Semi-Instrumental Variables: A Test for Instrument Admissibility

Abstract

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). For continuous variables, 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 continuous variable as an instrument. In this paper, we introduce the concept of semi-instrument, which generalizes the concept of instrument: each instrument is a semi-instrument, but the converse does not hold. 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 test whether a variable is semi-instrumental, and whether two semi-instruments are both instrumental. These algorithms can be used to test the experts' choice of instruments, or to identify instruments automatically.