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Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes

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Abstract

In this paper, we consider the problem of testing for an autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes. For a test, we propose a class of test statistics constructed by an iterated cumulative sums of squares of the difference between two adjacent observations. It is shown that each of the test statistics weakly converges to the supremum of the square of a Brownian bridge. The test statistics are evaluated by some empirical results.

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References

  1. Bai J. Weak convergence of the sequential empirical processes of residuals in ARMA models. Ann Statist, 1994, 22: 2051–2061

    Article  MathSciNet  MATH  Google Scholar 

  2. Barndorff-Nielsen O E. Processes of normal inverse Gaussian type. Finance Stoch, 1998, 2: 41–68

    Article  MathSciNet  MATH  Google Scholar 

  3. Barndorff-Nielsen O E. Superposition of Ornstein-Uhlenbeck type processes. Theory Probab Appl, 2001, 45: 175–194

    Article  MathSciNet  Google Scholar 

  4. Barndorff-Nielsen O E, Halgreen C. Infinite divisibility of the hyperbolic and generalized inverse gaussian distributions. Z Wahrscheinlichkeitstheor Verw Geb, 1977, 38: 309–311

    Article  MathSciNet  Google Scholar 

  5. Barndorff-Nielsen O E, Jensen J L, Søensen M. Some stationary processes in discrete and continuous time. Adv in Appl Probab, 1998, 30: 989–1007

    Article  MathSciNet  MATH  Google Scholar 

  6. Barndorff-Nielsen O E, Shephard N. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B Stat Methodol, 2001, 63: 167–241

    Article  MathSciNet  MATH  Google Scholar 

  7. Barndorff-Nielsen O E, Shephard N. Normal modified stable processes. Theory Probab Math Statist, 2001, 65: 1–19

    MathSciNet  MATH  Google Scholar 

  8. Barndorff-Nielsen O E, Shephard N. Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen O E, Mikosch T, Resnick S, eds. Lévy Processes-Theory and Applications. Boston: Birkhäuser, 2001, 283–318

    Chapter  Google Scholar 

  9. Barndorff-Nielsen O E, Shephard N. Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J R Stat Soc Ser B Stat Method, 2002, 64: 253–280

    Article  MathSciNet  MATH  Google Scholar 

  10. Barndorff-Nielsen O E, Shephard N. Integrated OU processes and non-Gaussian OU-based stochastic volatility models. Scand J Statist, 2003, 30: 277–295

    Article  MathSciNet  MATH  Google Scholar 

  11. Barndorff-Nielsen O E, Leonenko N N. Spectral properties of superpositions of Ornstein-Uhlenbeck type processes. Methodol Comput Appl Probab, 2005, 7: 335–352

    Article  MathSciNet  MATH  Google Scholar 

  12. Benth F E, Gorth M, Kufakunesu R. Valuing volatility and variance swaps for a non-Gaussian Ornstein-Uhlenbeck stochastic volatility model. Appl Math Finance, 2007, 14: 347–363

    Article  MathSciNet  MATH  Google Scholar 

  13. Billingsley P. Statistical Inference for Markov Processes. Chicago: The University of Chicago Press, 1961

    MATH  Google Scholar 

  14. Billingsley P. Convergence of Probability Measures. New York: John Wiley, 1968

    MATH  Google Scholar 

  15. Brockwell P J, Davis R A, Yang Y. Estimation for nonnegative Lévy-driven Ornstein-Uhlenbeck processes. J Appl Probab, 2007, 44: 977–989

    Article  MathSciNet  MATH  Google Scholar 

  16. Cao G, Wang Z, Wu Y, et al. Inference of change-point in single index models. Sci China Ser A, 2008, 51: 1855–1870

    Article  MathSciNet  MATH  Google Scholar 

  17. Cariboni C, Schoutens W. Jumps in intensity models: investigating the performance of Ornstein-Uhlenbeck processes in credit risk modeling. Metrika, 2009, 69: 173–198

    Article  MathSciNet  Google Scholar 

  18. Chen J, Gupta A K. Parametric Statistical Change Point Analysis. Boston: Birkhäuser, 2000

    MATH  Google Scholar 

  19. Csörgő M, Horváth L. L Limit Theorems in Change-Point Analysis. New York: Wiley, 1997

    Google Scholar 

  20. Dedecker J, Rio E. On the functional central limit theorem for stationary processes. Ann Inst H Poincaré Probab Statist, 2000, 36: 1–34

    Article  MathSciNet  MATH  Google Scholar 

  21. Durrett R. Probability: Theory and Examples, 3rd edition. Belmont: Thomson, 2004

    Google Scholar 

  22. Halgreen C. Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z Wahrscheinlichkeitstheor Verw Geb, 1979, 47: 13–17

    Article  MathSciNet  MATH  Google Scholar 

  23. Helland I S. Central limit theorems for martingales with discrete or continuous time. Scand J Statist, 1982, 9: 79–94

