Necessary and Probably Sufficient Test for Finding Valid Instrumental Variables

Can instrumental variables be found from data? While instrumental variable (IV) methods are widely used to identify causal effect, testing their validity from observed data remains a challenge. This is because validity of an IV depends on two assumptions, exclusion and as-if-random, that are largely believed to be untestable from data. In this paper, we show that under certain conditions, testing for instrumental variables is possible. We build upon prior work on necessary tests to derive a test that characterizes the odds of being a valid instrument, thus yielding the name "necessary and probably sufficient". The test works by defining the class of invalid-IV and valid-IV causal models as Bayesian generative models and comparing their marginal likelihood based on observed data. When all variables are discrete, we also provide a method to efficiently compute these marginal likelihoods. We evaluate the test on an extensive set of simulations for binary data, inspired by an open problem for IV testing proposed in past work. We find that the test is most powerful when an instrument follows monotonicity---effect on treatment is either non-decreasing or non-increasing---and has moderate-to-weak strength; incidentally, such instruments are commonly used in observational studies. Among as-if-random and exclusion, it detects exclusion violations with higher power. Applying the test to IVs from two seminal studies on instrumental variables and five recent studies from the American Economic Review shows that many of the instruments may be flawed, at least when all variables are discretized. The proposed test opens the possibility of data-driven validation and search for instrumental variables.