Flexible pair-copula estimation in D-vines using bivariate penalized splines

The paper presents a new method for flexible fitting of D-vines. Pair-copulas are estimated semi-parametrically using penalized Bernstein polynomials or constant and linear B-splines, respectively, as spline bases in each knot of the D-vine throughout each level. A penalty induce smoothness of the fit while the high dimensional spline basis guarantees flexibility. To ensure uniform univariate margins of each pair-copula, linear constraints are placed on the spline coefficients and quadratic programming is used to fit the model. The amount of penalizations for each pair-copula is driven by a penalty parameter which is selected in a numerically efficient way. Simulations and practical examples accompany the presentation.