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

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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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  • Rodríguez, Alejandro, 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
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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.
    2. Meredith Beechey & Pär Österholm, 2012. "The Rise and Fall of U.S. Inflation Persistence," International Journal of Central Banking, International Journal of Central Banking, vol. 8(3), pages 55-86, September.
    3. 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 Group Publishing Limited.
    4. Fresoli, Diego & Poncela, Pilar & Ruiz, Esther, 2023. "Ignoring cross-correlated idiosyncratic components when extracting factors in dynamic factor models," Economics Letters, Elsevier, vol. 230(C).
    5. 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.
    6. 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.

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