Journal

Correlation & Causality

Words by Calum Mitchell
28 October 2016

by Rethinking Economics Oxford’s Max Schroder

“Correlation does not imply causation” must be the most routinely thrown-around phraseology in all of economics. Its meaning: even a systematic co-occurrence (correlation) between two (or more) observed phenomena does not grant conclusive grounds for assuming that there exists a causal relationship between these phenomena. The truth of the phrase is widely revered: it is used to dismiss arguments deemed naive and it distinguishes the initiated – those who have taken Econometrics 101 –  from all those who “do not do proper data analysis”.

The problem with this phrase – as with so many others (e.g. “the free market exists in the absence of government intervention”) – is that it is primarily used by people who do not know what they are talking about. The statement seems obvious as long as one can maintain the belief that causation is something that occurs between certain events (causes and effects) and that we have a pretty solid ability to distinguish between mere accident (correlation) and true causation. Clearly however, such a folklore notion of causation cannot be the basis of any rigorous scientific inquiry. The aim therefore – if we want to retain our initial assumption – must be to arrive a definition of causation that does not rely on correlation.

In the following I will briefly sketch two of the prominent philosophical positions on the matter: The Regularity Theory and the Counterfactual Theory of Causation, I will elaborate how their pros and cons impact on this debate. I will show that any serious consideration of causality leads us to a conundrum with regards to correlation (and why that might not be that problematic) and finally I will change my mind completely in order to surprise everyone.

The philosophical treatment of causality[1] starts with David Hume (Entschuldigung, Herr Professor Kant), who writes: “we may define a cause to be an object, followed by another, and where all the objects similar to the first are followed by objects similar to the second.”“An Enquiry concerning Human Understanding” (Section 7).

Apart from dispensing with the notion that causation is something magical that can be intuited and doesn’t need to be defined, Hume sets out the first account of the Regularity Theory. We do only ever observe cause and effect (the white ball hitting the red ball and the red ball subsequently speeding away) but never causality. Or more specifically: we never even observe cause or effect – only two events that are related by some “spatiotemporal” proximity and thus infer (naively) that one must have caused the other. In short: causation is nothing but (albeit perfect) correlation! Although there have been several revisions of this theory over the centuries – the core idea has remained surprisingly untouched. One of the debates centres around the question of Fragility – of how general the “objects” in question should be defined. For example: does ignition of a flame have to follow the striking of a match in every case (regardless of other conditions e.g. dampness of the match) in order for the striking of the match to count as causal for the subsequent ignition; or are we allowed to include certain other items (dryness of the match, existence of oxygen, etc.) in our objects – set of conditions which are (forgive the language) “jointly sufficient and individually necessary”[2].

I will return to the fragility issue in due course, but for now I want to introduce the second causal theory for this essay. If we read onto the sentence following the last quote we find that Hume also (probably unknowingly) laid the foundations for the Counterfactual Theory of Causation:

“Or in other words where, if the first object had not been, the second never had existed.”“An Enquiry concerning Human Understanding” (Section 7).

The idea seems simple enough: a cause is some event that is such that had it not occurred the effect would not have occurred either. The problem herein lies with the modular “would not have” – how does anyone know what would have happened? The 20th century philosopher David Lewis makes a fantastically imaginative case for the existence of counterfactuals and uses them to back up a counterfactual theory of causation. Lewis says (I paraphrase badly): A causes B (in one world[3]) if a) in this world A occurs and B occurs and b) if in all those worlds closest (read most similar[4][5]) to it A doesn’t occur and B doesn’t occur. Basically imagine a world where you pick up a rock and throw it into a window – the window smashes. Now imagine a world which is just like ours (ceteris paribus) but this time you don’t pick up a rock and you don’t throw it into a window. If the window didn’t spontaneously break in this world, then your throwing of the rock caused the breaking of the window.

