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Adaptive estimation of mean and volatility functions in (auto-)regressive models

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  • Comte, F.
  • Rozenholc, Y.

Abstract

In this paper, we study the problem of nonparametric estimation of the mean and variance functions b and [sigma]2 in a model: Xi+1=b(Xi)+[sigma](Xi)[var epsilon]i+1. For this purpose, we consider a collection of finite dimensional linear spaces. We estimate b using a mean squares estimator built on a data driven selected linear space among the collection. Then an analogous procedure estimates [sigma]2, using a possibly different collection of models. Both data driven choices are performed via the minimization of penalized mean squares contrasts. The penalty functions are random in order not to depend on unknown variance-type quantities. In all cases, we state nonasymptotic risk bounds in empirical norm for our estimators and we show that they are both adaptive in the minimax sense over a large class of Besov balls. Lastly, we give the results of intensive simulation experiments which show the good performances of our estimator.

Suggested Citation

  • Comte, F. & Rozenholc, Y., 2002. "Adaptive estimation of mean and volatility functions in (auto-)regressive models," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 111-145, January.
    • Handle: RePEc:eee:spapps:v:97:y:2002:i:1:p:111-145
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    References listed on IDEAS

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    1. Hardle, W. & Tsybakov, A., 1997. "Local polynomial estimators of the volatility function in nonparametric autoregression," Journal of Econometrics, Elsevier, vol. 81(1), pages 223-242, November.
    2. Gourieroux, Christian & Monfort, Alain, 1992. "Qualitative threshold ARCH models," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 159-199.
    3. Fan, Jianqing & Yao, Qiwei, 1998. "Efficient estimation of conditional variance functions in stochastic regression," LSE Research Online Documents on Economics 6635, London School of Economics and Political Science, LSE Library.
    4. Hoffmann, Marc, 1999. "On nonparametric estimation in nonlinear AR(1)-models," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 29-45, August.
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    6. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Cited by:

    1. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    2. Nicolas Asin & Jan Johannes, 2017. "Adaptive nonparametric estimation in the presence of dependence," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(4), pages 694-730, October.
    3. Asin, Nicolas & Johannes, Jan, 2016. "Adaptive non-parametric estimation in the presence of dependence," LIDAM Discussion Papers ISBA 2016007, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Hildebrandt, Florian & Trabs, Mathias, 2023. "Nonparametric calibration for stochastic reaction–diffusion equations based on discrete observations," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 171-217.
    5. Schmisser, Émeline, 2019. "Non parametric estimation of the diffusion coefficients of a diffusion with jumps," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5364-5405.
    6. Francesco Audrino & Peter Bühlmann, 2009. "Splines for financial volatility," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 655-670, June.
    7. Emeline Schmisser, 2012. "Non-parametric estimation of the diffusion coefficient from noisy data," Statistical Inference for Stochastic Processes, Springer, vol. 15(3), pages 193-223, October.
    8. F. Comte & V. Genon-Catalot, 2020. "Regression function estimation on non compact support in an heteroscesdastic model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(1), pages 93-128, January.
    9. Comte, F. & Lacour, C. & Rozenholc, Y., 2010. "Adaptive estimation of the dynamics of a discrete time stochastic volatility model," Journal of Econometrics, Elsevier, vol. 154(1), pages 59-73, January.

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