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Prediction and nonparametric estimation for time series with heavy tails

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  • PETER HALL
  • LIANG PENG
  • QIWEI YAO

Abstract

Motivated by prediction problems for time series with heavy‐tailed marginal distributions, we consider methods based on `local least absolute deviations' for estimating a regression median from dependent data. Unlike more conventional `local median' methods, which are in effect based on locally fitting a polynomial of degree 0, techniques founded on local least absolute deviations have quadratic bias right up to the boundary of the design interval. Also in contrast to local least‐squares methods based on linear fits, the order of magnitude of variance does not depend on tail‐weight of the error distribution. To make these points clear, we develop theory describing local applications to time series of both least‐squares and least‐absolute‐deviations methods, showing for example that, in the case of heavy‐tailed data, the conventional local‐linear least‐squares estimator suffers from an additional bias term as well as increased variance.

Suggested Citation

  • Peter Hall & Liang Peng & Qiwei Yao, 2002. "Prediction and nonparametric estimation for time series with heavy tails," Journal of Time Series Analysis, Wiley Blackwell, vol. 23(3), pages 313-331, May.
    • Handle: RePEc:bla:jtsera:v:23:y:2002:i:3:p:313-331
    DOI: 10.1111/1467-9892.00266
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    Cited by:

    1. Ghosh, Yashowanto N. & Mukherjee, Bhramar, 2006. "On probabilistic properties of conditional medians and quantiles," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1775-1780, October.
    2. Toshio Honda, 2010. "Nonparametric estimation of conditional medians for linear and related processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 995-1021, December.
    3. Wang, Qiao, 2023. "A simple nonparametric conditional quantile estimator for time series with thin tails," Economics Letters, Elsevier, vol. 232(C).
    4. Toshio Honda, 2013. "Nonparametric quantile regression with heavy-tailed and strongly dependent errors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(1), pages 23-47, February.
    5. Honda, Toshio, 2013. "Nonparametric LAD cointegrating regression," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 150-162.
    6. Peng, Liang & Yao, Qiwei, 2017. "Estimating conditional means with heavy tails," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 14-22.
    7. Bonsoo Koo & Oliver Linton, 2013. "Let's get LADE: robust estimation of semiparametric multiplicative volatility models," CeMMAP working papers 11/13, Institute for Fiscal Studies.
    8. Peng, Liang & Yao, Qiwei, 2017. "Estimating conditional means with heavy tails," LSE Research Online Documents on Economics 73082, London School of Economics and Political Science, LSE Library.
    9. Peng, Liang & Yao, Qiwei, 2004. "Nonparametric regression under dependent errors with infinite variance," LSE Research Online Documents on Economics 22874, London School of Economics and Political Science, LSE Library.

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