Backtesting risk measure forecasts requires identifiability (for model calibration and validation) and elicitability (for model comparison). We show that the three widely-used systemic risk measures  conditional value-at-risk (CoVaR), conditional expected shortfall (CoES) and marginal expected shortfall (MES), which measure the risk of a position Y given that a reference position X is in distress, fail to be identifiable and elicitable on their own. As a remedy, we establish the joint identifiability of CoVaR, MES and (CoVaR, CoES) together with the value-at-risk (VaR) of the reference position X . While this resembles the situation of the classical risk measures expected shortfall (ES) and VaR concerning identifiability, a joint elicitability result fails. Therefore, we introduce a completely novel notion of multivariate scoring functions equipped with some order, which are therefore called multi-objective scores. We introduce and investigate corresponding notions of multi-objective elicitability, which may prove beneficial in various applications beyond finance. In particular, we prove that conditional elicitability of two functionals implies joint multi-objective elicitability with respect to the lexicographic order on \mathbb{R}^2 , which makes it applicable in the context of CoVaR, MES or (CoVaR, CoES), together with VaR. We describe corresponding comparative backtests of Diebold-Mariano type , for two-sided and 'one and a half'-sided hypotheses, which respect the particularities of the lexicographic order and which can be used in a regulatory setting . We demonstrate the viability of these backtesting approaches in simulations and in an empirical application to DAX 30 and S&P 500 returns.
Backtesting risk measure forecasts requires identifiability (for model validation) and elicitability (for model comparison). The systemic risk measures CoVaR ( conditional value-at-risk ), CoES ( conditional expected shortfall ) and MES ( marginal expected shortfall ), measuring the risk of a position Y given that a reference position X is in distress, fail to be identifiable and elicitable . We establish the joint identifiability of CoVaR, MES and (CoVaR, CoES) together with the value-at-risk (VaR) of the reference position X , but show that an analogue result for elicitability fails. The novel notion of multi-objective elicitability however, relying on multivariate scores equipped with an order, leads to a positive result when using the lexicographic order on \mathbb{R}^2 . We establish comparative backtests of Diebold--Mariano type for superior systemic risk forecasts and comparable VaR forecasts, accompanied by a traffic-light approach . We demonstrate the viability of these backtesting approaches in simulations and in an empirical application to DAX 30 and S&P 500 returns.
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