How to Compare Copula Forecasts?

TL;DR

This paper develops a principled framework for comparing copula forecasts using strictly consistent scores, addressing the challenge that copulas are generally non-elicitable, and proposes novel multi-objective scores and tests for forecast evaluation.

Contribution

It introduces new multi-objective scoring methods and two-step tests for copula and marginal forecast comparison, overcoming non-elicitability issues.

Findings

Two-step tests perform well in size and power.

Method effectively attributes forecast accuracy to copulas or marginals.

Empirical example demonstrates practical applicability.

Abstract

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 cannot sensibly be compared on their own. A notable exception is on Fr\'echet classes, that is, when the marginal distribution structure is given and fixed, 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 copula forecast along with marginal forecasts. They give rise to two-step tests of equal or superior predictive ability which admit attribution of the forecast ranking to the accuracy of 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 via an empirical example using copula forecasts for international stock market indices.