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Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models

Author

Listed:
  • David F. Findley

    (U.S. Census Bureau)

  • Demetra P. Lytras

    (U.S. Census Bureau)

  • Agustin Maravall

    (Bank of Spain)

Abstract

Our starting place is the first order seasonal autoregressive model. Its series are shown to have canonical model-based decompositions whose finite-sample estimates, filters, and error covariances have simple revealing formulas from basic linear regression. We obtain analogous formulas for seasonal random walks, extending some of the results of Maravall and Pierce (J Time Series Anal, 8:177–293, 1987). The seasonal decomposition filters of the biannual seasonal random walk have formulas that explicitly reveal which deterministic functions they annihilate and which they reproduce, directly illustrating very general results of Bell (J Off Stat, 28:441–461, 2012; Center for Statistical Research and Methodology, Research Report Series, Statistics #2015-03, U.S. Census Bureau, Washington, D.C. https://www.census.gov/srd/papers/pdf/RRS2015-03 , 2015). Other formulas express phenomena heretofore lacking such concrete expression, such as the much discussed negative autocorrelation at the first seasonal lag quite often observed in differenced seasonally adjusted series. An innovation that is also applied to airline model seasonal decompositions is the effective use of signs of lag one and first-seasonal-lag autocorrelations (after differencing) to indicate, in a formal way, where smoothness is increased by seasonal adjustment and where its effect is opposite.

Suggested Citation

  • David F. Findley & Demetra P. Lytras & Agustin Maravall, 2016. "Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 7(1), pages 11-52, March.
  • Handle: RePEc:spr:series:v:7:y:2016:i:1:d:10.1007_s13209-016-0139-4
    DOI: 10.1007/s13209-016-0139-4
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    References listed on IDEAS

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    1. Agustin Maravall & David A. Pierce, 1987. "A Prototypical Seasonal Adjustment Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(2), pages 177-193, March.
    2. William R. Bell & Donald E. K. Martin, 2004. "Computation of asymmetric signal extraction filters and mean squared error for ARIMA component models," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 603-623, July.
    3. McElroy, Tucker & Sutcliffe, Andrew, 2006. "An iterated parametric approach to nonstationary signal extraction," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2206-2231, May.
    4. Zinde-Walsh, Victoria, 1988. "Some Exact Formulae for Autoregressive Moving Average Processes," Econometric Theory, Cambridge University Press, vol. 4(3), pages 384-402, December.
    5. McElroy, Tucker, 2008. "Matrix Formulas For Nonstationary Arima Signal Extraction," Econometric Theory, Cambridge University Press, vol. 24(4), pages 988-1009, August.
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    Cited by:

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    More about this item

    Keywords

    ARIMA models; Signal extraction smoothness; Timeseries;
    All these keywords.

    JEL classification:

    • C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs

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