, Volume 75, Issue 8, pp 1111–1127 | Cite as

A new fluctuation test for constant variances with applications to finance

  • Dominik Wied
  • Matthias Arnold
  • Nicolai Bissantz
  • Daniel Ziggel


We present a test to determine whether variances of time series are constant over time. The test statistic is a suitably standardized maximum of cumulative first and second moments. We apply the test to time series of various assets and find that the test performs well in applications. Moreover, we propose a portfolio strategy based on our test which hedges against potential financial crises and show that it works in practice.


Econometric modeling Finance Portfolio optimization Structural breaks Variance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andreou E, Ghysels E (2002) Detecting multiple breaks in financial market volatility dynamics. J Appl Econom 17: 579–600CrossRefGoogle Scholar
  2. Andreou E, Ghysels E (2006) Monitoring disruptions in financial markets. J Econom 135: 77–124MathSciNetCrossRefGoogle Scholar
  3. Andreou E, Ghysels E (2010) Structural breaks in financial time series. In: Andersen T, Davis RA, Kreiss JP, Mikosch T (eds) Handbook of financial time series. Springer, BerlinGoogle Scholar
  4. Andrews DWK (1997) A conditional Kolmogorov test. Econometrica 65(5): 1097–1128MathSciNetMATHCrossRefGoogle Scholar
  5. Aue A, Hörmann S, Horváth L, Reimherr M (2009) Break detection in the covariance structure of multivariate time series models. Ann Stat 37(6B): 4046–4087MATHCrossRefGoogle Scholar
  6. Billingsley P (1968) Convergence of probability measures. Wiley, New YorkMATHGoogle Scholar
  7. Bissantz N, Steinorth V, Ziggel D (2011) Stabilität von Diversifikationseffekten im Markowitz-Modell. AStA Wirtschafts- und Sozialstatistisches Archiv 5(2): 145–157CrossRefGoogle Scholar
  8. Campbell R, Forbes C, Koedijk K, Kofman P (2008) Increasing correlations or just fat tails?. J Empir Finance 15: 287–309CrossRefGoogle Scholar
  9. Carrasco M, Chen XH (2002) Mixing and moment properties of various garch and stochastic volatility models. Econom Theory 18: 17–39MathSciNetMATHCrossRefGoogle Scholar
  10. Chen J, Gupta AK (1997) Testing and locating variance changepoints with application to stock prices. J Am Stat Assoc 92: 739–747MathSciNetMATHCrossRefGoogle Scholar
  11. Chu C-SJ, Stinchcombe M, White H (1996) Monitoring structural change. Econometrica 64(5): 1045–1065MATHCrossRefGoogle Scholar
  12. Chu CS (1995) Detecting parameter shift in GARCH models. Econom Rev 14: 241–266MATHCrossRefGoogle Scholar
  13. Davidson J (1994) Stochastic limit theory. Oxford University Press, OxfordCrossRefGoogle Scholar
  14. Davidson J, de Jong RM (2000) Consistency of kernel estimators of heteroscedastic and autocorrelated covariance matrices. Econometrica 68(2): 407–424MathSciNetMATHCrossRefGoogle Scholar
  15. Dias A, Embrechts P (2004) Change point analysis for dependence structures in finance and insurance. In: Szegö G (ed) Risk measures of the 21st century. Wiley, New York, pp 321–335Google Scholar
  16. Galeano P, Peña D (2007) Covariance changes detection in multivariate time series. J Stat Plan Inference 137(1): 194–211MATHCrossRefGoogle Scholar
  17. Goetzmann WN, Li L, Rouwenhorst KG (2005) Long-term global market correlations. J Bus 78(1): 1–38CrossRefGoogle Scholar
  18. Hansen BE (1991) GARCH(1,1) processes are near-epoch dependent. Econ Lett 36: 181–186MATHCrossRefGoogle Scholar
  19. Inoue A (2001) Testing for distributional change in time series. Econom Theory 17: 156–187MATHCrossRefGoogle Scholar
  20. Jennrich RI (1970) An asymptotic chi-square test for the equality of two correlation matrices. J Am Stat Assoc 65: 904–912MathSciNetMATHGoogle Scholar
  21. Kokoszka P, Leipus R (2000) Change-point estimation in ARCH models. Bernoulli 6: 513–539MathSciNetMATHCrossRefGoogle Scholar
  22. Krishan CNV, Petkova R, Ritchken P (2009) Correlation risk. J Empir Finance 16: 353–367CrossRefGoogle Scholar
  23. Krämer W, Schotman P (1992) Range vs. maximum in the OLS-based version of the CUSUM test. Econ Lett 40: 379–381MATHCrossRefGoogle Scholar
  24. Longin F, Solnik B (2002) Extreme correlation of international equity markets. J Finance 56: 649–675CrossRefGoogle Scholar
  25. Mikosch T, Starica C (2004) Changes of structure in financial time series and the GARCH model. Revstat Stat J 2: 41–73MathSciNetMATHGoogle Scholar
  26. Pearson ES, Wilks SS (1933) Methods of statistical analysis appropriate for k samples of two variables. Biometrika 25: 353–378Google Scholar
  27. Ploberger W, Krämer W (1990) The local power of the CUSUM and CUSUM of squares tests. Econom Theory 6: 335–347CrossRefGoogle Scholar
  28. Ploberger W, Krämer W (1992) The CUSUM-test with OLS residuals. Econometrica 60(2): 271–285MathSciNetMATHCrossRefGoogle Scholar
  29. Ploberger W, Krämer W, Kontrus K (1989) A new test for structural stability in the linear regression model. J Econom 40: 307–318MATHCrossRefGoogle Scholar
  30. Wied D, Krämer W, Dehling H (2011) Testing for a change in correlation at an unknown point in time using an extended functional delta method. Econom Theory, forthcomingGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Dominik Wied
    • 1
  • Matthias Arnold
    • 1
  • Nicolai Bissantz
    • 2
  • Daniel Ziggel
    • 3
  1. 1.Fakultät StatistikTU DortmundDortmundGermany
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  3. 3.quasol GmbHMünsterGermany

Personalised recommendations