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Spectral Density and Spectral Distribution Inference for Long Memory Time Series via Fixed-b Asymptotics

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  • McElroy, Tucker
  • Politis, Dimitris

Abstract

Recent work in econometrics has provided large bandwidth asymptotic theory for taper-based studentized estimates of the mean, in the context of nonparametric estimation for serially correlated time series data. These taper-based statistics can be viewed as estimates of the spectral density at frequency zero, and hence it is quite natural to extend the asymptotic theory to non-zero frequencies and thereby obtain a large bandwidth theory for spectarl estimation. This approach was developed by Hashimzade and Vogelsang (2008) for the case of a single frequency. This paper extends their work in several ways: (i) we treat multiple frequencies jointly; (ii) we allow for long-range dependence at differing frequencies; (iii) we allow for piecewise smooth tapers, such as trapezoidal tapers; (iv) we develop a theory of higher order accuracy by a novel expansion of the Laplace Transform of the limit distribution. The theoretical results are complemented by simulations of the limit distributions, an application to confidence band construction, and a discussion of the issue of optimal bandwidth selection.

Suggested Citation

  • McElroy, Tucker & Politis, Dimitris, 2013. "Spectral Density and Spectral Distribution Inference for Long Memory Time Series via Fixed-b Asymptotics," University of California at San Diego, Economics Working Paper Series qt6164c110, Department of Economics, UC San Diego.
  • Handle: RePEc:cdl:ucsdec:qt6164c110
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    References listed on IDEAS

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    1. Yixiao Sun & Peter C. B. Phillips & Sainan Jin, 2008. "Optimal Bandwidth Selection in Heteroskedasticity-Autocorrelation Robust Testing," Econometrica, Econometric Society, vol. 76(1), pages 175-194, January.
    2. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory For Heteroskedasticity-Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 21(6), pages 1130-1164, December.
    3. Nigar Hashimzade & Timothy J. Vogelsang, 2008. "Fixed‐b asymptotic approximation of the sampling behaviour of nonparametric spectral density estimators," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(1), pages 142-162, January.
    4. Velasco, Carlos & Robinson, Peter M., 2001. "Edgeworth Expansions For Spectral Density Estimates And Studentized Sample Mean," Econometric Theory, Cambridge University Press, vol. 17(3), pages 497-539, June.
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    6. Findley, David F, et al, 1998. "New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(2), pages 127-152, April.
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    8. Peter C. B. Phillips & Yixiao Sun & Sainan Jin, 2006. "Spectral Density Estimation And Robust Hypothesis Testing Using Steep Origin Kernels Without Truncation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(3), pages 837-894, August.
    9. McElroy, Tucker & Politis, Dimitris N., 2012. "Fixed-B Asymptotics For The Studentized Mean From Time Series With Short, Long, Or Negative Memory," Econometric Theory, Cambridge University Press, vol. 28(2), pages 471-481, April.
    10. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
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    Cited by:

    1. Tucker S McElroy & Agnieszka Jach, 2019. "Testing collinearity of vector time series," Econometrics Journal, Royal Economic Society, vol. 22(2), pages 97-116.
    2. McElroy, Tucker S. & Holan, Scott H., 2016. "Computation of the autocovariances for time series with multiple long-range persistencies," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 44-56.
    3. van Delft, Anne & Eichler, Michael, 2020. "A note on Herglotz’s theorem for time series on function spaces," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3687-3710.

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    More about this item

    Keywords

    Social and Behavioral Sciences; Cyclical Long Memory; Kernel Spectral Estimator; Long Range Dependence; Spectral Confidence Bands;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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