Predictive properties of forecast combination, ensemble methods, and Bayesian predictive synthesis

This paper studies the theoretical predictive properties of classes of forecast combination methods. A novel strategy based on continuous time stochastic processes is proposed and developed, where the combined predictive error processes are expressed as stochastic differential equations, evaluated using Ito's lemma. We identify a class of forecast combination methods, which we categorize as non-linear synthesis, and find that it entails an extra term in the predictive error process that "corrects" the bias from misspecification and dependence amongst forecasts, effectively improving forecasts. We show that a subclass of the recently developed framework of Bayesian predictive synthesis fits within this class. Theoretical properties are examined and we show that non-linear synthesis improves the expected squared forecast error over any and all linear combination, averaging, and ensemble of forecasts, under mild conditions that are met in most real applications. We discuss the conditions for which non-linear synthesis outperforms linear combinations, and its implications for developing further strategies. A finite sample simulation study is presented to illustrate our results.