(1998) On the optimal performance of forecasting systems: The prequential approach. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Consider a forecaster who observes a sequence of data on-line and after each new observation makes a forecast (a point estimate or a full probability distribution) for the next observation. A general theory of assessment of such prequential (predictive-sequential) forecasting systems was introduced by Dawid (1984). Within this framework the notion of efficiency of probability forecasting systems was introduced, and it was shown that Bayesian probability forecasting systems are efficient. In this thesis the concept of prequential efficiency is studied further by presenting some new results. We focus especially on a class of non-Bayesian statistical forecasting systems, the plug-in systems, and we study their efficiency. We show that under suitable conditions the plug-in systems are efficient, but we also show, using counterexamples, that for some models no plug-in system is efficient. Next, we extend the notion of efficiency to point prediction systems. The efficiency of Bayesian point prediction systems is established, and sufficient conditions are presented for the efficiency of plug-in systems. The results are applied to time series forecasting. By adopting a predictive point of view, we also study the consistency of extremum estimators for possibly misspecified models. We show, using martingale arguments, that an estimator defined as the minimize of a statistical criterion measuring predictive performance, converges to the value of the parameter indexing the model that issues the "best" one step ahead predictions for the data at hand. In order to prove our results we establish a martingale version of the uniform law of large numbers.

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