Generalized instrumental inequalities: testing the instrumental variable independence assumption

This paper proposes a new set of testable implications for the instrumental variable independence assumption for discrete treatment, but unrestricted outcome and instruments: generalized instrumental inequalities. When outcome and treatment are both binary, but instruments are unrestricted, we show that the generalized instrumental inequalities are necessary and sufficient to detect all observable violations of the instrumental variable independence assumption. To test the generalized instrumental inequalities, we propose an approach combining a sample splitting procedure and an inference method for intersection bounds. This idea allows one to easily implement the test using existing Stata packages. We apply our proposed strategy to assess the validity of the instrumental variable independence assumption for various instruments used in the returns to college literature.