Are Copula-GoF-tests of any practical use? Empirical evidence for stocks, commodities and FX futures

Abstract In this paper, the optimality of bivariate copula-VaR models and the usefulness of several goodness-of-fit tests for copulas are analysed in a comprehensive empirical study using data for stocks, commodities and FX futures. In particular, I try to answer two questions: (1) which parametric copula is optimal for estimating the VaR and Expected Shortfall (ES) of a given portfolio consisting of linear assets? (2) How can the VaR- or ES-optimal parametric copula be identified in-sample? To answer these questions, the VaR and ES for a total of 12,000 bivariate portfolios are estimated from 435 linear assets over eight different time windows. The results show that although copula-models with GARCH-margins yield considerably better VaR-estimates than correlation-based models, the identification of the optimal parametric copula form is a serious unsolved problem. The analysis of three state-of-the-art approaches for testing a copula-model's goodness-of-fit showed that none of the tests is able to identify the optimal parametric form unequivocally. In addition to this result, for more than 80% of all portfolios considered, all five parametric copula models yielded worse ES-estimates than the correlation-based benchmark or underestimated actual portfolio risk. Moreover, the backtesting results show that the optimal parametric copula is both dependent on the risk measure and time-variant.