IDEAS home Printed from https://ideas.repec.org/a/spr/series/v7y2016i1d10.1007_s13209-016-0139-4.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13209-016-0139-4
    File Function: Abstract
    Download Restriction: no

    File URL: https://libkey.io/10.1007/s13209-016-0139-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Zinde-Walsh, Victoria, 1988. "Some Exact Formulae for Autoregressive Moving Average Processes," Econometric Theory, Cambridge University Press, vol. 4(3), pages 384-402, December.
    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. 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.
    5. McElroy, Tucker, 2008. "Matrix Formulas For Nonstationary Arima Signal Extraction," Econometric Theory, Cambridge University Press, vol. 24(4), pages 988-1009, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zomchak, Larysa & Umrysh, Hryhorii, 2017. "Моделювання Й Прогнозування Виробництва М’Яса Та Яєць В Україні За Допомогою Сезонної Arima-Моделі," Agricultural and Resource Economics: International Scientific E-Journal, Agricultural and Resource Economics: International Scientific E-Journal, vol. 3(3), September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. McElroy Tucker S. & Maravall Agustin, 2014. "Optimal Signal Extraction with Correlated Components," Journal of Time Series Econometrics, De Gruyter, vol. 6(2), pages 237-273, July.
    2. Stephen Pollock, 2014. "Trends Cycles and Seasons: Econometric Methods of Signal Extraction," Discussion Papers in Economics 14/04, Division of Economics, School of Business, University of Leicester.
    3. McElroy, Tucker S. & Wildi, Marc, 2020. "The Multivariate Linear Prediction Problem: Model-Based and Direct Filtering Solutions," Econometrics and Statistics, Elsevier, vol. 14(C), pages 112-130.
    4. Tucker McElroy & Thomas Trimbur, 2015. "Signal Extraction for Non-Stationary Multivariate Time Series with Illustrations for Trend Inflation," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(2), pages 209-227, March.
    5. Proietti, Tommaso, 2007. "Signal extraction and filtering by linear semiparametric methods," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 935-958, October.
    6. Agustín Maravall Herrero & Domingo Pérez Cañete, 2011. "Applying and interpreting model-based seasonal adjustment. The euro-area industrial production series," Working Papers 1116, Banco de España.
    7. Dias, Maria Helena Ambrosio & Dias, Joilson, 2010. "Measuring the Cyclical Component of a Time Series: a New Proposed Methodology," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 30(1), October.
    8. McElroy Tucker S, 2010. "A Nonlinear Algorithm for Seasonal Adjustment in Multiplicative Component Decompositions," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 14(4), pages 1-23, September.
    9. Wildi, Marc & McElroy, Tucker S., 2019. "The trilemma between accuracy, timeliness and smoothness in real-time signal extraction," International Journal of Forecasting, Elsevier, vol. 35(3), pages 1072-1084.
    10. Victor Gomez & Jorg Breitung, 1999. "The Beveridge–Nelson Decomposition: A Different Perspective with New Results," Journal of Time Series Analysis, Wiley Blackwell, vol. 20(5), pages 527-535, September.
    11. Guy Mélard, 2016. "On some remarks about SEATS signal extraction," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 7(1), pages 53-98, March.
    12. Maravall, Agustin, 2006. "An application of the TRAMO-SEATS automatic procedure; direct versus indirect adjustment," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2167-2190, May.
    13. Tommaso Proietti, 2012. "Seasonality, Forecast Extensions And Business Cycle Uncertainty," Journal of Economic Surveys, Wiley Blackwell, vol. 26(4), pages 555-569, September.
    14. Thornton, Michael A., 2013. "Removing seasonality under a changing regime: Filtering new car sales," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 4-14.
    15. McElroy Tucker & Wildi Marc, 2010. "Signal Extraction Revision Variances as a Goodness-of-Fit Measure," Journal of Time Series Econometrics, De Gruyter, vol. 2(1), pages 1-32, June.
    16. Paulina Granados Z., 2004. "Chilean Household Income Function: Life Cycle and Persistence of Shocks," Journal Economía Chilena (The Chilean Economy), Central Bank of Chile, vol. 7(1), pages 51-89, April.
    17. Breitung, Jörg, 1998. "On model based seasonal adjustment procedures," SFB 373 Discussion Papers 1998,12, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    18. Bloechl, Andreas, 2014. "Penalized Splines, Mixed Models and the Wiener-Kolmogorov Filter," Discussion Papers in Economics 21406, University of Munich, Department of Economics.
    19. Tucker McElroy, 2013. "Forecasting continuous-time processes with applications to signal extraction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(3), pages 439-456, June.
    20. Pollock Stephen D.S.G., 2009. "Statistical Fourier Analysis: Clarifications and Interpretations," Journal of Time Series Econometrics, De Gruyter, vol. 1(1), pages 1-49, April.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:series:v:7:y:2016:i:1:d:10.1007_s13209-016-0139-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.