Bootstrap prediction intervals in state–space models

Abstract. Prediction intervals in state–space models can be obtained by assumingGaussian innovations and using the prediction equations of the Kalman filter, with thetrue parameters substituted by consistent estimates. This approach has two limitations.First, it does not incorporate the uncertainty caused by parameter estimation. Second,the Gaussianity of future innovations assumption may be inaccurate. To overcome thesedrawbacks, Wall and Stoffer [Journal of Time Series Analysis (2002) Vol. 23, pp.733–751] propose a bootstrap procedure for evaluating conditional forecast errors thatrequires the backward representation of the model. Obtaining this representationincreases the complexity of the procedure and limits its implementation to models forwhich it exists. In this article, we propose a bootstrap procedure for constructingprediction intervals directly for the observations, which does not need the backwardrepresentation of the model. Consequently, its application is much simpler, withoutlosing the good behaviour of bootstrap prediction intervals. We study its finite-sampleproperties and compare them with those of the standard and the Wall and Stofferprocedures for the local level model. Finally, we illustrate the results byimplementing the new procedure to obtain prediction intervals for future values of areal time series.Keywords. Backward representation; Kalman filter; local level model; unobservedcomponents.