A prediction model is any statement of a probability distribution for an outcome not yet observed. This study considers the properties of weighted linear combinations of n prediction models, or linear pools, evaluated using the conventional log predictive scoring rule. The log score is a concave function of the weights and, in general, an optimal linear combination will include several models with positive weights despite the fact that exactly one model has limiting posterior probability one. The paper derives several interesting formal results: for example, a prediction model with positive weight in a pool may have zero weight if some other models are deleted from that pool. The results are illustrated using S&P 500 returns with prediction models from the ARCH, stochastic volatility and Markov mixture families. In this example models that are clearly inferior by the usual scoring criteria have positive weights in optimal linear pools, and these pools substantially outperform their best components.
forecasting GARCH log scoring Markov mixture model combination