Endogeneity

A regressor is endogenous when it is correlated with the error term: $\text{cov}(X_i, u_i) \neq 0$. Under endogeneity, OLS is both biased and inconsistent — more data does not heal the bias because OLS converges to the wrong target. The three classic sources are: omitted variables (a confounder drives both $X$ and $Y$), reverse causality ($Y$ causes $X$), and selection (who is treated is not random).

When to use

Flag endogeneity whenever your regressor of interest could plausibly be tangled up with anything in $u$. PS_2's education-fertility regression is the textbook case: cultural norms drive both schooling and family size (omitted variable), and having children early forces women to leave school (reverse causality). The standard fixes are Instrumental Variables, Fixed Effects for time-invariant confounders, Regression Discontinuity for sharp cutoffs, and randomised experiments when feasible.