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

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  • Alejandro Rodríguez

    ()

  • Esther Ruiz

    ()

Abstract

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.

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Bibliographic Info

Paper provided by Universidad Carlos III, Departamento de Estadística y Econometría in its series Statistics and Econometrics Working Papers with number ws100301.

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Date of creation: Jan 2010
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Handle: RePEc:cte:wsrepe:ws100301

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Keywords: NAIRU; Output gap; Parameter uncertainty; Prediction Intervals; State Space Models;

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  1. James H. Stock & Mark W. Watson, 2006. "Why Has U.S. Inflation Become Harder to Forecast?," NBER Working Papers 12324, National Bureau of Economic Research, Inc.
  2. 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.
  3. Athanasios Orphanides & Simon van Norden, 2001. "The Unreliability of Output Gap Estimates in Real Time," CIRANO Working Papers 2001s-57, CIRANO.
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  6. Frank Smets, 2002. "Output gap uncertainty: Does it matter for the Taylor rule?," Empirical Economics, Springer, vol. 27(1), pages 113-129.
  7. Alejandro Rodriguez & Esther Ruiz, 2008. "Bootstrap prediction intervals in State Space models," Statistics and Econometrics Working Papers ws081104, Universidad Carlos III, Departamento de Estadística y Econometría.
  8. Jesus Fernandez-Villaverde & Juan F. Rubio-Ramirez & Thomas J. Sargent, 2005. "A, B, C’s (And D’s) For Understanding VARS," PIER Working Paper Archive 05-018, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  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. Harvey, Andrew & Ruiz, Esther & Shephard, Neil, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Wiley Blackwell, vol. 61(2), pages 247-64, April.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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.
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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.

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