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Asymptotics for Autocovariances and Integrated Periodograms for Linear Processes Observed at Lower Frequencies

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  • Tobias Niebuhr
  • Jens-Peter Kreiss

Abstract

type="main" xml:id="insr12019-abs-0001"> One of the most frequently used class of processes in time series analysis is the one of linear processes. For many statistical quantities, among them sample autocovariances and sample autocorrelations, central limit theorems are available in the literature. We investigate classical linear processes under a nonstandard observation pattern; namely, we assume that we are only able to observe the linear process at a lower frequency. It is shown that such observation pattern destroys the linear structure of the observations and leads to substantially different asymptotic results for standard statistical quantities. Central limit theorems are given for sample autocovariances and sample autocorrelations as well as more general integrated periodograms and ratio statistics. Moreover, for specific autoregressive processes, the possibilities to estimate the parameters of the underlying autoregression from lower frequency observations are addressed. Finally, we suggest for autoregressions of order 2 a valid bootstrap procedure. A small simulation study demonstrates the performance of the bootstrap proposal for finite sample size.

Suggested Citation

  • Tobias Niebuhr & Jens-Peter Kreiss, 2014. "Asymptotics for Autocovariances and Integrated Periodograms for Linear Processes Observed at Lower Frequencies," International Statistical Review, International Statistical Institute, vol. 82(1), pages 123-140, April.
    • Handle: RePEc:bla:istatr:v:82:y:2014:i:1:p:123-140
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    File URL: http://hdl.handle.net/10.1111/insr.12019
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    References listed on IDEAS

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    1. Dahlhaus, Rainer, 1985. "Asymptotic normality of spectral estimates," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 412-431, June.
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    Cited by:

    1. Dennis D. Boos & Jason A. Osborne, 2015. "Assessing Variability of Complex Descriptive Statistics in Monte Carlo Studies Using Resampling Methods," International Statistical Review, International Statistical Institute, vol. 83(2), pages 228-238, August.
    2. Sourav Das & Suhasini Subba Rao & Junho Yang, 2021. "Spectral methods for small sample time series: A complete periodogram approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 597-621, September.

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