Testing the impossible: identifying exclusion restrictions

Method of moment estimators are generally obtained by adopting orthogonality conditions, in which particular functions in terms of the observed data and unknown parameters are supposed to have zero expectation. For regression models this implies exploiting presumed uncorrelatedness of the model disturbances and identifying instrumental variables. Here, utilizing non-orthogonality conditions is examined for linear cross-section multiple regression models. Employing flexible bounds on the correlations between disturbances and regressors one avoids: (i) adoption of often incredible and unverifiable strictly zero correlation assumptions, and (ii) imprecise inference due to possibly weak or invalid instruments. The asymptotic validity of the suggested alternative form of inference is proved and its finite sample accuracy is demonstrated by simulation. It enables to produce inference on coefficient values that within constraints is endogeneity robust. Also a sensitivity analysis of standard least-squares or instrument-based inference is possible, and even a test of the in the standard approach unavoidable though "non-testable" exclusion restrictions regarding external instruments. The practical relevance is illustrated in a few applications borrowed from the textbook literature.