John Geweke and Gianni Amisano
CQER Working Paper 09-05
October 2009
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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.
JEL classification: C11, C53
Key words: forecasting, GARCH, log scoring, Markov mixture, model combination, S&P 500 returns, stochastic volatility
This paper was originally prepared for the Forecasting in Rio Conference, Graduate School of Economics, Getulio Vargas Foundation, Rio de Janeiro, July 2008. The authors acknowledge helpful comments from James Chapman, Frank Diebold, Joel Horowitz, James Mitchell, Luke Tierney, Mattias Villani, Kenneth Wallis, Robert Winkler, Arnold Zellner, and participants in presentations at the Bank of Canada, Erasmus University, the NBER-NSF Time Series Conference, Princeton University, Rimini Centre for Economic Analysis, Sveriges Riksbank, Rice University, and the University of Queensland. The authors gratefully acknowledge financial support from NSF grant SBR-0720547. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility.
Please address questions regarding content to John Geweke, Departments of Statistics and Economics, W210 Pappajohn Business Bldg., University of Iowa, Iowa City, IA 52242-1000, john-geweke@uiowa.edu, and Gianni Amisano, University of Brescia and European Central Bank, amisano@eco.unibs.it.
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