I'm currently reading on the Granger causality test but different sources seem to be contradicting.
In the original paper Using the mutual information coefficient to identify lags in nonlinear models (1994) it says:
An ideal measure of functional dependence for a pair of random variables x, y should [...] (iii) have a value of unity for the modulus of the measure if there is an exact nonlinear relationship between the variables, such as x = h(y).
Now suppose we have the following relationship: $X_t = \lvert Y_{t-1} \rvert$, where $Y_t$ are i.i.d. random variables from a $N(0,1)$ distribution. According to the above definition, the Granger causality should equal 1.
However, according to Wikipedia, Granger causality is calculated via a VAR model and one rejects the null-hypothesis that $Y_t$ does not Granger cause $X_t$ if and only if no lagged values of $Y_t$ are contained in the VAR model. For the example above, one would therefore not reject the null hypothesis, even though $X_t$ is completely determined by $Y_{t-1}$.
In essence and to my understanding, this question boils down to whether non-linear effects are captured when testing for Granger causality. Are there maybe different implementations to test for Granger causality, some of which do capture nonlinear effects?