The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study

Abstract

The effect on the estimation of the Value at Risk when dealing with multivariate portfolios when there is a misspecification both in the marginals and in the copulas is investigated. It is first shown that, when there is skewness in the data and symmetric marginals are used, the estimated elliptical (normal or t) copula correlations are negatively biased, reaching values as high as 70% of the true values. Besides, the bias almost doubles if negative correlations are considered, compared to positive correlations. As for the t copula degrees of freedom parameter, the use of wrong marginals delivers large positive biases, instead. If the dependence structure is represented by a copula which is not elliptical, e.g. the Clayton copula, the effects of marginal misspecifications on the copula parameter estimation can be rather different, depending on the sign of marginal skewness. Extensive Monte Carlo studies then show that the misspecifications in the marginal volatility equation more than offset the biases in copula parameters when VaR forecasting is of concern, small samples are considered and the data are leptokurtic. The biases in the volatility parameters are much smaller, whereas those ones in the copula parameters remain almost unchanged or even increase when the sample dimension increases. In this case, copula misspecifications do play a role for VaR estimation. However, these effects depend heavily on the sign of the dependence.