IDEAS home Printed from https://ideas.repec.org/a/bpj/strimo/v21y2003i1-2003p47-68n6.html
   My bibliography  Save this article

Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models

Author

Listed:
  • Woerner Jeannette H. C.

Abstract

In the framework of general semimartingale models we provide limit theorems for variational sums including the p-th power variation, i.e. the sum of p-th absolute powers of increments of a process. This gives new insight in the use of quadratic and realised power variation as an estimate for the integrated volatility in finance. It also provides a criterion to decide from high frequency data, whether a jump component should be included in the model. Furthermore, results on the asymptotic behaviour of integrals with respect to Lévy processes, estimates for integrals with respect to Lévy measures and non-parametric estimation for Lévy processes will be derived and viewed in the framework of variational sums.

Suggested Citation

  • Woerner Jeannette H. C., 2003. "Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models," Statistics & Risk Modeling, De Gruyter, vol. 21(1), pages 47-68, January.
    • Handle: RePEc:bpj:strimo:v:21:y:2003:i:1/2003:p:47-68:n:6
    DOI: 10.1524/stnd.21.1.47.20316
    as

    Download full text from publisher

    File URL: https://doi.org/10.1524/stnd.21.1.47.20316
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1524/stnd.21.1.47.20316?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Andersen T. G & Bollerslev T. & Diebold F. X & Labys P., 2001. "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 42-55, March.
    2. John M. Maheu & Thomas H. McCurdy, 2002. "Nonlinear Features of Realized FX Volatility," The Review of Economics and Statistics, MIT Press, vol. 84(4), pages 668-681, November.
    3. John W. Galbraith & Victoria Zinde-Walsh, 2000. "Properties of Estimates of Daily GARCH Parameters Based on Intra-Day Observations," Econometric Society World Congress 2000 Contributed Papers 1800, Econometric Society.
    4. C. W. J. Granger & Zhuanxin Ding, 1995. "Some Properties of Absolute Return: An Alternative Measure of Risk," Annals of Economics and Statistics, GENES, issue 40, pages 67-91.
    5. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    6. Andersen, Torben G. & Bollerslev, Tim & Diebold, Francis X. & Ebens, Heiko, 2001. "The distribution of realized stock return volatility," Journal of Financial Economics, Elsevier, vol. 61(1), pages 43-76, July.
    7. West, Kenneth D. & Cho, Dongchul, 1995. "The predictive ability of several models of exchange rate volatility," Journal of Econometrics, Elsevier, vol. 69(2), pages 367-391, October.
    8. Jorion, Philippe, 1995. "Predicting Volatility in the Foreign Exchange Market," Journal of Finance, American Finance Association, vol. 50(2), pages 507-528, June.
    9. Ole E. Barndorff-Nielsen & Neil Shephard, 2001. "Realised power variation and stochastic volatility models," Economics Papers 2001-W18, Economics Group, Nuffield College, University of Oxford.
    10. Andersen, Torben G. & Bollerslev, Tim, 1997. "Intraday periodicity and volatility persistence in financial markets," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 115-158, June.
    11. Yacine Aït‐Sahalia, 2002. "Telling from Discrete Data Whether the Underlying Continuous‐Time Model Is a Diffusion," Journal of Finance, American Finance Association, vol. 57(5), pages 2075-2112, October.
    12. Granger, Clive W.J. & Sin, Chor-yiu, 1999. "Modelling the Absolute Returns of Different Stock Indices: Exploring the Forecastability of an Alternative Measure of Risk," University of California at San Diego, Economics Working Paper Series qt48r4781r, Department of Economics, UC San Diego.
    13. Andersen, Torben G & Bollerslev, Tim, 1998. "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 885-905, November.
    14. Bollerslev, Tim & Zhou, Hao, 2002. "Estimating stochastic volatility diffusion using conditional moments of integrated volatility," Journal of Econometrics, Elsevier, vol. 109(1), pages 33-65, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ole E. Barndorff-Nielsen & José Manuel Corcuera & Mark Podolskij, 2009. "Multipower Variation for Brownian Semistationary Processes," CREATES Research Papers 2009-21, Department of Economics and Business Economics, Aarhus University.
    2. Liu, Guangying & Zhang, Xinsheng, 2011. "Power variation of fractional integral processes with jumps," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 962-972, August.
    3. Tim Bollerslev & Viktor Todorov, 2011. "Estimation of Jump Tails," Econometrica, Econometric Society, vol. 79(6), pages 1727-1783, November.
    4. Figueroa-López, José E. & Houdré, Christian, 2009. "Small-time expansions for the transition distributions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3862-3889, November.
