18.2 When the assumptions fail

The classical assumptions of Section 6 are linearity (A1), \(E(U \mid X) = 0\), a spherical disturbance \(\Var (U \mid X) = \sigma ^2I\), full rank, and, for exact inference, normality. Their failures fall into two classes, and the distinction is the single most useful idea in applied econometrics. If \(E(U \mid X) \neq 0\), least squares is biased and inconsistent, and no correction of the standard errors helps: a different estimator or a different specification is needed. If only the variance assumption fails, least squares remains unbiased and consistent, and the problem is efficiency and the validity of the standard errors, which robust methods repair.

Two further lessons recur. First, a diagnostic test that rejects says that something is wrong, not what: a significant Durbin–Watson statistic may mean omitted dynamics rather than AR(1) disturbances (Theorem 11.4.2), a significant White test may mean a wrong functional form, and a significant Sargan or \(J\) statistic may mean a wrong model rather than a wrong instrument. Second, every remedy has a price: IV trades bias for variance, GLS and SUR trade robustness for efficiency, and fixed effects trades the between variation for consistency. The applied econometrician’s job is to know which trade is being made.