Diks, C. and Wolski, M. (2016). Nonlinear Granger Causality: Guidelines for Multivariate Analysis Journal of Applied Econometrics, 31(7):1333--1351.

We propose an extension of the bivariate nonparametric Diks–Panchenko Granger non-causality test to multivariate settings. We first show that the asymptotic theory for the bivariate test fails to apply to the multivariate case, because the kernel density estimator bias and variance cannot both tend to zero at a sufficiently fast rate. To overcome this difficulty we propose to reduce the order of the bias by applying data sharpening prior to calculating the test statistic. We derive the asymptotic properties of the {\textquoteleft}sharpened{\textquoteright} test statistic and investigate its performance numerically. We conclude with an empirical application to the US grain market, using the price of futures on heating degree days as an additional conditioning variable.