Heterogeneous Effects and the Interpretation of OLS Coefficient Movements

Applied researchers commonly interpret coefficient movements across OLS and difference-in-differences specifications with varying controls as evidence of bias reduction or improved identification. This note shows that such interpretations generally fail under heterogeneous treatment effects. In this case, each specification estimates a differently weighted average treatment effect, where the weights depend on the set of included controls. Consequently, adding or removing controls can change the estimand even when the controls are irrelevant for the outcome conditional on treatment. The note further derives an augmented version of the canonical omitted variables bias formula that remains valid under heterogeneity by explicitly accounting for induced changes in weighting.