Vine copula is an important tool in modeling dependence structures of continuous-valued random variables. The maximum likelihood estimation (MLE) for vine copulas has long been considered computationally difficult in higher dimensions, even in 10 or 20 dimensions. Current computational practice, including the implementation in the state-of- the-art R package VineCopula, suffers from the bottleneck of slow calculations of gradients for the log-likelihood function. We show that by using techniques and tools developed in automatic differentiation, gradients of the log-likelihood function can be accurately calculated in orders of magnitude faster than the current practice. This change immediately accelerates the MLE for vine copulas by 1-2 orders of magnitudes in moderately high dimension. Furthermore, implementation with automatic differentiation is much simpler, e.g., only the objective function evaluation needs to be implemented, and higher order derivatives are readily available. We implement the case of D-Vines with Clayton copulas, and demonstrate such advantages over the current implementation in VineCopula. The purpose of this paper is to further advocate the usage of automatic differentiation tools, which appear to be less known or used in the statistical community.
Citation
Working paper, Department of Mathematics and Statistics, Washington State University, Feb, 2017.