Abstract. 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.

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) January 2010 AbstractIn the context of linear state space models with Gaussian errors and known parameters, the Kalman filter generates best linear unbiased predictions of the underlying components together with their corresponding prediction mean squared errors (PMSE). However, in practice, the filter is run by substituting some parameters of the model by consistent estimates. In these circumstances, the PMSEs given by the Kalman filter do not take into account the parameter uncertainty and, consequently, they underestimate the true PMSEs. In this paper, we propose two new bootstrap procedures to obtain PMSE of the unobserved states based on obtaining replicates of the underlying states conditional on the information available at each moment of time. By conditioning on the available information, we simplify the procedure with respect to alternative bootstrap proposals previously available in the literature. Furthermore, we show that the new procedures proposed in this paper have better finite sample properties than the alternatives. We implement the proposed procedures for estimating the PMSE of several key unobservable US macroeconomic variables as the output gap, the non accelerating inflation rate of unemployment (NAIRU), the long-run investment rate and the core inflation. We show how taking into account the parameter uncertainty may change the prediction intervals constructed for those unobservable macroeconomic variables and, in particular, change the conclusions about the utility of the NAIRU as a macroeconomic indicator for expansions and recessions.

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