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Proper scoring rules are commonly applied to quantify the accuracy ofdistribution forecasts. Given an observation they assign a scalar score to eachdistribution forecast, with the the lowest expected score attributed to thetrue distribution. The energy and variogram scores are two rules that haverecently gained some popularity in multivariate settings because theircomputation does not require a forecast to have parametric density function andso they are broadly applicable. Here we conduct a simulation study to comparethe discrimination ability between the energy score and three variogram scores.Compared with other studies, our simulation design is more realistic because itis supported by a historical data set containing commodity prices, currenciesand interest rates, and our data generating processes include a diverseselection of models with different marginal distributions, dependencestructure, and calibration windows. This facilitates a comprehensive comparisonof the performance of proper scoring rules in different settings. To comparethe scores we use three metrics: the mean relative score, error rate and ageneralised discrimination heuristic. Overall, we find that the variogram scorewith parameter p=0.5 outperforms the energy score and the other two variogramscores.

Evaluating the Discrimination Ability of Proper Multivariate Scoring Rules