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Uniform Convergence Rates of Kernel-Based Nonparametric Estimators for Continuous Time Diffusion Processes: A Damping Function Approach

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  • Shin Kanaya

    () (Aarhus University and CREATES)

Abstract

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya-Watson type with introducing a new technical device called a damping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also have an independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

Suggested Citation

  • Shin Kanaya, 2015. "Uniform Convergence Rates of Kernel-Based Nonparametric Estimators for Continuous Time Diffusion Processes: A Damping Function Approach," CREATES Research Papers 2015-50, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2015-50
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    File URL: ftp://ftp.econ.au.dk/creates/rp/15/rp15_50.pdf
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    Citations

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    Cited by:

    1. Debopam Bhattacharya & Shin Kanaya & Margaret Stevens, 2017. "Are University Admissions Academically Fair?," The Review of Economics and Statistics, MIT Press, vol. 99(3), pages 449-464, July.
    2. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    3. Shin Kanaya, 2016. "Convergence rates of sums of a-mixing triangular arrays: with an application to non-parametric drift function estimation of continuous-time processes," CREATES Research Papers 2016-24, Department of Economics and Business Economics, Aarhus University.
    4. Debopam Bhattacharya & Pascaline Dupas & Shin Kanaya, 2013. "Estimating the Impact of Means-tested Subsidies under Treatment Externalities with Application to Anti-Malarial Bednets," CREATES Research Papers 2013-06, Department of Economics and Business Economics, Aarhus University.
    5. Li, Degui & Lu, Zudi & Linton, Oliver, 2012. "Local Linear Fitting Under Near Epoch Dependence: Uniform Consistency With Convergence Rates," Econometric Theory, Cambridge University Press, vol. 28(5), pages 935-958, October.
    6. Kanaya, S. & Bhattacharya, D., 2017. "Uniform Convergence of Smoothed Distribution Functions with an Application to Delta Method for the Lorenz Curve," Cambridge Working Papers in Economics 1760, Faculty of Economics, University of Cambridge.

    More about this item

    Keywords

    Diffusion process; uniform convergence; kernel estimation; nonparametric.;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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