Correlation versus causation

Idea: Correlation is not cause

Correlation measures how tightly a pair lines up on a straight line. It has no direction.

• $\mathrm{Corr}(X,Y)=\mathrm{Corr}(Y,X)$
• It sees only lines: $Y=X^2$ can have zero covariance
• Cause: changing $X$ alone would change $Y$

Formula: Correlation

$\mathrm{Corr}(X,Y)=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$

Covariance in standard units, always between −1 and 1.

Idea: A confounder

A third variable $Z$ can drive both $X$ and $Y$.

• $X=3Z+E$, $Y=5Z+F$, noise sizes $4$ and $12$
• Neither touches the other, yet $\mathrm{Cov}(X,Y)=3\times 5=15$
• Correlation: $\frac{15}{5\times 13}=\frac{3}{13}$

Example: Simpson's paradox

New router: $\frac{720}{800}$ liquid and $\frac{100}{200}$ illiquid good fills. Old: $\frac{190}{200}$ and $\frac{480}{800}$. What is the new router's pooled rate?

1. Old pooled: $\frac{190+480}{1000}=0.67$
2. Per kind the new one is worse: $0.90<0.95$, $0.50<0.60$
3. New pooled: $\frac{720+100}{1000}=0.82$

Quick check: Hold the mix fixed

New router: $90\%$ liquid, $50\%$ illiquid. Old: $95\%$ and $60\%$. With an even mix of orders, how many points worse is the new one?

Formula: Omitted variable bias

$\text{slope}=\beta +\gamma \delta$

Leave out Z, which drives Y by γ and moves with X by δ, and the slope gains γδ.

Common mistake: A strong regression as proof

A high t-statistic says the slope is precise, not causal. More data only tightens the wrong number.

Instead Name the confounder and its push. Never control for a common result of $X$ and $Y$.

Idea: What does establish cause

Randomise: a coin flip picks the router, so $Z$ has no say and $\delta =0$.

• No experiment? Find an outside rule that moved $X$
• Tick size pilots, index cutoffs: only as good as random
• In the room: confounder, its push, your experiment