Despite the great work done by Lewis I do not believe that this account can convince many: there is simply no way for us to know how things would have been – at least not out of hand: To construct the counterfactual case we need to refer back to some sort of known regularity. We think that in a world that is very much like ours the “not throwing of rocks” results in the “not breaking of windows” simply because in our world we often observe that the “not throwing of rocks” is followed by the not breaking of windows. Our evaluations of counterfactuals are tainted by our experiences made in this particularly world – that is to say that the results of the counterfactual theory are subsumed under the regularity theory.

It seems that this analysis has left us with a “even worse than second best” result: Our best account of causation essentially involves correlation which renders the statement at the start of this essay an analytic contradiction. Yet we still want to make sense of this very intuitive notion that sometimes co-occurrence of two types of events just involve “coincidences” and sometimes more intimate relations. To address this problem one has to rigorously think about the restrictions put on the set of events that we think of as causes and of the generality (fragility) of these types. The good news is that this allows us to retain our notion of causality in simple everyday situations: we can be pretty sure that in the class of pool tables the hitting of one ball off another causes the second ball to move (provided the ball doesn’t spontaneously decide to roll away at exactly the right point). Since we can observe that one always (regularly) follows the other across a large set of circumstances. This allows us to weed[6] out those events which are merely accidental (i.e. whether Jimmy had 3 pints before the shot) and which are causal (that the ball is hit with sufficient force).

The bad news is that this approach will probably not help us with analysing causal forces in the economy. It is unlikely that we will ever have enough information about all the miniscule details that might matter. In essence chaos theory tells us that complex systems are superfragile[7] – a slight change in the initial conditions[8] will result in a vastly different end state. Under these circumstances we will probably never be able to come across enough cases that are “sufficiently similar & sufficiently different” to be able to distinguish between causality and correlation.

However, there might be a way out of this dilemma: abandon causation! Bertrand Russell suggested: “The concept “cause”, as it occurs in the works of most philosophers, is one which is apparently not used in any advanced science.” – Human Knowledge (1948). The aim (for physicists for example) is not to understand the interaction between different particles as relationships of cause and effect, but rather as simultaneous differential equations that preserve certain relationships between them. Like the expressions on the two sides of an equal sign preserve their equality, events in the real world maintain – trough all permutations – steady relations to each other. The purpose of science is to determine the laws of nature/economics/whatever – that is the essence – of these relationships. But where does this leave us in the case of economics?

The purpose of this essay was not to advocate purging the notion of causality form the minds of economists. Causality is a useful concept and I would hold that we often have a good intuitive grasp of when it applies and when it doesn’t. Like an experienced (cigar-chewing, coat-wearing, glass-eyed) detective who is able to pick out from among all the clues of the crime scene the one detail that will lead him to the murderer, a good economist is able to have “a feel” for the factors that might be relevant for a given problem. Fostering such a mindset however, requires a broad curiosity that covers a broad spectrum of theories and experiences; and an open mind for new ways of thinking[9].

Lastly we should be weary of the limitations of models that try to explain a system as complex as the economy (or even just a part of it). In the light of the flaw that Lucas pointed out in the Keynesian models of the late 60´s and early 70´s, and the similar comments made recently by economists such as Paul Romer about the succeeding DSGE[10] Models this worry seems particularly applicable to economics: It is incredibly difficult to find anything resembling causality in complex systems. It is too easy to miss the butterfly.

Sometimes correlation is the best we can hope for.

Footnotes

[1] Disclaimer: This essay is not trying to be a comprehensive account of all philosophical theories of causation. I am merely trying to make a point (and amuse).

[2] The fragility issue becomes even more complex if we consider how general the classes of „effects“ can be.

[3] Lewis thinks of worlds as something akin to paralel universes.

[4] Whatever that means.

[5] Economists might spot a „ceteris paribus“ in there.

[6] This process has a lot of similarities with regression analysis where differences in differences are used to determine individual effects.

[7] My phrase.

[8] The iconic „wings of a butterfly“.

[9] All great literarly detectives have a multitude of interests – think only of Sherlock Holmes.

[10] Dynamic Stochastic General Equilibrium.

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