# Granger causality test - What dependencies are being captured?

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?

• By original paper, you do not mean the paper that proposed Granger causality? (See Granger, C. W. J. (1969). "Investigating Causal Relations by Econometric Models and Cross-spectral Methods". Econometrica. 37 (3): 424–438. doi:10.2307/1912791.) – Richard Hardy Apr 2 at 12:05
• I honestly was not aware of this much earlier paper. So the term original might be misleading. I'm referring to the the paper that I linked where a statistic R is defined that can take on values between 0 (two variables are independent) and 1 (two variables are functionally related). – shiningPanther Apr 2 at 19:14

Hi : All Granger causality says is that, if including including a variable X_t as a variable in the forecast for $$Y_t$$, improves the forecast of $$Y_t$$, then $$X_t$$ Granger causes $$Y_t$$. As far as I know, it's not dependent on the use of a VAR. So my I don't see why it wouldn't apply to non-linear forecasts also. The point is that it's a very specific type of causality and has nothing to do with say the causal concept of Judea Pearl. Granger did a lot of work in non-linear modelling so I would confirm this by checking his work in that area out. I think that he may even have a book on non-linear modelling. ( Granger and Lee possibly ? ).
• It is worth noting for completeness that (in a two variable system) the history of $X$ needs to improve the prediction of $Y$ beyond the prediction that uses the history of $Y$ alone. (I have seen this being omitted and resulting in mistaken interpretations frequently enough to care pointing out.) – Richard Hardy Apr 2 at 19:18