Causality: Some New Thoughts on an Old Topic
Traditional "Granger-Causality" (henceforth just G-causality) concerned the conditional mean. It required that the causal variable Yt preceded the causal variable Xt+1 in time and also that Yt contained special information about Xt+1 which would be shown in the conditional mean E[Xt+1|Yt]. There is an immediate forecasting implication. Later, in terms of conditional distributions, Yt did not cause Xt=1 in distributions, if Yt was conditionally independent of Xt+1. Some new implications of this definition will be presented and the links between the distributions in mean and distribution explored. Since the appearance of these definitions a number of alternative forms have appeared, due to Hoover, Pearl, White, and others and they will be discussed and compared. There will be no conclusion
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