Combining predictive distributions

Abstract

In probabilistic forecasting combination formulas for the aggregation of predictive distributions need to be estimated based on past experience and training data. We study combination formulas and aggregation methods for predictive cumulative distribution functions from the perspectives of calibration and dispersion, taking an original prediction space approach that applies to discrete, mixed discrete-continuous and continuous predictive distributions alike. The key idea is that aggregation methods ought to be parsimonious, yet sufficiently flexible to accommodate any type of dispersion in the component distributions. Both linear and non-linear aggregation methods are investigated, including generalized, spread-adjusted and beta-transformed linear pools. The effects and techniques are demonstrated theoretically, in simulation examples, and in case studies, where we fit combination formulas for density forecasts of S&P 500 returns and daily maximum temperature at Seattle-Tacoma Airport.