Conditional expectation formula of copulas for higher dimensions and its application

Abstract

Due to its simple form, linear regression is the most commonly used model when dealing with a predictive model. However, there are some limitations to the model, such as the constraint of only being able to model variables that have a linear relationship, the assumption of normality on its error, and the multi-collinearity between independent variables which should not occur. One of the alternative models that is free from these limitations is the copula-based regression model defined by the conditional expectation formula of copulas. Leong and Valdez [Claims prediction using copula models, Insurance Math. Econom., 2005] [15] developed a conditional expectation formula of copulas for higher dimensions in the implicit form with bivariate case examples. Crane and Hoek [Conditional expectation formulae for copulas, Aust. N.Z.J. Stat, 2008] [5] provided conditional expectation formula of copulas explicitly for two dimensions with its examples. However, in practice, a predictive model often involves more than two variables, i.e. one dependent variable with more than one independent variable, including a copula-based regression model. With regard to these problems and the limitations of dimension in previous studies, our contribution in this study is extending the copula-based regression model for higher dimensions for class of Farlie-Gumbel-Morgenstern, elliptical, and Archimedean copula. We obtain a closed-form of conditional expectation formula of Farlie-Gumbel-Morgenstern, Gaussian, Student-t, and Clayton copula for n dimensions and provide the formula for Gumbel copula up to four dimensions. We apply our extended formula to estimate KRW/USD currency based on its association with CNY/USD and JPY/USD, and found that the extended function can be used quite accurately.