How to understand mutual Granger causality As stated in title, when we have a time series model of two variables, e.g x and y, and conduct the Granger causality test to examine the "causal relationship" between two variables, we can be in a situation that x Granger causes y and y also Granger causes x, that is, a mutual Granger causal relationship. How to understand and interpret such relationship in real case? I know that the result of Granger causality test does not equal to real causal relationship in philosophy sense, but how to make sense to such result?
 A: Consider the following examples:


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*Your time series are the gross domestic products (GDP) of France and Germany. As the two country’s economics are strongly interacting, a strong French economy is likely to give rise to an improvement in the German economy and vice versa. Thus, knowing the GDP for France allows you to better predict the German GDP and vice versa. The time series Granger-cause each other.

*Your time series are the populations of two species in a complex ecosystem, which are predator and prey towards each other. A high population of the prey is good for the predator and a high population of the predator is bad for the prey. Again, knowing either population helps to predict the other and both time series Granger-cause each other.
In general, mutual Granger causality occurs whenever two systems are mutually interacting with each other, which is the default interaction.
One-directional Granger causality occurs in the rare case where you have a one-directional interaction between systems (for example the weather influences the performance of your wind turbine, but not vice versa).
Also note that two systems that do not interact or only interact in one direction but are influenced by a third one may be mutual Granger-causal.
