Score tests for covariate effects in conditional copulas

We consider copula modeling of the dependence between two or more random variables in the presence of a multivariate covariate. The dependence parameter of the conditional copula possibly depends on the value of the covariate vector. In this paper we develop a new testing methodology for some important parametric specifications of this dependence parameter: constant, linear, quadratic, etc. in the covariate values, possibly after transformation with a link function. The margins are left unspecified. Our novel methodology opens plenty of new possibilities for testing how the conditional copula depends on the multivariate covariate and also for variable selection in copula model building. The suggested test is based on a Rao-type score statistic and regularity conditions are given under which the test has a limiting chi-square distribution under the null hypothesis. For small and moderate sample sizes, a permutation procedure is suggested to assess significance. In simulations it is shown that the test performs well (even under misspecification of the copula family and/or the dependence parameter structure) in comparison to available tests designed for testing for constancy of the dependence parameter. The test is illustrated on a real data set on concentrations of chemicals in water samples.