Semi-Instrumental Variables: A Test for Instrument Admissibility
This addresses a gap in causal inference for researchers and practitioners by offering a method to test instrument admissibility, though it is incremental as it builds on existing instrumental variable theory.
The paper tackles the problem of validating instrumental variables in causal inference by introducing semi-instruments and providing statistical tests to check if a variable is semi-instrumental or if two semi-instruments are instrumental, enabling automated or expert-validated instrument selection.
In a causal graphical model, an instrument for a variable X and its effect Y is a random variable that is a cause of X and independent of all the causes of Y except X. (Pearl (1995), Spirtes et al (2000)). Instrumental variables can be used to estimate how the distribution of an effect will respond to a manipulation of its causes, even in the presence of unmeasured common causes (confounders). In typical instrumental variable estimation, instruments are chosen based on domain knowledge. There is currently no statistical test for validating a variable as an instrument. In this paper, we introduce the concept of semi-instrument, which generalizes the concept of instrument. We show that in the framework of additive models, under certain conditions, we can test whether a variable is semi-instrumental. Moreover, adding some distribution assumptions, we can test whether two semi-instruments are instrumental. We give algorithms to estimate the p-value that a random variable is semi-instrumental, and the p-value that two semi-instruments are both instrumental. These algorithms can be used to test the experts' choice of instruments, or to identify instruments automatically.