Kernel Estimation of Copula Densities and Applications

In this paper, we study the kernel estimation of the copula density on unit square [0,1]X[0,1], and demonstrate the implementation of this methodology to equity and bond markets. There are two crucial problems associated with this estimator. First, the kernel estimator is biased at the boundaries. Second, the kernel estimator is sensitive to both kernel and bandwidth. To correct the boundary effects, we propose a Gaussian copula (GC) kernel and a logit transformation (LT) kernel estimators, and derive their asymptotic properties. Moreover, we introduce a Bayesian approach to bandwidth and kernel selection, and an L2-type goodness-of-fit test. We conduct a simulation study to assess the finite sample performance of the GC and LT kernels, and two comparable kernels based on Gaussian transformation (GT) and mirror-reflection (MR), with the Bayesian and likelihood cross-validation (LCV) bandwidths. The results show that the performances of the Bayesian and LCV bandwidths are more or less the same. The performance of the kernel functions depends largely on the shapes of the underlying copula densities. The GC kernel density estimate fits the copula density of All Ords and S&P 500 returns well. The t-copula fits the copula density of sovereign bond yield spreads of Greece and Spain well