The Modified Instrumental Variable (MIV) for Endogenous Instrumental Variables

Abstract This article introduces an estimator aimed at reducing the inconsistency of instrumental variable estimates when the instrument is not fully exogenous. The degree of improvement depends on how much the instrument deviates from exogeneity: if the exogenous component outweighs the endogenous one, the proposed estimator can fully correct the inconsistency. Importantly, when the instrument is only weakly exogenous, the Modified Instrumental Variables (MIV) estimator does not alter the estimates—an outcome that provides a useful diagnostic for assessing whether the instrument was truly exogenous. To demonstrate the practical utility of the method, we apply it in two empirical settings: (i) a Mincer earnings equation using quarter of birth as an instrument for years of education, and (ii) an earnings regression using Vietnam War draft lottery eligibility as an instrument for military service. The results indicate that the quarter of birth requires modification due to endogeneity, while the draft lottery behaves as an exogenous instrument, validating its use in causal inference.