Publication: Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters
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2010-01
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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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NAIRU, Output gap, Parameter uncertainty, Prediction Intervals, State Space Models