Threshold Martingales and the Evolution of Forecasts

This paper introduces a martingale that characterizes two properties of evolving forecast distributions. Ideal forecasts of a future event behave as martingales, sequentially updating the forecast to leverage the available information as the future event approaches. The threshold martingale introduced here measures the proportion of the forecast distribution lying below a threshold. In addition to being calibrated, a threshold martingale has quadratic variation that accumulates to a total determined by a quantile of the initial forecast distribution. Deviations from calibration or total volatility signal problems in the underlying model. Calibration adjustments are well-known, and we augment these by introducing a martingale filter that improves volatility while guaranteeing smaller mean squared error. Thus, post-processing can rectify problems with calibration and volatility without revisiting the original forecasting model. We apply threshold martingales first to forecasts from simulated models and then to models that predict the winner in professional basketball games. ∗All correspondence regarding this manuscript should be directed to Prof. Stine at the address shown with the title. He can be reached via e-mail at stine@wharton.upenn.edu.