  The multivariate nonlinear Granger causality developed by Bai et al. (2010)
plays an important role in detecting the dynamic interrelationships between two
groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by
Hiemstra and Jones (1994), they attempt to establish a central limit theorem
(CLT) of their test statistic by applying the asymptotical property of
multivariate $U$-statistic. However, Bai et al. (2016) revisit the HJ test and
find that the test statistic given by HJ is NOT a function of $U$-statistics
which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor
the one extended by Bai et al. (2010) is valid for statistical inference. In
this paper, we re-estimate the probabilities and reestablish the CLT of the new
test statistic. Numerical simulation shows that our new estimates are
consistent and our new test performs decent size and power.
