A Transformation-Based Nonparametric Estimator of Multivariate Densities with an Application to Global Financial Markets

We propose a probability-integral-transformation-based estimator of multivariate densities. Given a sample of random vectors, we first transform the data into their corresponding marginal distributions. We then estimate the density of the transformed data via the exponential series estimator. The density of the original data is constructed as the product of the density of the transformed data and marginal densities of the original data. This construction coincides with the copula decomposition of multivariate densities. We decompose the Kullback-Leibler Information Criterion (KLIC) between the true and estimated densities into the KLIC of the marginal densities and that between the true copula density and a variant of the estimated copula density. This result is of independent interest in itself, and facilitates our asymptotic analysis. We derive the large sample properties of the proposed estimator, and further propose a hierarchical model specification method guided by stepwise preliminary subset selections. Monte Carlo simulations demonstrate the superior performance of the proposed method. We employ the proposed method to explore the joint distribution of four major stock markets and some conditional distributions of interest. The estimated copula density function, a by-product of our estimation, provides useful insight into the conditional dependence structure between the US and UK markets, and suggests a certain resilience against financial contagions originated from the Asian market.