Copulas are extensively used for dependence modeling. In many cases the data does
not reveal how the dependence can be modeled using a particular parametric copula.
Nonparametric copulas do not share this problem since they are entirely data based.
This paper proposes nonparametric estimation of the density copula for α-mixing data
using Bernstein polynomials. We study the asymptotic properties of the Bernstein
density copula, i.e., we provide the exact asymptotic bias and variance, we establish
the uniform strong consistency and the asymptotic normality.
This paper proposes a new nonparametric test for conditional independence, which is
based on the comparison of Bernstein copula densities using the Hellinger distance.
The test is easy to implement because it does not involve a weighting function in the
test statistic, and it can be applied in general settings since there is no restriction on the
dimension of the data. In fact, to apply the test, only a bandwidth is needed for the
nonparametric copula. We prove that the test statistic is asymptotically pivotal under
the null hypothesis, establish local power properties, and motivate the validity of the
bootstrap technique that we use in finite sample settings. A simulation study illustrates
the good size and power properties of the test. We illustrate the empirical relevance of
our test by focusing on Granger causality using financial time series data to test for
nonlinear leverage versus volatility feedback effects and to test for causality between
stock returns and trading volume. In a third application, we investigate Granger
causality between macroeconomic variables
We study the asymptotic properties of the Bernstein estimator for unbounded density copula
functions. We show that the estimator converges to infinity at the corner. We establish its relative
convergence when the copula is unbounded and we provide the uniform strong consistency of the
estimator on every compact in the interior region. We also check the finite simple performance of the
estimator via an extensive simulation study and we compare it with other well known nonparametric
methods. Finally, we consider an empirical application where the asymmetric dependence between
international equity markets (US, Canada, UK, and France) is re-examined.
We propose a nonparametric estimator and a nonparametric test for Granger causality measures that quantify linear and nonlinear Granger causality in distribution between random variables. We first show how to write the Granger causality measures in terms of copula densities. We suggest a consistent estimator for these causality measures based on nonparametric estimators of copula densities. Further, we prove that the nonparametric estimators are asymptotically normally distributed and we discuss the validity of a local smoothed bootstrap that we use in finite sample settings to compute a bootstrap bias-corrected estimator and test for our causality measures. A simulation study reveals that the bias-corrected bootstrap estimator of causality measures behaves well and the corresponding test has quite good finite sample size and power properties for a variety of typical data generating processes and different sample sizes. Finally, we illustrate the practical relevance of nonparametric causality measures by quantifying the Granger causality between S&P500 Index returns and many exchange rates (US/Canada, US/UK and US/Japen exchange rates).
Nonparametric estimation of the copula function using Bernstein polynomials is studied. Convergence in the uniform topology is established. From the nonparametric Bernstein copula, the nonparametric Bernstein copula density is derived. It is shown that the nonparametric Bernstein copula density is closely related to the histogram estimator, but has the smoothing properties of kernel estimators. The optimal order of polynomial under the L2 norm is shown to be closely related to the inverse of the optimal smoothing factor for common nonparametric estimator. In order of magnitude, this estimator has variance equal to the square root of other common nonparametric estimators, e.g. kernel smoothers, but it is biased as a histogram estimator.