Global jump filters and quasi-likelihood analysis for volatility

Abstract

We propose a new estimation scheme for estimation of the volatility parameters of a semimartingale with jumps based on a jump detection filter. Our filter uses all of the data to analyze the relative size of increments and to discriminate jumps more precisely. We construct quasi-maximum likelihood estimators and quasi-Bayesian estimators and show limit theorems for them including \(L^p\)-estimates of the error and asymptotic mixed normality based on the framework of the quasi-likelihood analysis. The global jump filters do not need a restrictive condition for the distribution of the small jumps. By numerical simulation, we show that our “global” method obtains better estimates of the volatility parameter than the previous “local” methods.

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References

  1. Dohnal, G. (1987). On estimating the diffusion coefficient. Journal of Applied Probability, 105–114.

  2. Genon-Catalot, V., Jacod, J. (1993). On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Annales de l’Institut Henri Poincaré Probability and Statistics, 29(1), 119–151.

    MathSciNet  MATH  Google Scholar 

  3. Iacus, S. M., Yoshida, N. (2018). Simulation and inference for stochastic processes with YUIMA. Basel: Springer.

    Google Scholar 

  4. Kamatani, K., Uchida, M. (2014). Hybrid multi-step estimators for stochastic differential equations based on sampled data. Statistical Inference for Stochastic Processes, 18(2), 177–204.

    MathSciNet  Article  Google Scholar 

  5. Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scandinavian Journal of Statistics, 24(2), 211–229.

    MathSciNet  Article  Google Scholar 

  6. Ogihara, T., Yoshida, N. (2011). Quasi-likelihood analysis for the stochastic differential equation with jumps. Statistical Inference for Stochastic Processes, 14(3), 189.

    MathSciNet  Article  Google Scholar 

  7. Ogihara, T., Yoshida, N. (2014). Quasi-likelihood analysis for nonsynchronously observed diffusion processes. Stochastic Processes and Their Applications, 124(9), 2954–3008.

    MathSciNet  Article  Google Scholar 

  8. Prakasa Rao, B. (1988). Statistical inference from sampled data for stochastic processes. Statistical Inference from Stochastic Processes (Ithaca, NY, 1987), 80, 249–284.

    MathSciNet  Article  Google Scholar 

  9. Prakasa Rao, B. L. S. (1983). Asymptotic theory for nonlinear least squares estimator for diffusion processes. Mathematische Operationsforschung und Statistik Series Statistics, 14(2), 195–209.

    MathSciNet  MATH  Google Scholar 

  10. Shimizu, Y. (2009). A practical inference for discretely observed jump-diffusions from finite samples. Journal of the Japan Statistical Society, 38(3), 391–413.

    MathSciNet  Article  Google Scholar 

  11. Shimizu, Y., Yoshida, N. (2006). Estimation of parameters for diffusion processes with jumps from discrete observations. Statistical Inference for Stochastic Processes, 9(3), 227–277.

    MathSciNet  Article  Google Scholar 

  12. Uchida, M., Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Processes and Their Applications, 122(8), 2885–2924.

    MathSciNet  Article  Google Scholar 

  13. Uchida, M., Yoshida, N. (2013). Quasi likelihood analysis of volatility and nondegeneracy of statistical random field. Stochastic Processes and Their Applications, 123(7), 2851–2876.

    MathSciNet  Article  Google Scholar 

  14. Uchida, M., Yoshida, N. (2014). Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations. Statistical Inference for Stochastic Processes, 17(2), 181–219.

    MathSciNet  Article  Google Scholar 

  15. Yoshida, N. (1992). Estimation for diffusion processes from discrete observation. Journal of Multivariate Analysis, 41(2), 220–242.

    MathSciNet  Article  Google Scholar 

  16. Yoshida, N. (2011). Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Annals of the Institute of Statistical Mathematics, 63(3), 431–479.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors thank the reviewers for their valuable comments to improve this article.

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Correspondence to Nakahiro Yoshida.

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This work was in part supported by CREST JPMJCR14D7 Japan Science and Technology Agency; Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H01702 (Scientific Research); and a Cooperative Research Program of the Institute of Statistical Mathematics.

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Inatsugu, H., Yoshida, N. Global jump filters and quasi-likelihood analysis for volatility. Ann Inst Stat Math (2021). https://doi.org/10.1007/s10463-020-00768-x

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Keywords

  • Volatility
  • Jump
  • Global filter
  • High-frequency data
  • Quasi-likelihood analysis
  • Stochastic differential equation
  • Order statistic
  • Asymptotic mixed normality
  • Polynomial-type large deviation
  • Moment