This paper develops asymptotic prediction functions that approximate the shape of the density of future observations and correct for parameter uncertainty. The functions are based on extensions to a definition of predictive likelihood originally suggested by Lauritzen and Hinkley. The prediction function is shown to possess efficiency properties based on the Kullback-Leibler measure of information loss. Examples of the application of the prediction function and the derivation of relative efficiency are shown for linearnormal models, nonnormal models, and ARCH models.

[In this paper we consider the estimation of a model with time varying structure. The parameters of the model are assumed to be subject to permanent and transitory changes over time. Estimation methods are developed, and the asymptotic properties of the estimates are derived.]