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Nonparametric estimation of a periodic sequence in the presence of a smooth trend


  • Oliver Linton

    () (Institute for Fiscal Studies and University of Cambridge)

  • Michael Vogt

    (Institute for Fiscal Studies)


In this paper, we study a nonparametric regression model including a periodic component, a smooth trend function, and a stochastic error term. We propose a procedure to estimate the unknown period and the function values of the periodic component as well as the nonparametric trend function. The theoretical part of the paper establishes the asymptotic properties of our estimators. In particular, we show that our estimator of the period is consistent. In addition, we derive the convergence rates as well as the limiting distributions of our estimators of the periodic component and the trend function. The asymptotic results are complemented with a simulation study that investigates the small sample behaviour of our procedure. Finally, we illustrate our method by applying it to a series of global temperature anomalies.

Suggested Citation

  • Oliver Linton & Michael Vogt, 2012. "Nonparametric estimation of a periodic sequence in the presence of a smooth trend," CeMMAP working papers CWP23/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:23/12

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    References listed on IDEAS

    1. Kristensen, Dennis, 2009. "Uniform Convergence Rates Of Kernel Estimators With Heterogeneous Dependent Data," Econometric Theory, Cambridge University Press, vol. 25(05), pages 1433-1445, October.
    2. D'Andrade, Kendall, 1992. "The End of an Era," Business Ethics Quarterly, Cambridge University Press, vol. 2(03), pages 379-389, July.
    3. Peter Hall & Jiying Yin, 2003. "Nonparametric methods for deconvolving multiperiodic functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 869-886.
    4. Robert M. De Jong & James Davidson, 2000. "Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices," Econometrica, Econometric Society, vol. 68(2), pages 407-424, March.
    5. Peter Hall & Ming Li, 2006. "Using the periodogram to estimate period in nonparametric regression," Biometrika, Biometrika Trust, vol. 93(2), pages 411-424, June.
    6. Marc G. Genton & Peter Hall, 2007. "Statistical inference for evolving periodic functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(4), pages 643-657.
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    Cited by:

    1. Seok Young Hong & Oliver Linton & Hui Jun Zhang, 2014. "Multivariate Variance Ratio Statistics," Cambridge Working Papers in Economics 1459, Faculty of Economics, University of Cambridge.
    2. Peter Malec, 2016. "A Semiparametric Intraday GARCH Model," Cambridge Working Papers in Economics 1633, Faculty of Economics, University of Cambridge.
    3. Michael Vogt & Oliver Linton, 2015. "Classification of nonparametric regression functions in heterogeneous panels," CeMMAP working papers CWP06/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    More about this item


    Nonparametric estimation; penalized least squares; periodic sequence; temperature anomaly data.;

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