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AStA Advances in Statistical Analysis

, Volume 101, Issue 3, pp 289–308 | Cite as

Fourier methods for analyzing piecewise constant volatilities

  • Max Wornowizki
  • Roland FriedEmail author
  • Simos G. Meintanis
Original Paper
  • 161 Downloads

Abstract

We develop procedures for testing whether a sequence of independent random variables has constant variance. If this is fulfilled, the modulus of a Fourier-type transformation of the volatility process is identically equal to one. Our approach takes advantage of this property considering a canonical estimator for the modulus under the assumption of piecewise identically distributed zero mean observations. Using blockwise variance estimation, we introduce several test statistics resulting from different weight functions. All of them are given by simple explicit formulae. We prove the consistency of the corresponding tests and compare them to alternative procedures on extensive Monte Carlo experiments. According to the results, our proposals offer fairly high power, particularly in the case of multiple structural breaks. They also allow for an adequate estimation of the change point positions. We apply our procedure to gold mining data and also briefly discuss how it can be modified to test for the stationarity of other distributional parameters.

Keywords

Change point analysis Variance Piecewise identical distribution Independence Weight function 

Notes

Acknowledgements

The work was supported in part by the Collaborative Research Center 823, Project C3 (Analysis of Structural Change in Dynamic Processes), of the German Research Foundation and by Grant No.11699 of the Special Account for Research Grants (ELKE) of the National and Kapodistrian University of Athens. We also thank the anonymous referees for their valuable remarks which helped us to improve this work.

References

  1. Brooks, C., Burke, S.P., Heravi, S., Persand, G.: Autoregressive conditional kurtosis. J. Financ. Econom. 3(3), 399–421 (2005)CrossRefGoogle Scholar
  2. Davies, L., Höhenrieder, C., Krämer, W.: Recursive computation of piecewise constant volatilities. Comput. Stat. Data Anal. 56(11), 3623–3631 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Epps, T.W.: Characteristic functions and their empirical counterparts: geometrical interpretations and applications to statistical inference. Am. Stat. 47(1), 33–38 (1993)Google Scholar
  4. Epps, T.W.: Limiting behavior of the ICF test for normality under Gram–Charlier alternatives. Stat. Prob. Lett. 42(2), 175–184 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Feunou, B., Tédongap, R.: A stochastic volatility model with conditional skewness. J. Bus. Econ. Stat. 30(4), 576–591 (2012)MathSciNetCrossRefGoogle Scholar
  6. Fisher, R.A.: The design of experiments. Oliver & Boyd, Oxford (1935)Google Scholar
  7. Fried, R.: On the online estimation of local constant volatilities. Comput. Stat. Data Anal. 56(11), 3080–3090 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Good, P.: Multiple tests. Permutation, parametric and bootstrap tests of hypotheses. Springer, New York (2005)zbMATHGoogle Scholar
  9. Guégan, D.: Non-stationary samples and meta-distribution. In: Basu, A., Samanta, T., Sen Gupta, A. (eds.) Statistical Paradigms Recent Advances and Reconciliations, vol. 14. World Scientific, New Jersey (2015)Google Scholar
  10. Harvey, C.R., Siddique, A.: Autoregressive conditional skewness. J. Financ. Quant. Anal. 34(4), 465–487 (1999)CrossRefGoogle Scholar
  11. Hlávka, Z., Hušková, M., Kirch, C., Meintanis, S.G.: Monitoring changes in the error distribution of autoregressive models based on Fourier methods. Test 21(4), 605–634 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hlávka, Z., Hušková, M., Kirch, C., Meintanis, S.G.: Fourier-type tests involving martingale difference processes. Econ. Rev. (2015). doi: 10.1080/07474938.2014.977074
  13. Hsu, D.A.: Tests for variance shift at an unknown time point. J. R. Stat. Soc. 26(3), 279–284 (1977)Google Scholar
  14. Hušková, M., Meintanis, S.G.: Change point analysis based on empirical characteristic functions. Metrika 63(2), 145–168 (2006a)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hušková, M., Meintanis, S.G.: Change-point analysis based on empirical characteristic functions of ranks. Seq. Anal. 25(4), 421–436 (2006b)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Jandhyala, V.K., Fotopoulos, S.B., Hawkins, D.M.: Detection and estimation of abrupt changes in the variability of a process. Comput. Stat. Data Anal. 40(1), 1–19 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jiménez-Gamero, M.D., Alba-Fernández, V., Muñoz-García, J., Chalco-Cano, Y.: Goodness-of-fit tests based on empirical characteristic functions. Comput. Stat. Data Anal. 53(12), 3957–3971 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Loéve, M.: Probability Theory I. Springer, New York (1977)zbMATHGoogle Scholar
  19. Matteson, D.S., James, N.A.: A nonparametric approach for multiple change point analysis of multivariate data. J. Am. Stat. Assoc. 109(505), 334–345 (2014)MathSciNetCrossRefGoogle Scholar
  20. Meintanis, S.G.: Permutation tests for homogeneity based on the empirical characteristic function. Nonparametr. Stat. 17(5), 583–592 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Meintanis, S.G., Swanepoel, J., Allison, J.: The probability weighted characteristic function and goodness-of-fit testing. J. Stat. Plan. Inference 146, 122–132 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Mercurio, D., Spokoiny, V.: Statistical inference for time-inhomogeneous volatility models. Ann. Stat. 32(2), 577–602 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Mood, A.M.: On the asymptotic efficiency of certain nonparametric two-sample tests. Ann. Math. Stat. 25(3), 514–522 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Pardo-Fernández, J.C., Jiménez-Gamero, M.D., El Ghouch, A.: A nonparametric ANOVA-type test for regression curves based on characteristic functions. Scand. J. Stat. 42, 197–213 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Peña, D.: Análisis de Series Temporales, vol. 319. Alianza Editorial, Madrid (2005)Google Scholar
  26. Potgieter, C.J., Genton, M.G.: Characteristic function-based semiparametric inference for skew-symmetric models. Scand. J. Stat. 40(3), 471–490 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Ross, G.J.: Modelling financial volatility in the presence of abrupt changes. Phys. A Stat. Mech. Appl. 392(2), 350–360 (2013)CrossRefGoogle Scholar
  28. Rowland, R.S.J., Sichel, H.: Statistical quality control of routine underground sampling. J. S. Afr. Inst. Min. Metal 60, 251–284 (1960)Google Scholar
  29. Spokoiny, V.: Multiscale local change point detection with applications to value-at-risk. Ann. Stat. 37(3), 1405–1436 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Steland, A., Rafajłowicz, E.: Decoupling change-point detection based on characteristic functions: methodology, asymptotics, subsampling and application. J. Stat. Plan. Inference 145, 49–73 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Tenreiro, C.: On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Comput. Stat. Data Anal. 53, 1038–1053 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Vostrikova, L.J.: Detecting disorder in multidimensional random process. Sov. Math. Dokl. 24, 55–59 (1981)zbMATHGoogle Scholar
  33. Wied, D., Arnold, M., Bissantz, N., Ziggel, D.: A new fluctuation test for constant variances with applications to finance. Metrika 75(8), 1111–1127 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Max Wornowizki
    • 1
  • Roland Fried
    • 1
    Email author
  • Simos G. Meintanis
    • 2
    • 3
  1. 1.Department of StatisticsTU Dortmund UniversityDortmundGermany
  2. 2.Department of EconomicsNational and Kapodistrian University of AthensAthensGreece
  3. 3.Unit for Business Mathematics and InformaticsNorth-West UniversityPotchefstroomSouth Africa

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