The information-theoretical concept transfer entropy is an ideal measure for detecting conditional independence, or Granger causality in a time series setting. The recent literature indeed witnesses an increased interest in applications of entropy-based tests in this direction. However, those tests are typically based on nonparametric entropy estimates for which the development of formal asymptotic theory turns out to be challenging. In this paper, we provide numerical comparisons for simulation-based tests to gain some insights into the statistical behavior of nonparametric transfer entropy-based tests. In particular, surrogate algorithms and smoothed bootstrap procedures are described and compared. We conclude this paper with a financial application to the detection of spillover effects in the global equity market.

To date, testing for Granger non-causality using kernel density-based nonparametric estimates of the transfer entropy has been hindered by the intractability of the asymptotic distribution of the estimators. We overcome this by shifting from the transfer entropy to its first-order Taylor expansion near the null hypothesis, which is also non-negative and zero if and only if Granger causality is absent. The estimated Taylor expansion can be expressed in terms of a U-statistic, demonstrating asymptotic normality. After studying its size and power properties numerically, the resulting test is illustrated empirically with applications to stock indices and exchange rates.