Instrumental Variables and the Sign of the Average Treatment Effect
57 Pages Posted: 14 Feb 2019
Date Written: January 16, 2018
This paper considers identiﬁcation and inference about the sign of the average eﬀect of a binary endogenous regressor (or treatment) on a binary outcome of interest when a binary instrument is available. In this setting, the average eﬀect of the endogenous regressor on the outcome is sometimes referred to as the average treatment eﬀect (ATE). We consider four diﬀerent sets of assumptions: instrument exogeneity, instrument exogeneity and monotonicity on the outcome equation, instrument exogeneity and monotonicity on the equation for the endogenous regressor, or instrument exogeneity and monotonicity on both the outcome equation and the equation for the endogenous regressor. For each of these sets of conditions, we characterize when (i) the distribution of the observed data is inconsistent with the assumptions and (ii) the distribution of the observed data is consistent with the assumptions and the sign of ATE is identiﬁed. A distinguishing feature of our results is that they are stated in terms of a reduced form parameter from the population regression of the outcome on the instrument. In particular, we ﬁnd that the reduced form parameter being far enough, but not too far, from zero, implies that the distribution of the observed data is consistent with our assumptions and the sign of ATE is identiﬁed, while the reduced form parameter being too far from zero implies that the distribution of the observed data is inconsistent with our assumptions. For each set of restrictions, we also develop methods for simultaneous inference about the consistency of the distribution of the observed data with our restrictions and the sign of the ATE when the distribution of the observed data is consistent with our restrictions. We show that our inference procedures are valid uniformly over a large class of possible distributions for the observed data that include distributions where the instrument is arbitrarily “weak.” A novel feature of the methodology is that the null hypotheses involve unions of moment inequalities.
Keywords: Average Treatment Eﬀect, Endogeneity, Instrumental Variables, Union of Moment In-equalities, Bootstrap, Uniform Validity, Multiple Testing, Familywise Error Rate, Gatekeeping
JEL Classification: C12, C31, C35, C36
Suggested Citation: Suggested Citation