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Nonlinear functionals of the periodogram

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  • GILLES FAY
  • ERIC MOULINES
  • PHILIPPE SOULIER

Abstract

A central limit theorem is stated for a wide class of triangular arrays of nonlinear functionals of the periodogram of a stationary linear sequence. Those functionals may be singular and not‐bounded. The proof of this result is based on Bartlett decomposition and an existing counterpart result for the periodogram of an independent and identically distributed sequence, here taken to be the driving noise. The main contribution of this paper is to prove the asymptotic negligibility of the remainder term from Bartlett decomposition, feasible under short dependence assumption. As it is highlighted by applications (to estimation of nonlinear functionals of the spectral density, robust spectral estimation, local polynomial approximation and log‐periodogram regression), this extends may results until then tied to Gaussian assumption.

Suggested Citation

  • Gilles Fay & Eric Moulines & Philippe Soulier, 2002. "Nonlinear functionals of the periodogram," Journal of Time Series Analysis, Wiley Blackwell, vol. 23(5), pages 523-553, September.
    • Handle: RePEc:bla:jtsera:v:23:y:2002:i:5:p:523-553
    DOI: 10.1111/1467-9892.00277
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    Cited by:

    1. Tommaso Proietti & Alessandra Luati, 2013. "The Exponential Model for the Spectrum of a Time Series: Extensions and Applications," CREATES Research Papers 2013-34, Department of Economics and Business Economics, Aarhus University.
    2. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    3. Proietti, Tommaso & Lütkepohl, Helmut, 2013. "Does the Box–Cox transformation help in forecasting macroeconomic time series?," International Journal of Forecasting, Elsevier, vol. 29(1), pages 88-99.
    4. Bhansali, R.J. & Giraitis, L. & Kokoszka, P.S., 2007. "Convergence of quadratic forms with nonvanishing diagonal," Statistics & Probability Letters, Elsevier, vol. 77(7), pages 726-734, April.

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