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Matrix Formulas For Nonstationary Arima Signal Extraction

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  • McElroy, Tucker

Abstract

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. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Webel, Karsten, 2022. "A review of some recent developments in the modelling and seasonal adjustment of infra-monthly time series," Discussion Papers 31/2022, Deutsche Bundesbank.
    9. Víctor M. Guerrero & Adriana Galicia‐Vázquez, 2010. "Trend estimation of financial time series," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(3), pages 205-223, May.
    10. 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.
    11. 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.
    12. 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.
    13. D. Stephen G. Pollock & Emi Mise, 2022. "A Wiener–Kolmogorov Filter for Seasonal Adjustment and the Cholesky Decomposition of a Toeplitz Matrix," Computational Economics, Springer;Society for Computational Economics, vol. 59(3), pages 913-933, March.
    14. Irene Nandutu & Marcellin Atemkeng & Nokubonga Mgqatsa & Sakayo Toadoum Sari & Patrice Okouma & Rockefeller Rockefeller & Theophilus Ansah-Narh & Jean Louis Ebongue Kedieng Fendji & Franklin Tchakount, 2022. "Error Correction Based Deep Neural Networks for Modeling and Predicting South African Wildlife–Vehicle Collision Data," Mathematics, MDPI, vol. 10(21), pages 1-31, October.
    15. 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.
    16. 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.
    17. 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.
    18. McElroy, Tucker S. & Jach, Agnieszka, 2023. "Identification of the differencing operator of a non-stationary time series via testing for zeroes in the spectral density," Computational Statistics & Data Analysis, Elsevier, vol. 177(C).
    19. 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.
    20. Bloechl, Andreas, 2014. "Penalized Splines, Mixed Models and the Wiener-Kolmogorov Filter," Discussion Papers in Economics 21406, University of Munich, Department of Economics.
    21. McElroy, Tucker & Wildi, Marc, 2013. "Multi-step-ahead estimation of time series models," International Journal of Forecasting, Elsevier, vol. 29(3), pages 378-394.
    22. 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.
    23. 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.
    24. 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.
    25. 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.

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