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The adjustment of prediction intervals to account for errors in parameter estimation

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  • Paul Kabaila
  • Zhisong He

Abstract

. Standard approximate 1 − α prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1 − α. Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1 − α, conditional on an appropriate statistic T. For an autoregressive process of order p, it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1 − α. In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1 − α. We illustrate the application of this new criterion to a Gaussian zero‐mean autoregressive process of order 1 and one‐step‐ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1 − α conditional on the last observation is the better of the two adjustments.

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  • Paul Kabaila & Zhisong He, 2004. "The adjustment of prediction intervals to account for errors in parameter estimation," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(3), pages 351-358, May.
    • Handle: RePEc:bla:jtsera:v:25:y:2004:i:3:p:351-358
    DOI: 10.1111/j.1467-9892.2004.01848.x
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    References listed on IDEAS

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    1. Phillips, Peter C. B., 1979. "The sampling distribution of forecasts from a first-order autoregression," Journal of Econometrics, Elsevier, vol. 9(3), pages 241-261, February.
    2. Paul Kabaila, 1993. "On Bootstrap Predictive Inference For Autoregressive Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 14(5), pages 473-484, September.
    3. Paul Kabaila, 1999. "The Relevance Property For Prediction Intervals," Journal of Time Series Analysis, Wiley Blackwell, vol. 20(6), pages 655-662, November.
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    Cited by:

    1. Daniel W. Apley & Hyun Cheol Lee, 2010. "The effects of model parameter deviations on the variance of a linearly filtered time series," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(5), pages 460-471, August.
    2. Beutner, Eric & Heinemann, Alexander & Smeekes, Stephan, 2017. "A Justification of Conditional Confidence Intervals," Research Memorandum 023, Maastricht University, Graduate School of Business and Economics (GSBE).
    3. Paolo Vidoni, 2017. "Improved multivariate prediction regions for Markov process models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(1), pages 1-18, March.
    4. Kabaila, Paul & Syuhada, Khreshna, 2010. "The asymptotic efficiency of improved prediction intervals," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1348-1353, September.
    5. Paolo Vidoni, 2009. "A simple procedure for computing improved prediction intervals for autoregressive models," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(6), pages 577-590, November.

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