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A Range-Based Test for the Parametric Form of the Volatility in Diffusion Models

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  • Mark Podolskij
  • Daniel Ziggel

    ()
    (School of Economics and Management, University of Aarhus, Denmark and CREATES)

Abstract

We propose a new test for the parametric form of the volatility function in continuous time diffusion models of the type dXt = a(t;Xt)dt + (t;Xt)dWt. Our approach involves a range-based estimation of the integrated volatility and the integrated quarticity, which are used to construct the test statistic. Under rather weak assumptions on the drift and volatility we prove weak convergence of the test statistic to a centered mixed Gaussian distribution. As a consequence we obtain a test, which is consistent for any fixed alternative. We also provide a test for neighborhood hypotheses. Moreover, we present a parametric bootstrap procedure which provides a better approximation of the distribution of the test statistic. Finally, it is demonstrated by means of Monte Carlo study that the range-based test is more powerful than the return-based test when comparing at the same sampling frequency.

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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2008-22.

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Length: 23
Date of creation: 14 May 2008
Date of revision:
Handle: RePEc:aah:create:2008-22

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Web page: http://www.econ.au.dk/afn/

Related research

Keywords: Bipower Variation; Central Limit Theorem; Diffusion Models; Goodness-Of- Fit Testing; High-Frequency Data; Integrated Volatility; Range-Based Bipower Variation; Semimartingale Theory;

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References

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  1. Asger Lunde & Peter Reinhard Hansen, 2004. "Realized Variance and IID Market Microstructure Noise," Econometric Society 2004 North American Summer Meetings 526, Econometric Society.
  2. Neil Shephard, 2004. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," Economics Series Working Papers 2004-FE-21, University of Oxford, Department of Economics.
  3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  4. Dette, Holger & Podolskij, Mark, 2005. "Testing the parametric form of the volatility in continuous time diffusion models: an empirical process approach," Technical Reports 2005,50, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  5. Dette, Holger & Podolskij, Mark, 2008. "Testing the parametric form of the volatility in continuous time diffusion models--a stochastic process approach," Journal of Econometrics, Elsevier, vol. 143(1), pages 56-73, March.
  6. Holger Dette & Mark Podolskij & Mathias Vetter, 2006. "Estimation of Integrated Volatility in Continuous-Time Financial Models with Applications to Goodness-of-Fit Testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 259-278.
  7. Chan, K C, et al, 1992. " An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, American Finance Association, vol. 47(3), pages 1209-27, July.
  8. Constantinides, George M, 1992. "A Theory of the Nominal Term Structure of Interest Rates," Review of Financial Studies, Society for Financial Studies, vol. 5(4), pages 531-52.
  9. Christensen, Kim & Podolskij, Mark, 2006. "Range-Based Estimation of Quadratic Variation," Technical Reports 2006,37, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  10. Christensen, Kim & Podolskij, Mark, 2007. "Realized range-based estimation of integrated variance," Journal of Econometrics, Elsevier, vol. 141(2), pages 323-349, December.
  11. Barndorff-Nielsen, Ole Eiler & Graversen, Svend Erik & Jacod, Jean & Podolskij, Mark, 2004. "A central limit theorem for realised power and bipower variations of continuous semimartingales," Technical Reports 2004,51, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  12. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, Econometric Society, vol. 53(2), pages 385-407, March.
  13. Martens, M.P.E. & van Dijk, D.J.C., 2006. "Measuring volatility with the realized range," Econometric Institute Research Papers EI 2006-10, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  14. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
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