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Efficient estimation for ergodic diffusions sampled at high frequency

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  • Michael Sørensen

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

Abstract

A general theory of efficient estimation for ergodic diffusions sampled at high fre- quency is presented. High frequency sampling is now possible in many applications, in particular in finance. The theory is formulated in term of approximate martingale estimating functions and covers a large class of estimators including most of the pre- viously proposed estimators for diffusion processes, for instance GMM-estimators and the maximum likelihood estimator. Simple conditions are given that ensure rate optimality, where estimators of parameters in the diffusion coefficient converge faster than estimators of parameters in the drift coefficient, and for efficiency. The conditions turn out to be equal to those implying small delta-optimality in the sense of Jacobsen and thus gives an interpretation of this concept in terms of classical sta- tistical concepts. Optimal martingale estimating functions in the sense of Godambe and Heyde are shown to be give rate optimal and efficient estimators under weak conditions.

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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 2007-46.

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Length: 32
Date of creation: 22 Jan 2008
Date of revision:
Handle: RePEc:aah:create:2007-46

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

Related research

Keywords: Approximate martingale estimating functions; discrete time observation of a diffusion; efficiency; Euler approximation; generalized method of moments; optimal estimating function; optimal rate; small delta-optimality;

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References

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  1. Hansen, Lars Peter & Alexandre Scheinkman, Jose & Touzi, Nizar, 1998. "Spectral methods for identifying scalar diffusions," Journal of Econometrics, Elsevier, Elsevier, vol. 86(1), pages 1-32, June.
  2. Martin Jacobsen, 2001. "Discretely Observed Diffusions: Classes of Estimating Functions and Small Δ-optimality," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 123-149.
  3. Yacine Ait-Sahalia & Per A. Mykland, 2002. "The Effects of Random and Discrete Sampling When Estimating Continuous-Time Diffusions," NBER Technical Working Papers, National Bureau of Economic Research, Inc 0276, National Bureau of Economic Research, Inc.
  4. Yacine Ait-Sahalia, 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Econometrica, Econometric Society, Econometric Society, vol. 70(1), pages 223-262, January.
  5. Asger Roer Pedersen, 2000. "Estimating the Nitrous Oxide Emission Rate from the Soil Surface by Means of a Diffusion Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(3), pages 385-403.
  6. Overbeck, Ludger & Rydén, Tobias, 1997. "Estimation in the Cox-Ingersoll-Ross Model," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 13(03), pages 430-461, June.
  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, American Finance Association, vol. 47(3), pages 1209-27, July.
  8. Yoshida, Nakahiro, 1992. "Estimation for diffusion processes from discrete observation," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 41(2), pages 220-242, May.
  9. Bent Jesper Christensen & Michael Sørensen, 2008. "Optimal inference in dynamic models with conditional moment restrictions," CREATES Research Papers, School of Economics and Management, University of Aarhus 2008-51, School of Economics and Management, University of Aarhus.
  10. Leah Kelly & Eckhard Platen, 2003. "Estimating for Discretely Observed Diffusions Using Transform Functions," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 96, Quantitative Finance Research Centre, University of Technology, Sydney.
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