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Matrix Formulas For Nonstationary Arima Signal Extraction


  • McElroy, Tucker


The paper provides general matrix formulas for minimum mean squared error signal extraction for a finitely sampled time series whose signal and noise components are nonstationary autoregressive integrated moving average processes. These formulas are quite practical; in addition to being simple to implement on a computer, they make it possible to easily derive important general properties of the signal extraction filters. We also extend these formulas to estimates of future values of the unobserved signal, and we show how this result combines signal extraction and forecasting.

Suggested Citation

  • McElroy, Tucker, 2008. "Matrix Formulas For Nonstationary Arima Signal Extraction," Econometric Theory, Cambridge University Press, vol. 24(4), pages 988-1009, August.
  • Handle: RePEc:cup:etheor:v:24:y:2008:i:04:p:988-1009_08

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    Cited by:

    1. 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.
    2. Dermoune Azzouz & Djehiche Boualem & Rahmania Nadji, 2009. "Multivariate Extension of the Hodrick-Prescott Filter-Optimality and Characterization," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 13(3), pages 1-35, May.
    3. Bloechl, Andreas, 2014. "Reducing the Excess Variability of the Hodrick-Prescott Filter by Flexible Penalization," Discussion Papers in Economics 17940, University of Munich, Department of Economics.
    4. Blöchl, Andreas, 2014. "Penalized Splines as Frequency Selective Filters - Reducing the Excess Variability at the Margins," Discussion Papers in Economics 20687, University of Munich, Department of Economics.
    5. 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.
    6. 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.
    7. Bloechl, Andreas, 2014. "Penalized Splines, Mixed Models and the Wiener-Kolmogorov Filter," Discussion Papers in Economics 21406, University of Munich, Department of Economics.
    8. Dimitrios D. Thomakos, 2008. "Optimal Linear Filtering, Smoothing and Trend Extraction for Processes with Unit Roots and Cointegration," Working Paper series 14_08, Rimini Centre for Economic Analysis.
    9. 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.
    10. McElroy, Tucker & Wildi, Marc, 2013. "Multi-step-ahead estimation of time series models," International Journal of Forecasting, Elsevier, vol. 29(3), pages 378-394.
    11. 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.
    12. 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.
    13. Andreas Blöchl & Gebhard Flaig, 2014. "The Hodrick-Prescott Filter with a Time-Varying Penalization Parameter. An Application for the Trend Estimation of Global Temperature," CESifo Working Paper Series 4577, CESifo Group Munich.
    14. 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.
    15. McElroy Tucker S. & Maravall Agustin, 2014. "Optimal Signal Extraction with Correlated Components," Journal of Time Series Econometrics, De Gruyter, vol. 6(2), pages 1-37, July.
    16. Flaig Gebhard, 2015. "Why We Should Use High Values for the Smoothing Parameter of the Hodrick-Prescott Filter," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 235(6), pages 518-538, December.
    17. Irma Hindrayanto & John A.D. Aston & Siem Jan Koopman & Marius Ooms, 2013. "Modelling trigonometric seasonal components for monthly economic time series," Applied Economics, Taylor & Francis Journals, vol. 45(21), pages 3024-3034, July.
    18. 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;Working Papers Homepage.
    19. Bell, William R., 2011. "REGCMPNT A Fortran Program for Regression Models with ARIMA Component Errors," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 41(i07).
    20. 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.
    21. Tucker S. McElroy & Thomas M. Trimbur, 2012. "Signal extraction for nonstationary multivariate time series with illustrations for trend inflation," Finance and Economics Discussion Series 2012-45, Board of Governors of the Federal Reserve System (US).

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