Using Misspecified Marginals and Misspecified Copulas to Compute the Value at Risk: When Do We Have to Care?
Computational Statistics and Data Analysis, Forthcoming
79 Pages Posted: 14 Dec 2007 Last revised: 23 Dec 2011
In this paper we investigate how misspecification both in the marginals and in the copulas may affect the estimation of the Value at Risk when dealing with multivariate portfolios. We first show 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 the 70 % of the true values. We also find that 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 marginals misspecifications on the copula parameter estimation can be rather different, depending on the sign of marginal skewness. We then show that, when small samples are considered and the data are leptokurtic, the misspecifications in the marginal volatility equation are so large to offset the biases in copula parameters when VaR forecasting is of concern. When the sample dimension increases, the biases in the volatility parameters are much smaller, whereas those ones in the copula parameters remain almost unchanged or even increase. In this case, copula misspecifications do play a role for VaR estimation. However, these effects depend heavily on the sign of dependence.
Keywords: Marginals, Copulas, copula VAR-GARCH models, VaR, Simulation
JEL Classification: C15, C32, C51
Suggested Citation: Suggested Citation