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Multi-step-ahead estimation of time series models

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  • McElroy, Tucker
  • Wildi, Marc

Abstract

We study the fitting of time series models via the minimization of a multi-step-ahead forecast error criterion that is based on the asymptotic average of squared forecast errors. Our objective function uses frequency domain concepts, but is formulated in the time domain, and allows the estimation of all linear processes (e.g., ARIMA and component ARIMA). By using an asymptotic form of the forecast mean squared error, we obtain a well-defined nonlinear function of the parameters that is proven to be minimized at the true parameter vector when the model is correctly specified. We derive the statistical properties of the parameter estimates, and study the asymptotic impact of model misspecification on multi-step-ahead forecasting. The method is illustrated through a forecasting exercise, applied to several time series.

Suggested Citation

  • McElroy, Tucker & Wildi, Marc, 2013. "Multi-step-ahead estimation of time series models," International Journal of Forecasting, Elsevier, vol. 29(3), pages 378-394.
    • Handle: RePEc:eee:intfor:v:29:y:2013:i:3:p:378-394
    DOI: 10.1016/j.ijforecast.2012.08.003
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    References listed on IDEAS

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    1. Proietti, Tommaso, 2011. "Direct and iterated multistep AR methods for difference stationary processes," International Journal of Forecasting, Elsevier, vol. 27(2), pages 266-280.
    2. Gersch, Will & Kitagawa, Genshiro, 1983. "The Prediction of Time Series with Trends and Seasonalities," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(3), pages 253-264, July.
    3. Marc Wildi & Bernd Schips, 2004. "Signal Extraction: How (In)efficient Are Model-Based Approaches? An Empirical Study Based on TRAMO/SEATS and Census X-12-ARIMA," KOF Working papers 04-96, KOF Swiss Economic Institute, ETH Zurich.
    4. J. Haywood & G. Tunnicliffe Wilson, 1997. "Fitting Time Series Models by Minimizing Multistep‐ahead Errors: a Frequency Domain Approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(1), pages 237-254.
    5. Findley, David F, et al, 1998. "New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(2), pages 127-152, April.
    6. Tucker McElroy, 2008. "Exact formulas for the Hodrick-Prescott filter," Econometrics Journal, Royal Economic Society, vol. 11(1), pages 209-217, March.
    7. Stock, James H. & Watson, Mark W., 1999. "Forecasting inflation," Journal of Monetary Economics, Elsevier, vol. 44(2), pages 293-335, October.
    8. Marcellino, Massimiliano & Stock, James H. & Watson, Mark W., 2006. "A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 499-526.
    9. Findley, David F. & Potscher, Benedikt M. & Wei, Ching-Zong, 2004. "Modeling of time series arrays by multistep prediction or likelihood methods," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 151-187.
    10. Findley, David F, et al, 1998. "New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(2), pages 169-177, April.
    11. Diebold, Francis X & Mariano, Roberto S, 2002. "Comparing Predictive Accuracy," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 134-144, January.
    12. 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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    2. Shahedul A. Khan, 2018. "Exponentiated Weibull regression for time-to-event data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 24(2), pages 328-354, April.
    3. Wildi, Marc, 2010. "Real-Time Signal Extraction: a Shift of Perspective/Extracción de señal en tiempo real: un cambio de perspectiva," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 28, pages 497-518, Diciembre.
    4. 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.
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    6. Ivan Svetunkov & Nikolaos Kourentzes & Rebecca Killick, 2024. "Multi-step estimators and shrinkage effect in time series models," Computational Statistics, Springer, vol. 39(3), pages 1203-1239, May.

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