I propose principles and methods for the construction of a time-simultaneous prediction band for a univariate time series. The methods are entirely based on a learning sample of time trajectories, and make no parametric assumption about its distribution. Hence, the methods are general and widely applicable. The expected coverage probability of a band can be estimated by a bootstrap procedure. The estimate is likely to be less than the nominal level. Expected lack of coverage can be compensated for by increasing the coverage in the learning sample. Applications to simulated and empirical data illustrate the methods.  Copyright © 2007 John Wiley & Sons, Ltd.

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Ceteris paribus, Common random numbers, Counterfactual analysis, Stochastic simulation, Super exogeneity, Variance reduction, C15, C53, C90, E63,

A sample-based method in Kolsrud (Journal of Forecasting 2007; 26(3): 171-188) for the construction of a time-simultaneous prediction band for a univariate time series is extended to produce a variable-and time-simultaneous prediction box for a multivariate time series. A measure of distance based on the L 1 -norm is applied to a learning sample of multivariate time trajectories, which can be mean-and/or variance-nonstationary. Based on the ranking of distances to the centre of the sample, a subsample of the most central multivariate trajectories is selected. A prediction box is constructed by circumscribing the subsample with a hyperrectangle. The fraction of central trajectories selected into the subsample can be calibrated by bootstrap such that the expected coverage of the box equals a prescribed nominal level. The method is related to the concept of data depth, and thence modified to increase coverage. Applications to simulated and empirical data illustrate the method, which is also compared to several other methods in the literature adapted to the multivariate setting.