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This paper proposes and analyses the Kullback-Leibler information criterion (KLIC) as a unified statistical tool to evaluate, compare and combine density forecasts. Use of the KLIC is particularly attractive, as well as operationally convenient, given its equivalence with the widely used Berkowitz likelihood ratio test for the evaluation of individual density forecasts that exploits the probability integral transforms. Parallels with the comparison and combination of point forecasts are made. This and related Monte Carlo experiments help draw out properties of combined density forecasts. We illustrate the uses of the KLIC in an application to two widely used published density forecasts for UK inflation, namely the Bank of England and NIESR 'fan' charts.

This paper addresses the problem of testing for the presence of a stochastic bubble in a time series in the case that the bubble is periodically collapsing so that the asset price keeps returning to the level implied by the market fundamentals. As this is essentially a problem of identifying the collapsing periods from the expanding ones, we propose using a generalization of the Dickey–Fuller test procedure which makes use of the class of Markov regime‐switching models. The potential of the new methodology is illustrated via simulation, and an empirical example is given. Copyright © 1999 John Wiley & Sons, Ltd.