It is a commonplace that correlation does not imply causality, however eyebrow-wagglingly suggestive it may be of causal hypotheses. It is less commonly noted that causality does not imply correlation either. It is quite possible for two variables to have zero correlation, and yet for one of them to be completely determined by the other.

The causal analysis of statistical information is the subject of several major books, including Judea Pearl's Causality and Probabilistic reasoning in intelligent systems, and Spirtes et al's Causation, Prediction, and Search. One of the axioms used in the last-mentioned is the Faithfulness Axiom. See the book for the precise formulation; informally put it amounts to saying that if two variables are uncorrelated, then they are causally independent. As support for this, the book offers a theorem to the effect that while counterexamples are theoretically possible, they have measure zero in the space of causal systems, and anecdotal evidence that people find fault with causal explanations violating the axiom.

The purpose of this article is to argue that this is not the case.

The counterexample consists of just two variables A and B. The time series data can be found here, a text file in which each line contains a pair of values for A and B. Here is a scatter-plot:

The correlation is not significantly different from zero. Consider the possible causal relationships there might be between two variables, assuming there are no other variables involved. A causes B; B causes A; each causes the other; neither causes the other. Which of these describes the relationship between A and B for the above data?

The correct answer is that none of the four hypotheses can be rejected by these data alone. The actual relationship is: A causes B. Furthermore, there is no noise in the process. A is varying randomly, but B is deterministically caused by A and nothing else, and not by a complex process either. The process is robust: it is not by an accident that the correlation is zero. Every physical process that is modelled by the very simple mathematical relation at work here (to be revealed below) has the same property.

Because I know the process that generated these data, I can confidently predict that it is not possible for anyone to discover from them the true dynamical relation between A and B. So I'll make it a little ea

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