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A test for improved multi-step forecasting


  • John Haywood
  • Granville Tunnicliffe Wilson


We propose a general test of whether a time-series model, with parameters estimated by minimizing the single-step forecast error sum of squares, is robust with respect to multi-step prediction, for some specified lead time. The test may be applied to a, possibly seasonal, autoregressive integrated moving average (ARIMA) model using the parameters and residuals following maximum likelihood estimation. It is based on a score statistic, evaluated at these estimated parameters, which measures the sensitivity of the multi-step forecast error variance with respect to the parameters. We derive the large sample properties of the test and show by a simulation study that it has acceptable small sample size properties for higher lead times when applied to the integrated moving average or IMA model that gives rise to the exponentially weighted moving average predictor. We investigate the power of the test when the IMA(1,1) model has been fitted to an ARMA(1,1) process. Further, we demonstrate the high power of the test when an AR is fitted to a process generated as the sum of a stochastic trend and cycle plus noise. We use frequency domain methods for the derivation and sampling properties of the test, and to give insight into its application. The test is illustrated on two real series, and an R function for its general application is available from . Copyright 2009 Blackwell Publishing Ltd

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  • John Haywood & Granville Tunnicliffe Wilson, 2009. "A test for improved multi-step forecasting," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(6), pages 682-707, November.
  • Handle: RePEc:bla:jtsera:v:30:y:2009:i:6:p:682-707

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    References listed on IDEAS

    1. R. Bhansali, 1996. "Asymptotically efficient autoregressive model selection for multistep prediction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 577-602, September.
    2. Christopher A. Sims, 1986. "Are forecasting models usable for policy analysis?," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Win, pages 2-16.
    3. Guillaume Chevillon, 2007. "Direct Multi-Step Estimation And Forecasting," Journal of Economic Surveys, Wiley Blackwell, vol. 21(4), pages 746-785, September.
    4. Chevillon, Guillaume & Hendry, David F., 2005. "Non-parametric direct multi-step estimation for forecasting economic processes," International Journal of Forecasting, Elsevier, vol. 21(2), pages 201-218.
    5. Kang, In-Bong, 2003. "Multi-period forecasting using different models for different horizons: an application to U.S. economic time series data," International Journal of Forecasting, Elsevier, vol. 19(3), pages 387-400.
    6. 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.
    7. Ing, Ching-Kang, 2003. "Multistep Prediction In Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 19(02), pages 254-279, April.
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