When Regression Coefficients Are Causal

The one idea: A regression coefficient is a causal effect only under three conditions at once — the control set satisfies the backdoor criterion, nothing in it is a collider or mediator, and the functional form is roughly right. Miss any one of the three and the coefficient is still a real number, computed correctly, that answers a different question than the one being asked.

⚠️ The trap: "The coefficient is significant and the R² is high, so the model is capturing the real relationship." That feels right and it's wrong. **Significance and fit describe how precisely the software estimated a number — they say nothing about whether that number is the causal effect. A regression missing a confounder can be highly significant, well-fitting, and badly wrong, all at the same time.**

The last two lessons gave you the machinery: the bias equation names what contaminates a raw comparison, and the backdoor criterion tells you exactly which variables neutralize that contamination. This lesson puts them together into a checklist for the single most common causal-inference artifact in industry — the OLS regression table — and then breaks one on purpose, with real numbers, so you can see what a biased-but-convincing regression looks like from the inside.

The three conditions

Run a regression of outcome Y on treatment D and controls X. The coefficient on D, β̂, estimates the causal effect of D on Y — not "is correlated with," not "predicts," but causes — when all three of these hold:

1. X satisfies the backdoor criterion. Every backdoor path from D to Y is blocked, meaning every confounder — anything that…