2. Correlation does not imply causation

Level 4: Correlation, regression and the myth of causality

Among the roughly 500 shares of the S&P 500 there are more than 120,000 possible pairs.

Search through all of them with no prior criterion and you will find dozens of pairs with impressive correlations. Not because they are connected, but because there are 120,000 of them.

Three explanations for the same chart

When you observe two series moving together, there are basically three possible explanations:

• That one really does cause the other.
• That both respond to a third common variable, a "confounder".
• That it is pure statistical coincidence, all the more likely the more pairs of series you compare.

Confusing the first explanation with the other two is one of the most expensive and most frequent errors in quantitative finance.

The absurd example

The extreme, almost comic example serves to fix the idea.

There are historical data series entirely unrelated to one another that show correlations above 0.9 over certain periods, without any economic mechanism connecting them. For example, the per capita consumption of a food product in one country and the level of some distant equity index.

It is pure statistical coincidence, helped along by the fact that both series simply evolve over time.

The lesson is not that absurd coincidences are common. It is that a high correlation figure, on its own, is never proof of anything.

The hidden confounder

In the financial arena, the commonest case is that of the confounding variable.

Imagine a soft drinks company and a software company, businesses with no operational relationship between them, whose shares show a daily return correlation of 0.55.

The most plausible explanation is not that one influences the other. It is that both have a positive beta to the market as a whole: when the index rises or falls sharply, it drags most large shares with it, soft drinks and technology alike.

The market as a whole acts here as the confounding variable hidden behind the observed correlation.

The arrow pointing the other way

Another classic trap is reverse causation.

It is observed that shares tend to fall in the weeks before an interest rate cut. And a commentator concludes that "rate cuts sink the stock market".

The more plausible causal order is exactly the opposite. A prior deterioration in the economic data hits shares first, and it is that same deterioration that then pushes the central bank to cut rates.

The temporal correlation is real. What happens is that the arrow of causation points the other way from what it looks like at first glance.

When thousands of pairs are trawled

The problem gets worse when pairs of assets are searched for systematically, as happens when designing pairs trading strategies.

Let us go back to the figure we opened with: more than 120,000 possible pairs among the roughly 500 shares of the S&P 500.

If each one is examined in search of very high correlations with no prior economic criterion, it is mathematically to be expected that dozens of pairs with impressive correlations will turn up by chance alone.

It is exactly the same as tossing a coin thousands of times: it produces long runs of heads without the coin being rigged. The more comparisons you make, the greater the probability of stumbling on a coincidence that looks like signal.

Some much cited calendar anomalies —the January effect, or the idea of "sell in May"— probably owe part of their fame to this same problem. Decades of researchers testing different time windows and different markets on the same historical data end up finding, by pure statistics, some pattern that looks robust without really being so.

The practical rule

The rule that follows from all this is easy to state and hard to apply with discipline.

A correlation with no underlying economic story to explain it is a warning sign, not an opportunity. All the more so if it was found by trawling thousands of combinations.

Correlation is a descriptive tool. Demonstrating causation requires an additional structural or economic argument, not just a high coefficient.