This paper is concerned with the refined asymptotic properties of several tests for the admissibility of a subset of (overidentifying) instrumental variables. It derives maximum likelihood (ML) and linearized ML tests and calculates size corrections to the order 1/T. The local power function of the size-corrected tests is the same to the order 1/T, irrespective of the form of the test statistic or the limited information estimator used in its computation. Finally, it compares these tests with two previously proposed tests.