    5. Figueroa-López, José E., 2008. "Small-time moment asymptotics for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3355-3365, December.
    6. Ghysels, Eric & Santa-Clara, Pedro & Valkanov, Rossen, 2006. "Predicting volatility: getting the most out of return data sampled at different frequencies," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 59-95.
    7. Todorov, Viktor, 2013. "Power variation from second order differences for pure jump semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2829-2850.
    8. Diop, Assane & Jacod, Jean & Todorov, Viktor, 2013. "Central Limit Theorems for approximate quadratic variations of pure jump Itô semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 839-886.
    9. Vetter, Mathias, 2014. "Inference on the Lévy measure in case of noisy observations," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 125-133.
    10. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ole E. Barndorff-Nielsen & Neil Shephard, 2001. "Realised power variation and stochastic volatility models," Economics Papers 2001-W18, Economics Group, Nuffield College, University of Oxford.
    2. Nour Meddahi, 2002. "A theoretical comparison between integrated and realized volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 479-508.
    3. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2003. "Some Like it Smooth, and Some Like it Rough: Untangling Continuous and Jump Components in Measuring, Modeling, and Forecasting Asset Return Volatility," PIER Working Paper Archive 03-025, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 01 Sep 2003.
    4. Nour Meddahi, 2003. "ARMA representation of integrated and realized variances," Econometrics Journal, Royal Economic Society, vol. 6(2), pages 335-356, December.
    5. Ole E. Barndorff‐Nielsen & Neil Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(2), pages 253-280, May.
    6. Chun Liu & John M. Maheu, 2009. "Forecasting realized volatility: a Bayesian model-averaging approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(5), pages 709-733.
    7. Sévi, Benoît, 2014. "Forecasting the volatility of crude oil futures using intraday data," European Journal of Operational Research, Elsevier, vol. 235(3), pages 643-659.
    8. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2007. "Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility," The Review of Economics and Statistics, MIT Press, vol. 89(4), pages 701-720, November.
    9. Torben G. Andersen & Luca Benzoni, 2008. "Realized volatility," Working Paper Series WP-08-14, Federal Reserve Bank of Chicago.
    10. Degiannakis, Stavros & Xekalaki, Evdokia, 2007. "Assessing the Performance of a Prediction Error Criterion Model Selection Algorithm in the Context of ARCH Models," MPRA Paper 96324, University Library of Munich, Germany.
    11. Giot, Pierre & Laurent, Sebastien, 2004. "Modelling daily Value-at-Risk using realized volatility and ARCH type models," Journal of Empirical Finance, Elsevier, vol. 11(3), pages 379-398, June.
    12. Nielsen, Morten Ørregaard & Frederiksen, Per, 2008. "Finite sample accuracy and choice of sampling frequency in integrated volatility estimation," Journal of Empirical Finance, Elsevier, vol. 15(2), pages 265-286, March.
    13. Michael McAleer & Marcelo Medeiros, 2008. "Realized Volatility: A Review," Econometric Reviews, Taylor & Francis Journals, vol. 27(1-3), pages 10-45.
    14. Ho‐Chuan (River) Huang & Chien‐Chung Nieh, 2004. "Realize the Realized Stock Index Volatility," Asian Economic Journal, East Asian Economic Association, vol. 18(1), pages 59-80, March.
    15. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    16. Georgios Chortareas & John Nankervis & Ying Jiang, 2007. "Forecasting Exchange Rate Volatility with High Frequency Data: Is the Euro Different?," Money Macro and Finance (MMF) Research Group Conference 2006 79, Money Macro and Finance Research Group.
    17. Christophe Chorro & Florian Ielpo & Benoît Sévi, 2017. "The contribution of jumps to forecasting the density of returns," Post-Print halshs-01442618, HAL.
    18. Isao Ishida & Toshiaki Watanabe, 2009. "Modeling and Forecasting the Volatility of the Nikkei 225 Realized Volatility Using the ARFIMA-GARCH Model," CARF F-Series CARF-F-145, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    19. Bollerslev, Tim & Kretschmer, Uta & Pigorsch, Christian & Tauchen, George, 2009. "A discrete-time model for daily S & P500 returns and realized variations: Jumps and leverage effects," Journal of Econometrics, Elsevier, vol. 150(2), pages 151-166, June.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bpj:strimo:v:21:y:2003:i:1/2003:p:47-68:n:6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Peter Golla (email available below). General contact details of provider: https://www.degruyter.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.