Fitting Bivariate Cumulative Returns with Copulas

Abstract A copula based statistical method for fitting joint cumulative returns between a market index and a single stock to daily data is proposed. Modifying the method of inference functions for margins (IFM method), one performs two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chi-square statistic associated to an adequate bivariate version of the usual Pearson goodness-of-fit test. The copula fitting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate 1-year period are quite satisfactory. The best overall fits are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel–Hougaard copulas. Finally, a significant application to covariance estimation for the linear Spearman copula is discussed.

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