IDEAS home Printed from
   My bibliography  Save this paper

A Justification of Conditional Confidence Intervals


  • Beutner, Eric

    (QE / Mathematical economics and game the)

  • Heinemann, Alexander

    (QE / Econometrics)

  • Smeekes, Stephan

    (QE / Econometrics)


To quantify uncertainty around point estimates of conditional objects such as conditional means or variances, parameter uncertainty has to be taken into account. Attempts to incorporate parameter uncertainty are typically based on the unrealistic assumption of observing two independent processes, where one is used for parameter estimation, and the other for conditioning upon. Such unrealistic foundation raises the question whether these intervals are theoretically justified in a realistic setting. This paper presents an asymptotic justification for this type of intervals that does not require such an unrealistic assumption, but relies on a sample-split approach instead. By showing that our sample-split intervals coincide asymptotically with the standard intervals, we provide a novel, and realistic, justification for confidence intervals of conditional objects. The analysis is carried out for a general class of Markov chains nesting various time series models.

Suggested Citation

  • 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).
  • Handle: RePEc:unm:umagsb:2017023

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. Paolo Vidoni, 2009. "Improved Prediction Intervals and Distribution Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 735-748.
    2. Eric Beutner & Alexander Heinemann & Stephan Smeekes, 2017. "A Justification of Conditional Confidence Intervals," Papers 1710.00643,
    3. Kabaila, Paul & Syuhada, Khreshna, 2010. "The asymptotic efficiency of improved prediction intervals," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1348-1353, September.
    4. Francq, Christian & Zakoïan, Jean-Michel, 2015. "Risk-parameter estimation in volatility models," Journal of Econometrics, Elsevier, vol. 184(1), pages 158-173.
    5. Pan, Li & Politis, Dimitris N., 2016. "Bootstrap prediction intervals for Markov processes," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 467-494.
    6. Gospodinov, Nikolay, 2002. "Median unbiased forecasts for highly persistent autoregressive processes," Journal of Econometrics, Elsevier, vol. 111(1), pages 85-101, November.
    7. 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.
    8. Pascual, Lorenzo & Romo, Juan & Ruiz, Esther, 2006. "Bootstrap prediction for returns and volatilities in GARCH models," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2293-2312, May.
    9. Hansen, Bruce E., 2006. "Interval forecasts and parameter uncertainty," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 377-398.
    10. 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.
    11. Lewis, Richard & Reinsel, Gregory C., 1985. "Prediction of multivariate time series by autoregressive model fitting," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 393-411, June.
    12. Dufour, Jean-Marie & Taamouti, Abderrahim, 2010. "Short and long run causality measures: Theory and inference," Journal of Econometrics, Elsevier, vol. 154(1), pages 42-58, January.
    13. Blasques, Francisco & Koopman, Siem Jan & Łasak, Katarzyna & Lucas, André, 2016. "In-sample confidence bands and out-of-sample forecast bands for time-varying parameters in observation-driven models," International Journal of Forecasting, Elsevier, vol. 32(3), pages 875-887.
    14. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, August.
    15. 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.
    16. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    17. Pesaran, M. Hashem, 2015. "Time Series and Panel Data Econometrics," OUP Catalogue, Oxford University Press, number 9780198759980.
    18. Xiong, Shifeng & Li, Guoying, 2008. "Some results on the convergence of conditional distributions," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3249-3253, December.
    19. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    20. Paolo Vidoni, 2004. "Improved prediction intervals for stochastic process models," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(1), pages 137-154, January.
    21. Lorenzo Pascual & Juan Romo & Esther Ruiz, 2004. "Bootstrap predictive inference for ARIMA processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(4), pages 449-465, July.
    22. Schmidt, Peter, 1977. "Some Small Evidence on the Distribution of Dynamic Simulation Forecasts," Econometrica, Econometric Society, vol. 45(4), pages 997-1005, May.
    23. Paul Kabaila & Khreshna Syuhada, 2008. "Improved Prediction Limits For AR(p) and ARCH(p) Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(2), pages 213-223, March.
    24. Hirotugu Akaike, 1969. "Fitting autoregressive models for prediction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 243-247, December.
    25. Drew Creal & Siem Jan Koopman & André Lucas, 2013. "Generalized Autoregressive Score Models With Applications," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(5), pages 777-795, August.
    26. Giuseppe Cavaliere & Iliyan Georgiev & A. M. Robert Taylor, 2013. "Wild Bootstrap of the Sample Mean in the Infinite Variance Case," Econometric Reviews, Taylor & Francis Journals, vol. 32(2), pages 204-219, February.
    27. 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.
    28. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Eric Beutner & Alexander Heinemann & Stephan Smeekes, 2017. "A Justification of Conditional Confidence Intervals," Papers 1710.00643,
    2. Eric Beutner & Alexander Heinemann & Stephan Smeekes, 2018. "A Residual Bootstrap for Conditional Value-at-Risk," Papers 1808.09125,, revised Nov 2018.

    More about this item


    Conditional confidence intervals; Parameter Uncertainty; Markov chain; Sample-splitting; Prediction; Merging;

    JEL classification:

    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:unm:umagsb:2017023. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Leonne Portz). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.