I've discussed some of the issues associated with applying such tests in the presence of possibly integrated/cointegrated time-series data previously, here and here.
But can we justify limiting our attention to a linear environment?
Hiemstra and Jones (1994) proposed a nonparametric test for general (linear or non-linear) Granger non-causality - a test that was subsequently applied by a number of authors.
However, the Hiemstra and Jones (HJ) test has been brought into question by Diks and Panchenko (2006).
The difficulty with the HJ test arises for the following reason. The HJ test is a test of whether or not a particular conditional probability condition holds.
However, DP show that this condition that is being tested isn't necessarily implied by Granger non-causality (contrary to the assumption made by HJ).
So, a rejection of the HJ null hypothesis shouldn't really be taken as a rejection of Granger non-causality. Accordingly, the HJ test tends to lead to spurious "discoveries" of (nonlinear) Granger causality.
This leads Diks and Panchenko (DP) to suggest that the apparent "evidence" for the presence on nonlinear Granger causality in the empirical economics and finance literature (e.g., Brooks and Henry, 2000; Silvapulle and Moosa, 1999) should, at the very least, be questioned.
Moreover, DP show, through a simulation study, that the true size of the HJ test increases as the sample size increases.
As the size of the test approaches unity, this means that we are almost always rejecting Granger non-causality, when in fact no such causality exists!
DP provide simulation results that when the test is applied with a nominal significance level of 5%, the true significance level is virtually 100% by the time that the sample size is 5,000.
Such a sample size is quite common when working the high-frequency financial time-series data, for example, so this is a major practical issue.
Accordingly, my advice is to be very sceptical of studies based on the Hiemstra and Jones test for nonlinear Granger causality. Diks and Panchenko (2006) provide an alternative nonparametric test for nonlinear Granger causality that appears to circumvent the problems associated with the HJ test.
However, there's certainly room for more work to be done on this problem.
Brooks, C. and O. T. Henry, 2000. Linear and non-linear transmission of equity return volatility: Evidence from the US, Japan and Australia. Economic Modelling, 17, 497–513.
Diks, C. and V. Panchenko, 2006. A new statistic and practical guidelines for nonparametric Granger causality testing. Journal of Economic Dynamics & Control, 30, 1647-1669.
Hiemstra, C. and J. D. Jones, 1994. Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance, 49, 1639–1664.
Silvapulle, P. and I. A. Moosa, I. A., 1999. The relationship between spot and futures prices: Evidence from the crude oil market. Journal of Futures Markets, 19, 157–193.
© 2012, David E. Giles