle Scholar
  27. Karagiannis T, Faloutsos M, Riedi RH (2002) Long-range dependence: now you see it, now you don’t!. Global Telecommun Conf 3:2165–2169Google Scholar
  28. Kokoszka P, Leipus R (1998) Change-point in the mean of dependent observations. Stat Probab Lett 40:385–393MathSciNetCrossRefzbMATHGoogle Scholar
  29. Krämer W, Sibbertsen P (2002) Testing for structural changes in the persistence of long memory. Int J Bus Econ 1:235–242Google Scholar
  30. Künsch HR (1987) Statistical aspects of self-similar processes. In: Prohorov YuA, Sazonov VV (eds) Proceedings of the first world congres of the Bernoulli society, vol 1. VNU Science Press, Utrecht, pp 67–74Google Scholar
  31. Li Q, Mills DL (1998) On the long-range dependence of packet round-trip delays in internet. In: Proceedings of IEEE ICC98, pp 1185–1191 (1998)Google Scholar
  32. Lo AW (1991) Long-term memory in stock market prices. Econometrica 59:1279–1313CrossRefzbMATHGoogle Scholar
  33. McCloskey A, Perron P (2013) Memory parameter estimation in the presence of level shifts and deterministic trends. Econom Theory 29(06):1196–1237MathSciNetCrossRefzbMATHGoogle Scholar
  34. Montanari A, Rosso R, Taqqu MS (2000) A seasonal fractional ARIMA model applied to the Nile River monthly flows at Aswan. Water Resour Res 36(5):1249–1259CrossRefGoogle Scholar
  35. Percival DB, Walden AT (2006) Wavelet methods for time series analysis, vol 4. Cambridge University Press, CambridgezbMATHGoogle Scholar
  36. Phillips PC, Shimotsu K (2004) Local Whittle estimation in nonstationary and unit root cases. Ann Stat 32(2):656–692MathSciNetCrossRefzbMATHGoogle Scholar
  37. Rachinger H (2011) Multiple breaks in long memory time series. Job Market Paper, Universidad Carlos III de MadridGoogle Scholar
  38. Reisen V, Abraham B, Lopes S (2001) Estimation of parameters in ARFIMA processes: a simulation study. Commun Stat-Simul Comput 30(4):787–803MathSciNetCrossRefzbMATHGoogle Scholar
  39. Robinson PM (1995a) Gaussian semiparametric estimation of long range dependence. Ann Stat 5:1630–1661Google Scholar
  40. Robinson PM (1995b) Log-periodogram regression of time series with long range dependence. Ann Stat 3:1048–1072Google Scholar
  41. Shao X (2011) A simple test of changes in mean in the possible presence of long-range dependence. J Time Ser Anal 32(6):598–606MathSciNetCrossRefzbMATHGoogle Scholar
  42. Shimotsu K, Phillips PC (2005) Exact local Whittle estimation of fractional integration. Ann Stat 33(4):1890–1933MathSciNetCrossRefzbMATHGoogle Scholar
  43. Sibbertsen P (2004) Long memory versus structural breaks: an overview. Stat Papers 45:465–515MathSciNetCrossRefzbMATHGoogle Scholar
  44. Sibbertsen P, Willert J (2010) Testing for a break in persistence under long-range dependencies and mean shifts. Stat Papers 53:357–370MathSciNetCrossRefzbMATHGoogle Scholar
  45. Smith RL (1993) Long-range dependence and global warming. In: Barnett VD, Turkman KF (eds) Statistics for the environment. Wiley, Chichester, pp 141–161Google Scholar
  46. Taqqu MS, Teverovsky V (1998) On estimating the intensity of long-range dependence in finite and infinite variance time series. In: Adler R, Feldmann R, Taqqu MS (eds) A practical guide to heavy tails: statistical techniques and applications. Birkhäuser, Boston, pp 177–217Google Scholar
  47. Taqqu MS, Teverovsky V, Willinger W (1995) Estimators for long-range dependence: an emprical study. Fractals 3:785–798CrossRefzbMATHGoogle Scholar
  48. Velasco C (1999) Gaussian semiparametric estimation of non-stationary time series. J Time Ser Anal 20(1):87–127MathSciNetCrossRefGoogle Scholar
  49. Velasco C, Robinson PM (2000) Whittle pseudo-maximum likelihood estimation for nonstationary time series. J Am Stat Assoc 95(452):1229–1243MathSciNetCrossRefzbMATHGoogle Scholar
  50. Wang L (2008a) Change-in-mean problem for long memory time series models with applications. J Stat Comput Simul 78(7):653–668MathSciNetCrossRefzbMATHGoogle Scholar
  51. Wang L (2008b) Change-point detection with rank statistics in long-memory time-series models. Aust N Z J Stat 50(3):241–256MathSciNetCrossRefzbMATHGoogle Scholar
  52. Wang L (2008c) Change-point estimation in long memory nonparametric models with applications. Commun Stat-Simul Comput 37(1):48–61MathSciNetCrossRefzbMATHGoogle Scholar
  53. Whittle P (1953) Estimation and information in stationary time series. Arkiv för matematik 2(5):423–434MathSciNetCrossRefzbMATHGoogle Scholar
  54. Willinger W, Taqqu MS, Teverovsky V (1999) Stock market prices and long-range dependence. Finance Stoch 3(1):1–13MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Fakultät für StatistikTechnische Universität DortmundDortmundGermany

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