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Computation of asymmetric signal extraction filters and mean squared error for ARIMA component models

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  • William R. Bell
  • Donald E. K. Martin

Abstract

. Standard signal extraction results for both stationary and nonstationary time series are expressed as linear filters applied to the observed series. Computation of the filter weights, and of the corresponding frequency response function, is relevant for studying properties of the filter and of the resulting signal extraction estimates. Methods for doing such computations for symmetric, doubly infinite filters are well established. This study develops an algorithm for computing filter weights for asymmetric, semi‐infinite signal extraction filters, including the important case of the concurrent filter (for signal extraction at the current time point). The setting is where the time series components being estimated follow autoregressive integrated moving‐average (ARIMA) models. The algorithm provides expressions for the asymmetric signal extraction filters as rational polynomial functions of the backshift operator. The filter weights are then readily generated by simple expansion of these expressions, and the corresponding frequency response function is directly evaluated. Recursive expressions are also developed that relate the weights for filters that use successively increasing amounts of data. The results for the filter weights are then used to develop methods for computing mean squared error results for the asymmetric signal extraction estimates.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jtsera:v:25:y:2004:i:4:p:603-623
    DOI: 10.1111/j.1467-9892.2004.01920.x
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    References listed on IDEAS

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    2. Koopman, Siem Jan & Harvey, Andrew, 2003. "Computing observation weights for signal extraction and filtering," Journal of Economic Dynamics and Control, Elsevier, vol. 27(7), pages 1317-1333, May.
    3. Ian McLeod, 1975. "Derivation of the Theoretical Autocovariance Function of Autoregressive–Moving Average Time Series," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 24(2), pages 255-256, June.
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    Cited by:

    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. 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.
    3. 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.
    4. Balakrishna, N. & Hareesh, G., 2009. "Statistical signal extraction using stable processes," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 851-856, April.
    5. Suhasini Subba Rao & Junho Yang, 2023. "A prediction perspective on the Wiener–Hopf equations for time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(1), pages 23-42, January.
    6. 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.
    7. Wildi Marc & McElroy Tucker, 2016. "Optimal Real-Time Filters for Linear Prediction Problems," Journal of Time Series Econometrics, De Gruyter, vol. 8(2), pages 155-192, July.
    8. 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.
    9. Terence Mills, 2007. "A Note on Trend Decomposition: The 'Classical' Approach Revisited with an Application to Surface Temperature Trends," Journal of Applied Statistics, Taylor & Francis Journals, vol. 34(8), pages 963-972.
    10. Tommaso Proietti, 2012. "Seasonality, Forecast Extensions And Business Cycle Uncertainty," Journal of Economic Surveys, Wiley Blackwell, vol. 26(4), pages 555-569, September.
    11. Proietti, Tommaso, 2007. "Signal extraction and filtering by linear semiparametric methods," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 935-958, October.
    12. 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.

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