Chapter 32 of 57

Endogeneity: The One Assumption OLS Quietly Needs

Created Jul 5, 2026 Updated Oct 25, 2026

Fit a line through a cloud of points and you always get a slope. The quiet question is what that slope means. Read it as "the effect of X on y" and you've smuggled in an assumption most people never say out loud: that the thing you're measuring and the things you forgot to measure don't move together. When that assumption fails, OLS doesn't return a noisy version of the right answer — it returns a different quantity, and no amount of data turns one into the other.

Write the model as plainly as it gets:

y = β·X + ε

Here ε is everything that shapes y besides X — every cause you didn't put in the regression. Ordinary least squares consistently estimates β only when X and ε are uncorrelated:

$\mathrm{Cov}(X, \varepsilon) = 0$

When that holds, X is exogenous and OLS recovers whatever β the equation claims. That is not the same as the slope being causal: β is a causal effect when the equation is a correctly specified structural model, and exogeneity is what lets a regression reach it. When it fails, X is endogenous: it carries some of the error's variation, OLS can't tell the two apart, and β̂ comes back as a blend of the effect you wanted and the confound you left out.

The part that surprises people: collecting more data doesn't rescue you. Sampling error shrinks as 1/√N, so a bigger dataset makes the estimate tighter — but endogeneity is a bias, not noise, and it doesn't shrink at all. You end up more and more confident about the wrong number.

Generates data from y = β₁X₁ + β₂X₂ + ε with β₁ = 1, then fits both the long regression (with X₂) and the misspecified short one (without). The violet tick is the closed-form prediction β₁ + β₂·Cov(X₁,X₂)/Var(X₁), which is β₁ + β₂·ρ here because both regressors are standardised; the orange bar is the actual β̂ from the short OLS. Press ×10 sample size and watch the bars get tighter while the orange one stays planted on the wrong value — biased and inconsistent, made tactile.

Endogeneity is one word for several everyday ways X and ε get entangled:

Omitted variable — a cause of y that also correlates with X sits in ε. Ability drives both schooling and wages; leave it out and schooling's coefficient absorbs it.
Reverse causality / simultaneity — y pushes back on X. Observed price and quantity are set jointly, where supply meets demand; regress one on the other and you trace out neither curve.
Measurement error — a noisily-measured X correlates with its own measurement error, which lives in ε. Under the classical assumption — additive noise, independent of the true value — that attenuates the estimated slope toward zero.
Self-selection — who ends up treated isn't random. People who take the drug differ from those who don't in ways the regression never sees.

This is the whole reason "we controlled for a bunch of stuff" isn't the same as "we measured a causal effect." The causal-inference toolkit — instrumental variables, regression discontinuity, panel fixed effects, the control function — is four answers to one question: where is there variation in X that the error term doesn't touch? IV borrows it from an instrument, RD takes it from around a cutoff, fixed effects find it within a unit, the control function conditions the rest away — each paying for identification with an assumption of its own.

For the long version — every source of endogeneity, worked examples, and the fixes end to end — see the Endogeneity note. The companion idea, and the assumption's flip side, is exogeneity.