How to Compare Copula Forecasts?

This paper lays out a principled approach to compare copula forecasts via strictly consistent scores. We first establish the negative result that, in general, copulas fail to be elicitable, implying that copula predictions alone do not allow for meaningful comparisons. A notable exception is on Fréchet classes, that is, when the marginals are given and identical across all elements of the class, in which case we give suitable scores for the copula forecast comparison. As a remedy for the general non-elicitability of copulas, we establish novel multi-objective scores for joint (copula, marginal) forecasts. These scores give rise to two-step tests of equal or superior predictive ability, which allow differences in predictive accuracy to be clearly attributed to the copulas or the marginals. Simulations show that our two-step tests work well in terms of size and power. We illustrate our new methodology with copula forecasts for international stock market indices, which are issued from several popular models.