    MathSciNet  MATH  Google Scholar 

  24. Hougaard P. Survival models for heterogeneous populations derived from stable distributions. Biometrika, 1986, 73: 387–396

    Article  MathSciNet  MATH  Google Scholar 

  25. Inclán C, Tiao G C. Use of cumulative sums of squares for retrospective detection of changes of variances. J Amer Statist Assoc, 1994, 89: 913–923

    Article  MathSciNet  MATH  Google Scholar 

  26. Jongbloed G, Van Der Meulen F H, Van Der Vaart A W. Non-parametric inference for Lévy driven Ornstein-Uhlenbeck processes. Bernoulli, 2005, 11: 759–791

    Article  MathSciNet  MATH  Google Scholar 

  27. Jongbloed G, Van Der Meulen F H. Parametric estimation for subordinators and induced OU processes. Scand J Statist, 2006, 33: 825–847

    Article  MathSciNet  MATH  Google Scholar 

  28. Lee S, Ha J, Na O, et al. The cusum test for parameter change in time series models. Scand J Statist, 2003, 30: 781–796

    Article  MathSciNet  MATH  Google Scholar 

  29. Lee S, Nishiyama Y, Yoshida N. Test for parameter change in diffusion processes by cusum statistics based on one-step estimator. Ann Inst Statist Math, 2006, 58: 211–222

    Article  MathSciNet  MATH  Google Scholar 

  30. Madan D B, Carr P, Chang E C. The variance gamma process and option pricing. Europ Finance Rev, 1998, 2: 79–105

    Article  MATH  Google Scholar 

  31. Madan D B, Milne F. Option pricing with V.G. martingale components. Math Finance, 1991, 1: 39–55

    Article  MATH  Google Scholar 

  32. Madan D B, Seneta E. The Variance-Gamma (V.G.) model for share market returns. J Bus, 1990, 63: 511–524

    Article  Google Scholar 

  33. Masuda H. On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process. Bernoulli, 2004, 10: 97–120

    Article  MathSciNet  MATH  Google Scholar 

  34. Sato K. Lévy processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999

    MATH  Google Scholar 

  35. Sato K, Yamazato M. Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Process Appl, 1984, 17: 73–100

    Article  MathSciNet  MATH  Google Scholar 

  36. Schoutens W. Lévy Processes in Finance. Chichester: Wiley, 2003

    Book  Google Scholar 

  37. Shi X, Miao B, Ge C. Consistency of change point estimators for symmetrical stable distribution with parameters shift. Sci China Ser A, 2008, 51: 842–850

    Article  MathSciNet  Google Scholar 

  38. Song J, Lee S. Test for parameter change in discretely observed diffusion processes. Stat Inference Stoch Process, 2009, 12: 165–183

    Article  MathSciNet  MATH  Google Scholar 

  39. Sun S, Zhang X. Empirical likelihood estimation of discretely sampled processes of OU type. Sci China Ser A, 2009, 52: 908–931

    Article  MathSciNet  MATH  Google Scholar 

  40. Tweedie M. An index which distinguishes between some important exponential families. In: Ghosh J, Roy J, eds. Statistics: Applications and New Directions, Proc. Indian Statistical Institute Golden Jubilee International Conference. Calcutta: Indian Statistical Institute, 1984, 579–604

    Google Scholar 

  41. Wolfe J. On a continuous analogue of the stochastic difference equation X n = ρX n−1 + B n. Stochastic Process Appl, 1982, 12: 301–312

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang S. Transition-law based simulation of generalized inverse Gaussian Ornstein-Uhlenbeck processes. Methodol Comput Appl Probab, 2011, 13: 619–656

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang S. Exact simulation of tempered stable Ornstein-Uhlenbeck processes. J Stat Comput Simul, 2011, 81: 1533–1544

    Article  MathSciNet  Google Scholar 

  44. Zhang S, Zhang X. Exact simulation of IG-OU processes. Methodol Comput Appl Probab, 2008, 10: 337–355

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang S, Zhang X. A least squares estimator for discretely observed Ornstein-Uhlenbeck processes driven by symmetric α-stable motions. Ann Inst Statist Math, 2012, doi: 10.1007/s10463-012-0362-0

  46. Zhang S, Zhang X, Sun S. Parametric estimation of discretely sampled Gamma-OU processes. Sci China Ser A, 2006, 49: 1231–1257

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China

    ShiBin Zhang

  2. Department of Statistics, Fudan University, Shanghai, 200433, China

    XinSheng Zhang

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  1. ShiBin Zhang
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  2. XinSheng Zhang
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Correspondence to ShiBin Zhang.

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Zhang, S., Zhang, X. Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes. Sci. China Math. 56, 339–357 (2013). https://doi.org/10.1007/s11425-012-4511-y

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  • DOI: https://doi.org/10.1007/s11425-012-4511-y

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