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Identifying the Brownian Covariation from the Co-Jumps Given Discrete Observations

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  • Cecilia Mancini

    ()
    (Dipartimento di Matematica per le Decisioni, Universita' degli Studi di Firenze)

  • Fabio Gobbi

    ()
    (Department of Mathematical Economics, University of Bologna)

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    Abstract

    In this paper we consider two semimartingales driven by Wiener processes and (possibly infinite activity) jumps. Given discrete observations we separately estimate the integrated covariation IC from the sum of the co-jumps. The Realized Covariation (RC) approaches the sum of IC with the co-jumps as the number of observations increases to infinity. Our threshold (or truncated) estimator \hat{IC}_n excludes from RC all the terms containing jumps in the finite activity case and the terms containing jumps over the threshold in the infinite activity case, and is consistent. To further measure the dependence between the two processes also the betas, \beta^{(1,2)} and \beta^{(2,1)}, and the correlation coefficient \rho^{(1,2)} among the Brownian semimartingale parts are consistently estimated. In presence of only finite activity jumps: 1) we reach CLTs for \hat{IC}_n, \hat\beta^{(i,j)} and \hat \rho^{(1,2)}; 2) combining thresholding with the observations selection proposed in Hayashi and Yoshida (2005) we reach an estimate of IC which is robust to asynchronous data. We report the results of an illustrative application, made in a web appendix (on www.dmd.unifi.it/upload/sub/persone/mancini/WebAppendix3.pdf), to two very different simulated realistic asset price models and we see that the finite sample performances of \hat{IC}_n and of the sum of the co-jumps estimator are good for values of the observation step large enough to avoid the typical problems arising in presence of microstructure noises in the data. However we find that the co-jumps estimators are more sensible than \hat{IC}_n to the choice of the threshold. Finding criteria for optimal threshold selection is object of further research.

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    Bibliographic Info

    Paper provided by Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa in its series Working Papers - Mathematical Economics with number 2010-05.

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    Length: 50 pages
    Date of creation: Mar 2010
    Date of revision:
    Handle: RePEc:flo:wpaper:2010-05

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    Keywords: co-jumps; integrated covariation; integrated variance; finite activity jumps; infinite activity jumps; threshold estimator;

    This paper has been announced in the following NEP Reports:

    References

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    1. Bollerslev, Tim & Law, Tzuo Hann & Tauchen, George, 2008. "Risk, jumps, and diversification," Journal of Econometrics, Elsevier, vol. 144(1), pages 234-256, May.
    2. Ole BARNDORFF-NIELSEN & Svend Erik GRAVERSEN & Jean JACOD & Mark PODOLSKIJ & Neil SHEPHARD, 2004. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," OFRC Working Papers Series 2004fe21, Oxford Financial Research Centre.
    3. Cecilia Mancini, 2009. "Non-parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 36(2), pages 270-296.
    4. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    5. Egloff, Daniel & Leippold, Markus & Vanini, Paolo, 2007. "A simple model of credit contagion," Journal of Banking & Finance, Elsevier, vol. 31(8), pages 2475-2492, August.
    6. Takaki Hayashi & Shigeo Kusuoka, 2008. "Consistent estimation of covariation under nonsynchronicity," Statistical Inference for Stochastic Processes, Springer, vol. 11(1), pages 93-106, February.
    7. Ole E. Barndorff-Nielsen & Neil Shephard, 2004. "Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics," Econometrica, Econometric Society, vol. 72(3), pages 885-925, 05.
    8. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 517-559, April.
    9. Xin Huang & George Tauchen, 2005. "The Relative Contribution of Jumps to Total Price Variance," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 3(4), pages 456-499.
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