The famous Hiemstra-Jones (HJ) test developed by Hiemstra and Jones (1994) plays a significant role in studying nonlinear causality. In the last two decades there are numerous applications and theoretical extensions based on this pioneering work. However several works pointed out that counter-intuitive results are obtained from the HJ test and some researchers found that the HJ test is seriously over rejecting through simulation study. In this paper, we reinvestigate HJ's creative work in 1994 and found that their proposed estimators of the probabilities over different time intervals were not consistent to the target ones proposed in their criterion. To test the HJ's novel hypothesis on Granger causality, we shall propose new estimators of the probabilities defined in HJ's paper and reestablish the asymptotic properties which will induce new tests similar to HJ's ones. Some simulation work will also be presented to support our new findings.