IDEAS home Printed from
   My bibliography  Save this paper

Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters


  • Rodríguez, Alejandro
  • Ruiz, Esther


Prediction intervals in State Space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, where the true parameters are substituted by consistent estimates. This approach has two limitations. First, it does not incorporate the uncertainty due to parameter estimation. Second, the Gaussianity assumption of future innovations may be inaccurate. To overcome these drawbacks, Wall and Stoffer (2002) propose to obtain prediction intervals by using a bootstrap procedure that requires the backward representation of the model. Obtaining this representation increases the complexity of the procedure and limits its implementation to models for which it exists. The bootstrap procedure proposed by Wall and Stoffer (2002) is further complicated by fact that the intervals are obtained for the prediction errors instead of for the observations. In this paper, we propose a bootstrap procedure for constructing prediction intervals in State Space models that does not need the backward representation of the model and is based on obtaining the intervals directly for the observations. Therefore, its application is much simpler, without loosing the good behavior of bootstrap prediction intervals. We study its finite sample properties and compare them with those of the standard and the Wall and Stoffer (2002) procedures for the Local Level Model. Finally, we illustrate the results by implementing the new procedure to obtain prediction intervals for future values of a real time series.

Suggested Citation

  • Rodríguez, Alejandro & Ruiz, Esther, 2010. "Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters," DES - Working Papers. Statistics and Econometrics. WS ws100301, Universidad Carlos III de Madrid. Departamento de Estadística.
  • Handle: RePEc:cte:wsrepe:ws100301

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. Jesús Fernández-Villaverde & Juan F. Rubio-Ramírez & Thomas J. Sargent & Mark W. Watson, 2007. "ABCs (and Ds) of Understanding VARs," American Economic Review, American Economic Association, vol. 97(3), pages 1021-1026, June.
    2. Athanasios Orphanides & Simon van Norden, 2002. "The Unreliability of Output-Gap Estimates in Real Time," The Review of Economics and Statistics, MIT Press, vol. 84(4), pages 569-583, November.
    3. Tommaso Proietti & Alberto Musso & Thomas Westermann, 2007. "Estimating potential output and the output gap for the euro area: a model-based production function approach," Empirical Economics, Springer, vol. 33(1), pages 85-113, July.
    4. Andrew Harvey & Esther Ruiz & Neil Shephard, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Oxford University Press, vol. 61(2), pages 247-264.
    5. James H. Stock & Mark W. Watson, 2007. "Erratum to "Why Has U.S. Inflation Become Harder to Forecast?"," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 39(7), pages 1849-1849, October.
    6. Harvey, Andrew & Ruiz, Esther & Sentana, Enrique, 1992. "Unobserved component time series models with Arch disturbances," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 129-157.
    7. Harvey, Andrew C. & Delle Monache, Davide, 2009. "Computing the mean square error of unobserved components extracted by misspecified time series models," Journal of Economic Dynamics and Control, Elsevier, vol. 33(2), pages 283-295, February.
    8. Frank Smets, 2002. "Output gap uncertainty: Does it matter for the Taylor rule?," Empirical Economics, Springer, vol. 27(1), pages 113-129.
    9. J. Durbin, 2002. "A simple and efficient simulation smoother for state space time series analysis," Biometrika, Biometrika Trust, vol. 89(3), pages 603-616, August.
    10. Alejandro Rodriguez & Esther Ruiz, 2009. "Bootstrap prediction intervals in state-space models," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(2), pages 167-178, March.
    11. James H. Stock & Mark W. Watson, 2007. "Why Has U.S. Inflation Become Harder to Forecast?," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 39(s1), pages 3-33, February.
    12. Douglas Staiger & James H. Stock & Mark W. Watson, 2001. "Prices, Wages and the U.S. NAIRU in the 1990s," NBER Working Papers 8320, National Bureau of Economic Research, Inc.
    13. Domenech, Rafael & Gomez, Victor, 2006. "Estimating Potential Output, Core Inflation, and the NAIRU as Latent Variables," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 354-365, July.
    14. Ray, W. D., 1989. "Rates of convergence to steady state for the linear growth version of a dynamic linear model (DLM)," International Journal of Forecasting, Elsevier, vol. 5(4), pages 537-545.
    15. Hamilton, James D., 1986. "A standard error for the estimated state vector of a state-space model," Journal of Econometrics, Elsevier, vol. 33(3), pages 387-397, December.
    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. Ng, Jason & Forbes, Catherine S. & Martin, Gael M. & McCabe, Brendan P.M., 2013. "Non-parametric estimation of forecast distributions in non-Gaussian, non-linear state space models," International Journal of Forecasting, Elsevier, vol. 29(3), pages 411-430.
    2. repec:eme:aecozz:s0731-905320150000035010 is not listed on IDEAS
    3. David Harris & Gael M. Martin & Indeewara Perera & Don S. Poskitt, 2017. "Construction and visualization of optimal confidence sets for frequentist distributional forecasts," Monash Econometrics and Business Statistics Working Papers 9/17, Monash University, Department of Econometrics and Business Statistics.
    4. Krieg, Sabine & van den Brakel, Jan A., 2012. "Estimation of the monthly unemployment rate for six domains through structural time series modelling with cointegrated trends," Computational Statistics & Data Analysis, Elsevier, vol. 56(10), pages 2918-2933.
    5. Pilar Poncela & Esther Ruiz, 2016. "Small- Versus Big-Data Factor Extraction in Dynamic Factor Models: An Empirical Assessment," Advances in Econometrics,in: Dynamic Factor Models, volume 35, pages 401-434 Emerald Publishing Ltd.

    More about this item



    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:cte:wsrepe:ws100301. 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: (Ana Poveda). 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.