Testing the IV Zero-Covariance Assumption and Identification with an Invalid Instrument

Abstract

This paper discusses the empirical content of the exogeneity (zero-covariance) assumption, the key assumption for identification in the linear IV model. Contrary to the general belief, we show that whenever the outcome is bounded, the exogeneity assumption imposes some testable restrictions on the observables, namely the linear instrumental inequalities (LII). The LII can therefore be used to test the instrument validity. The test can be easily implemented using existing inferential methods for testing multiple inequalities. Whenever, the causal effect parameter is random, an additional critical condition that trivially holds in the linear IV model is required for the IV estimand to identify the average treatment effect (ATE). We show that when this critical condition holds while the exogeneity assumption does not, the magnitude of the violations of the LII allows us to derive (informative) bounds on the ATE. We also derive testable implications of this critical condition which when violated implies that the usual IV estimand cannot be interpreted as the ATE or as the local ATE. We illustrate our results by testing the validity of various instruments present in the literature.