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Review on Goodness of Fit Tests for Ergodic Diffusion Processes by Different Sampling Schemes

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  • Ilia Negri
  • Yoichi Nishiyama

Abstract

We review some recent results on goodness of fit test for the drift coefficient of a one‐dimensional ergodic diffusion, where the diffusion coefficient is a nuisance function which however is estimated. Using a theory for the continuous observation case, we first present a test based on deterministic discrete time observations of the process. Then we also propose a test based on the data observed discretely in space, that is, the so‐called tick time sample scheme. In both sampling schemes the limit distribution of the test is the supremum of the standard Brownian motion, thus the test is asymptotically distribution free. The tests are also consistent under any fixed alternatives.

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  • Ilia Negri & Yoichi Nishiyama, 2010. "Review on Goodness of Fit Tests for Ergodic Diffusion Processes by Different Sampling Schemes," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 39(1‐2), pages 91-106, February.
    • Handle: RePEc:bla:ecnote:v:39:y:2010:i:1-2:p:91-106
    DOI: 10.1111/j.1468-0300.2010.00221.x
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    References listed on IDEAS

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    1. Masaaki Fukasawa, 2010. "Central limit theorem for the realized volatility based on tick time sampling," Finance and Stochastics, Springer, vol. 14(2), pages 209-233, April.
    2. Ilia Negri & Yoichi Nishiyama, 2009. "Goodness of fit test for ergodic diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 919-928, December.
    3. Yoshida, Nakahiro, 1992. "Estimation for diffusion processes from discrete observation," Journal of Multivariate Analysis, Elsevier, vol. 41(2), pages 220-242, May.
    4. Harry van Zanten, 2003. "On Empirical Processes for Ergodic Diffusions and Rates of Convergence of M‐estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(3), pages 443-458, September.
    5. Jim Griffin & Roel Oomen, 2008. "Sampling Returns for Realized Variance Calculations: Tick Time or Transaction Time?," Econometric Reviews, Taylor & Francis Journals, vol. 27(1-3), pages 230-253.
    6. Ilia Negri, 1998. "Stationary Distribution Function Estimation for Ergodic Diffusion Process," Statistical Inference for Stochastic Processes, Springer, vol. 1(1), pages 61-84, January.
    7. Delgado, Miguel A. & Stute, Winfried, 2008. "Distribution-free specification tests of conditional models," Journal of Econometrics, Elsevier, vol. 143(1), pages 37-55, March.
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