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Statistical Fourier Analysis: Clarifications and Interpretations

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  • D.S.G. Pollock

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Abstract

This paper expounds some of the results of Fourier theory that are essential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences. An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain. The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to antialiasing filtering and sub sampling. These techniques are also described. It is argued that band-limited processes are more prevalent in statistical and econometric time series than is commonly recognised.

Suggested Citation

  • D.S.G. Pollock, 2008. "Statistical Fourier Analysis: Clarifications and Interpretations," Discussion Papers in Economics 08/36, Department of Economics, University of Leicester.
  • Handle: RePEc:lec:leecon:08/36
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    1. Pollock, D S G, 2001. "Filters for Short Non-stationary Sequences," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 20(5), pages 341-355, August.
    2. McCoy, E. J. & Stephens, D. A., 2004. "Bayesian time series analysis of periodic behaviour and spectral structure," International Journal of Forecasting, Elsevier, vol. 20(4), pages 713-730.
    3. Granger, Clive W. J. & Hyung, Namwon, 2004. "Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns," Journal of Empirical Finance, Elsevier, vol. 11(3), pages 399-421, June.
    4. Pollock, D. S. G., 2000. "Trend estimation and de-trending via rational square-wave filters," Journal of Econometrics, Elsevier, vol. 99(2), pages 317-334, December.
    5. Granger, Clive W. J. & Ding, Zhuanxin, 1996. "Varieties of long memory models," Journal of Econometrics, Elsevier, vol. 73(1), pages 61-77, July.
    6. Adrian Pagan, 1997. "Towards an Understanding of Some Business Cycle Characteristics," Australian Economic Review, The University of Melbourne, Melbourne Institute of Applied Economic and Social Research, vol. 30(1), pages 1-15.
    7. Nerlove, Marc & Grether, David M. & Carvalho, José L., 1979. "Analysis of Economic Time Series," Elsevier Monographs, Elsevier, edition 1, number 9780125157506 edited by Shell, Karl.
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    1. repec:pal:marecl:v:19:y:2017:i:2:d:10.1057_s41278-016-0052-6 is not listed on IDEAS

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