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The Increment Ratio statistic under deterministic trends

  • K. Bružaitė
  • M. Vaičiulis
Article

Abstract

The Increment Ratio (IR) statistic (see (1.1) below) was introduced in Surgailis et al. [16]. The IR statistic can be used for testing nonparametric hypotheses for d-integrated (−1/2 < d < 5/4) behavior of time series, including short memory (d = 0), (stationary) long-memory (0 < d < 1/2), and unit roots (d = 1). For stationary/stationary increment Gaussian observations, in [16], a rate of decay of the bias of the IR statistic and a central limit theorem are obtained. In this paper, we study the asymptotic distribution of the IR statistic under the model X t = X t 0 + g N(t) (t = 1, …, N), where X t 0 is a stationary/stationary increment Gaussian process as in [16], and g N(t) is a slowly varying deterministic trend. In particular, we obtain sufficient conditions on X t 0 and g N(t) under which the IR test has the same asymptotic confidence intervals as in the absence of the trend. We also discuss the asymptotic distribution of the IR statistic under change-points in mean and scale parameters.

Keywords

central limit theorem increment ratio statistic fractional Brownian motion Gaussian processes long memory 

References

  1. 1.
    M.A. Arcones, Limit theorems for nonlinear functionals of a stationary gaussian sequence of vectors, Ann. Probab., 22:2242–2274, 1994.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe, and M.S. Taqqu, Generators of long-range dependent processes: A survey, in P. Doukhan, G. Oppenheim, and M.S. Taqqu (Eds.), Theory and Applications of Long-Range Dependence, Birkhauser, Boston, 2003, pp. 557–577.Google Scholar
  3. 3.
    J.-M. Bardet and D. Surgailis, Measuring the roughness of random paths by increment ratios, Preprint, 2008.Google Scholar
  4. 4.
    R.N. Bhattacharya, V.K. Gupta, and E. Waymire, The Hurst effect under trends, J. Appl. Probab., 20:649–662, 1983.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    K. Bružaitė and M. Vaičiulis, Asymptotic independence of distant partial sums of linear process, Lith. Math. J., 45(4):387–404, 2005.MATHCrossRefGoogle Scholar
  6. 6.
    L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1996.Google Scholar
  7. 7.
    F.X. Diebold and A. Inoue, Long memory and regime switching, J. Econom., 105:131–159, 2001.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    L. Giraitis, P. Kokoszka, and R. Leipus, Testing for long memory in the presence of a general trend, J. Appl. Prob., 38:1033–1054, 2001.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    L. Giraitis, P. Kokoszka, R. Leipus, and G. Teyssière, Rescaled variance and related tests for long memory in volatility and levels, J. Econom., 112:265–294, 2003.MATHCrossRefGoogle Scholar
  10. 10.
    L. Giraitis, R. Leipus, and A. Philippe, A test for stationarity versus trends and unit roots for a wide class of dependent errors, Econom. Theory, 22:989–1029, 2006.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    H. Kunsch, Discrimination between monotonic trends and long-range dependence, J. Appl. Probab., 23:1025–1030, 1996.CrossRefMathSciNetGoogle Scholar
  12. 12.
    R. Leipus and M.-C. Viano, Long memory and stochastic trend, Stat. Probab. Lett., 61:177–190, 2003.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    I.N. Lobato and N.E. Savin, Real and spurious long-memory properties of stock-market data (with comments), J. Business Econ. Stat., 16:261–283, 1998.CrossRefMathSciNetGoogle Scholar
  14. 14.
    K. Shimotsu, Simple (but effective) tests of long memory versus structural breaks, Queen’s Economics Department Working Paper No. 1101, 2006.Google Scholar
  15. 15.
    M. Stoncelis and M. Vaičiulis, Numerical approximations of some infinite Gaussian series and integrals, Preprint, 2008.Google Scholar
  16. 16.
    D. Surgailis, G. Teyssière, and M. Vaičiulis, The increment ratio statistic, J. Multivariate Anal., 99:510–541, 2008.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    V. Teverovsky and M.S. Taqqu, Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator, J. Time Ser. Anal., 18:279–304, 1997.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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