Diks, C. and Panchenko, V. (2006). A new statistic and practical guidelines for nonparametric Granger causality testing Journal of Economic Dynamics and Control, 30(9-10):1647--1669.

In this paper we introduce a new nonparametric test for Granger non-causality which avoids the over-rejection observed in the frequently used test proposed by Hiemstra and Jones [1994. Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance 49, 1639-1664]. After illustrating the problem by showing that rejection probabilities under the null hypothesis may tend to one as the sample size increases, we study the reason behind this phenomenon analytically. It turns out that the Hiemstra-Jones test for the null of Granger non-causality, which can be rephrased in terms of conditional independence of two vectors X and Z given a third vector Y, is sensitive to variations in the conditional distributions of X and Z that may be present under the null. To overcome this problem we replace the global test statistic by an average of local conditional dependence measures. By letting the bandwidth tend to zero at appropriate rates, the variations in the conditional distributions are accounted for automatically. Based on asymptotic theory we formulate practical guidelines for choosing the bandwidth depending on the sample size. We conclude with an application to historical returns and trading volumes of the Standard and Poor's index which indicates that the evidence for volume Granger-causing returns is weaker than suggested by the Hiemstra-Jones test.