Correlation vs Causation

Ice cream sales and drownings rise together every summer. Nobody thinks cones cause drownings, obviously, hot weather drives both. That's the whole lesson in one line: two series can march in lockstep without either one causing the other. Correlation is just a measurement of co-movement; causation is a claim that pushing on one lever moves the other. Interviewers love this topic because it's easy to state and easy to violate, and quant research lives and dies by getting it right.

When $X$ and $Y$ correlate but $X$ does not cause $Y$, there are three usual suspects.

1. A hidden common cause (confounder). Some third thing $Z$ drives both $X$ and $Y$. They move together because they share a puppet-master, not because they touch each other. This is the case in the diagram.
2. Reversed direction. Maybe $Y$ causes $X$, not the other way around. Successful firms advertise more, so ad-spend correlates with profit, but flip the arrow before concluding ads make the profit.
3. Pure coincidence or selection. Test enough unrelated pairs and some will correlate by luck. Filter your sample on an outcome and you can manufacture correlations that vanish in the full population.

Worked example: two stocks that "move together"

Take two stocks whose returns are each just the market plus their own private noise:

$$A = M + \varepsilon_A, \qquad B = M + \varepsilon_B,$$

where the market $M$ and the two noise terms are independent, each with variance $1$. Stock $A$ has nothing to do with stock $B$, no product, no supply chain, no shared anything except exposure to the market $M$ (their Beta (β)). Yet they correlate. Because the variances add,

$$\operatorname{Var}(A) = \operatorname{Var}(B) = 1 + 1 = 2, \qquad \operatorname{Cov}(A,B) = \operatorname{Var}(M) = 1,$$

so the correlation is

$$\rho_{A,B} = \frac{\operatorname{Cov}(A,B)}{\sqrt{\operatorname{Var}(A)\operatorname{Var}(B)}} = \frac{1}{\sqrt{2 \cdot 2}} = 0.5 .$$

A solid $0.5$ correlation out of thin air, purely from the shared driver $M$. The fix is to condition on the confounder: regress each return on $M$ and correlate the leftovers $\varepsilon_A$ and $\varepsilon_B$. Since those are independent by construction, the residual correlation is $0$. The apparent link was the market all along, exactly the trap in the diagram.

What it actually takes to claim causation

Since a raw correlation can't distinguish "$X$ causes $Y$" from "$Z$ causes both," you need something more:

• A randomized experiment, assign $X$ by a coin flip so nothing else can be tangled up with it. This is the gold standard and is why A/B tests exist.
• A natural experiment or instrument, when you can't randomize, find some outside nudge that shifts $X$ but couldn't affect $Y$ except through $X$.
• Controlling for the confounders you can name, add $Z$ to the regression. This only works for the confounders you know and measure; the dangerous ones are the ones you never thought of.

Common pitfalls

• Assuming the arrow points the way you'd like. Higher volatility "predicts" lower returns, but which causes which, and is a third factor (risk aversion, leverage) driving both? Always ask if the direction could be reversed.
• Ignoring the lurking variable. Most spurious finance correlations trace back to an unmeasured common driver, the market, a sector, a macro regime. If two signals correlate, suspect a shared factor before inventing a story.
• Mining for coincidences. Scan thousands of variables and some will correlate with your target by luck alone. Those "relationships" evaporate out of sample. See p-values and Multiple Testing.
• Trending series faking a bond. Any two things that both drift over time (prices, GDP, a growing index) will show a huge Correlation with no relationship at all. Work with changes or returns, not raw levels.

The reflex to build: whenever you see two things move together, before telling a causal story, ask "what could be causing both?", "could the arrow run the other way?", and "did I go looking until I found this?" Answer those three honestly and most fake causation collapses.