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AbstractIt is shown that many of the normal theory prediction intervals for statistics of random samples are still valid when the samples are correlated and have a specified covariance structure.

Prediction intervals in state space models can be obtained by assuming Gaussian innovations and using the prediction equations of the Kalman filter, with the true 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 these drawbacks, Wall and Stoffer [Journal of Time Series Analysis (2002) Vol. 23, pp. 733 751] propose a bootstrap procedure for evaluating conditional forecast errors 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. In this article, we propose a bootstrap procedure for constructing prediction intervals directly for the observations, which does not need the backward representation of the model. Consequently, its application is much simpler, without losing the good behaviour of bootstrap prediction intervals. We study its finite sample properties and compare them with those of the standard and the Wall and Stoffer 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. ; Financial support from Project SEJ2006-03919 by the Spanish Government is gratefully acknowledged ; Publicado