Confidence sets based on inverting Anderson–Rubin tests

Economists are often interested in the coefficient of a single endogenous explanatory variable in a linear simultaneous‐equations model. One way to obtain a confidence set for this coefficient is to invert the Anderson–Rubin (AR) test. The AR confidence sets that result have correct coverage under classical assumptions. However, AR confidence sets also have many undesirable properties. It is well known that they can be unbounded when the instruments are weak, as is true of any test with correct coverage. However, even when they are bounded, their length may be very misleading, and their coverage conditional on quantities that the investigator can observe (notably, the Sargan statistic for overidentifying restrictions) can be far from correct. A similar property manifests itself, for similar reasons, when a confidence set for a single parameter is based on inverting an F‐test for two or more parameters.