Semi-Instrumental Variables: A Test for Instrument Admissibility

In a causal graphical model, an instrument for a variable X and its effect Y is a ran­ dom variable that is a cause of X and in­ dependent of all the causes of Y except X (Pearl 1995). For continuous variables, in­ strumental 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 (con­ founders). 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 cer­ tain conditions, we can test whether a vari­ able is semi-instrumental. Moreover, adding some distribution assumptions, we can test whether two semi-instruments are instrumen­ tal. 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 ex­ perts' choice of instruments, or to identify in­ struments automatically.