Big Bias from Small Treatment Heterogeneity: When Controlling for Selection Backfires

Applied work routinely estimates treatment effects by linear regression of theoutcome on a treatment indicator and controls, with no interactions between them. We show that, even when selection is entirely on observables and the linear model is correctly specified, minimal unmodeled treatment effect heterogeneity can produce large biases for the average treatment effect on the treated. We decompose the asymptotic bias, control by control, into the product of an unobservable heterogeneity component and an observable amplifier, estimable without outcome data, that grows without bound as the controls better explain treatment. Covariates irrelevant for the outcome are not innocuous: they can generate large biases while their spuriously significant estimates make them self-validating. We propose breakdown diagnostics: the minimal correlated heterogeneity that overturns an estimate. In difference-in-differences, group-specific trends, a standard robustness check, absorb the dynamics of the treatment effect, producing a bias that accumulates with the length of the panel. Applications to 401(k) eligibility and unilateral divorce laws illustrate.