Causation does not even imply correlation

Correlation does not imply causation.

Correlation does not imply causation!

What this saying means is that just because two variables correlate, doesn't mean that there is any kind of causal relationship that is responsible. Correlation is, at best, only weak evidence that a causal relationship might exist.

This is one of the statistical fallacies that everybody learns about. You know this; I know this; everybody knows this. People still make this mistake all the time, because causal relationships can be so hard to reliably infer from data, and a strong correlation feels like such a smoking gun.

But surely at least the inverse is true? After all, "I am sick" may not imply "I have the flu", but "I am not sick" does imply "I do not have the flu". So does the absence of correlation imply the absence of causation?

No, in fact, it does not. Not even close. And when it comes to feedback systems, causation can actually imply a lack of correlation. Even worse, it can even create anti-correlation: that is, a correlation that goes in exactly the opposite direction you’d first expect.

That means that even the complete absence of correlation is not enough to conclude that there isn't a strong, direct causal relationship between two variables. It isn't necessarily even weak evidence to that effect.

At the risk of starting an argument, this essay by Monetary Mechanics is a recent example of this fallacy at work in the wild.

According to traditional economic theory, high levels of interest rates lead to reduced levels of credit issuance (through loans and bonds) because high levels of interest rates increase the cost of borrowing. According to the figure above, however, it seems that markets are totally indifferent to the level of interest rates. In fact, the volume of credit issuance and the level of interest rates seem to exhibit a positive, not negative, correlation.

[emphasis mine, and I’ve recreated the primary figure from the article.]

That sure sounds like a shot fired against the traditional notion that high interest rates reduce borrowing, doesn’t it?

I’ve taken the liberty of making the correlation more apparent by plotting the credit issuance and interest rates directly, and calculating the correlation coefficient. Monetary Mechanics is correct that they are positively correlated, which might seem surprising at first given that higher interest rates are supposed to reduce borrowing.

The problem is that Monetary Mechanics is treating this positive correlation as evidence against the notion that higher interest rates reduce credit issuance. It isn’t evidence of that at all.

It’s hot in here. Is the air conditioner running?

If you don't have an economics background, all this credit issuance stuff might sound a bit abstract and hard to follow. Here is the same claim, rephrased in the context of a more familiar household feedback system:

According to classical thermodynamic theory, air conditioning leads to reduced indoor temperatures because air conditioners transport heat from indoors to the exterior. According to the figure below, however, it seems that buildings are totally indifferent to the level of air conditioning. In fact, the indoor temperature and the air conditioner activity seem to exhibit a positive, not negative, correlation.

I’ve gone ahead and simulated an air-conditioned room, controlled by a thermostat with a simple PI (proportional-integral) controller. The summer temperature outside swings about between day and night, and the air conditioner tries to keep the room temperature at a stable 20°C. (I’m going to set aside any air conditioning → outdoor temperature feedback loops here).

We can see that, just as in the economics example, there’s a positive correlation between the air conditioner running and the indoor temperature.

So what’s going on here? Have we at last undermined the laws of thermodynamics? Is our air conditioner causing the room to heat up? Should we switch it off to cool down?

Well, you already know what’s really going on. The warm room is causing our thermostat to activate the air conditioner. In fact, the slope of our correlation is almost exactly the proportional gain that I programmed into the thermostat’s PI controller, but with its sign flipped.

Our correlation didn’t tell us what the air conditioner was doing. It told us what the thermostat was doing.

Let’s look more generally at the structure of this feedback loop.

You see here that the measured temperature T appears in two places: at the output on the right hand side, but also in the feedback signal, where it gets subtracted from the requested temperature setpoint in order to calculate the error.

When we correlate the measured temperature T and the air conditioner activation signal y, we are looking at exactly the transfer function T→y. And because the path from T to y goes through the subtraction junction and then the thermostat control law, that transfer function happens to be precisely -C, the control law I programmed into the thermostat. With its sign flipped.

(A follow up post will deep-dive into reverse engineering the control law from the observed correlation, but I’ll set that aside for now. Suffice to say that the correlation we observe depends very directly on this control law).

So now let’s go back and think about the macroeconomics example again.

This is a very basic model, but I think it captures the main role that a central bank has. Macro Mechanics uses US data but here I’ll just post some excerpts from the Bank of Canada’s website:

  • [The central bank’s] responsibilities include conducting monetary policy to achieve an inflation target agreed upon by the Bank and the Government of Canada.

  • The Bank influences the supply of money circulating in the economy[…] the main tool is the key policy rate.

[emphasis mine]

Once you have some experience in a systems mindset, that should, in broad strokes, sound exactly like a description of a feedback controller.

(This isn’t meant to be a political post, so let’s set aside any questions of whether the central bank is doing a good job or not this year.)

The physics at work are entirely different, but math is math, and systems analysis is agnostic to whether the system is made from spinning gears or flowing electrons or real live humans.

Although the constitutive equations will be different (and perhaps nonlinear/unknown/fundamentally unknowable), the basic structure is still very much like our temperature controller, and the laws of causality still apply. The transfer function from T→y is still -C.

Now everything should be falling into place. All that we’re missing is that central banks (theoretically) feed back on inflation, while Monetary Mechanics was looking at credit issuance. The final puzzle piece is just supposing that credit issuance has some positive correlation with inflation, which is (if not accepted by everyone) at least a fairly mainstream view, because credit forms a big part of the money supply.

Then, interestingly, the correlation that we observed is actually telling us something about the "control law" underlying the actions of the central bank — that is, the algorithm that approximates their decisionmaking process.

More on that later!

This post isn’t intended as professional engineering advice. If you are looking for professional engineering advice, please contact me with your requirements.

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