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Bootstrap prediction intervals in State Space models

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  • Alejandro Rodriguez

    ()

  • 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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Paper provided by Universidad Carlos III, Departamento de Estadística y Econometría in its series Statistics and Econometrics Working Papers with number ws081104.

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Date of creation: Mar 2008
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Handle: RePEc:cte:wsrepe:ws081104

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Keywords: Backward representation; Kalman filter; Local Level Model; Unobserved Components;

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  1. Durbin, James & Koopman, Siem Jan, 2001. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, number 9780198523543, September.
  2. Cavaglia, Stefano, 1992. "The persistence of real interest differentials: A Kalman filtering approach," Journal of Monetary Economics, Elsevier, vol. 29(3), pages 429-443, June.
  3. Harvey, Andrew & Ruiz, Esther & Shephard, Neil, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Wiley Blackwell, vol. 61(2), pages 247-64, April.
  4. Laurence Ball & Stephen G. Cecchetti, 1990. "Inflation and Uncertainty at Long and Short Horizons," Brookings Papers on Economic Activity, Economic Studies Program, The Brookings Institution, vol. 21(1), pages 215-254.
  5. Carmen Broto & Esther Ruiz, 2006. "Using Auxiliary Residuals To Detect Conditional Heteroscedasticity In Inflation," Statistics and Econometrics Working Papers ws060402, Universidad Carlos III, Departamento de Estadística y Econometría.
  6. Evans, Martin, 1991. "Discovering the Link between Inflation Rates and Inflation Uncertainty," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 23(2), pages 169-84, May.
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Cited by:
  1. Rodríguez, Alejandro & Ruiz, Esther, 2012. "Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 62-74, January.
  2. 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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