Computational Economics

, Volume 25, Issue 1–2, pp 103–113 | Cite as

Tests of Long Memory: A Bootstrap Approach

  • Pilar Grau-CarlesEmail author


Many time series in diverse fields have been found to exhibit long memory. This paper analyzes the behaviour of some of the most used tests of long memory: the R/S analysis, the modified R/S, the Geweke and Porter-Hudak (GPH) test and the detrended fluctuation analysis (DFA). Some of these tests exhibit size distortions in small samples. It is well known that the bootstrap procedure may correct this fact. Here I examine the size and power of those tests for finite samples and different distributions, such as the normal, uniform, and lognormal. In the short-memory processes such as AR, MA and ARCH and long memory ones such as ARFIMA, p-values are calculated using the post-blackening moving-block bootstrap. The Monte Carlo study suggests that the bootstrap critical values perform better. The results are applied to financial return time series.


long-memory tests bootstrap time series 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andersson, M.K. and Gedenhoff, M.P. (1997). Bootstrap testing for fractional integration. Working papers Series in Economics and Finance No. 188. Department of Economic Statistic, Stockhlom School of Economics, Sweden.Google Scholar
  2. Beran, J. (1994). Statistics for Long Memory Processes, Chapman and Hall, London.Google Scholar
  3. Cheung, Y.-W. (1993). Tests for fractional integration: A Monte Carlo investigation. Journal of Time Series Analysis, 14(4), 331–345.MathSciNetGoogle Scholar
  4. Davidson, R. and MacKinnon, J. (1996). The size distortion of bootstrap tests, GREQAM Document de Travail No 96A15.Google Scholar
  5. Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and Their Application, Cambridge Univesity Press, Cambridge.Google Scholar
  6. Efron, B. (1979). Bootstrap methods: Another look at the Jackknife. Annals of Statistics, 7, 1–26.Google Scholar
  7. Efron, B. and Tibshirani, R.J. (1993). An Introduction to Bootstrap, Chapman and Hall, New York.Google Scholar
  8. Geweke, J. and Porter-Hudak, S. (1983). The estimation and appication of long memory time series models. Journal of Time Series Analysis, 4, 221–238.Google Scholar
  9. Granger, C.W.J. and Ding, Z. (1993). Some Properties fo Absolute Return: An Alternative Measure of Risk. University of California at San Diego, Economic Working Paper Series, pp. 93–38.Google Scholar
  10. Hosking, J. (1981). Fractional differencing. Biometrika, 68, 165–176.Google Scholar
  11. Hu, K., Ivanov, P.C., Chen, Z., Carpena, P. and Stanley, E. (2001). Effects of trends on detrended fluctuation analysis. Physical Review E, 64, 01114–01119.Google Scholar
  12. Hurst, H.E. (1951). Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, 116, 770–199.Google Scholar
  13. Hurvich, C.M., Deo, R. and Brodsky, J. (1998). The mean squared error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time series. Journal of Time Series Analysis, 19, 19–46.CrossRefGoogle Scholar
  14. Kokoszka, A.J. and Taqqu, M.S. (1995). Fractional ARIMA with stable innovations. Stochastic Processes and Their Applications, 60, 19–47.Google Scholar
  15. Lo, A. (1991). Long term memory in stock market prices. Econometrica, 59, 451–474.Google Scholar
  16. Mandelbrot, B.B. and Wallis, J.R. (1968). Noah, Joseph and operational hydrology. Water Resources Research, 4, 909–918.Google Scholar
  17. Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E. and Goldberger, A.L. (1994). Mosaic organization of DNA sequences. Physical Review E, 49, 1684–1989.CrossRefGoogle Scholar
  18. Robinson, P.M. (1995). Log-periodogram regression of time series with long range dependence. Annals of Statistics, 23, 1048–1072.Google Scholar
  19. Srinivas, V.V. and Srinivasan, K. (2000). Post-blackening approach for modeling dependent annual streamflows, Journal of Hydrology, 230, 86–126.CrossRefGoogle Scholar
  20. Taquu, M.S., Teverovsky, V., Alder, R., Feldman, R. and Taquu, M.S. (eds.) (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhauser, Boston, pp. 177–217.Google Scholar
  21. Taquu, M.S., Teverovsky, V. and Willinger, W. (1995). Estimators for long-range dependence: An empirical study. Fractals, 3(4), 785–788.Google Scholar
  22. Teverovsky, V., Taqqu, M.S. and Willinger, W. (1999). A critical look at Lo modified R/S statistic. Journal of Statistical Planning and Inference, 80, 211–227.CrossRefMathSciNetGoogle Scholar
  23. Xiao, Z. (2003). Note on the bandwidth selection in testing for long range dependence. Economic Letters, 78, 33–39.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Departamento de Economía AplicadaUniversidad Rey Juan CarlosMadridSpain

Personalised recommendations