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Testing for mediation effects is empirically important and theoretically interesting. It is empirically important in psychology, medicine, and marketing for instance, and leads to a theoretically interesting and long-standing problem in statistics that this paper solves. The no-mediation effect hypothesis, expressed as H₀:θ₁θ₂=0, defines a manifold that is non-regular in the origin where the size and power of standard tests, including the Wald, LR, and LM tests, are extremely low. The statistical challenge is to find a similar test with correct size independent of θ₁ and θ₂. We solve this problem by considering the critical region directly, rather than defining a test statistic and trying to correct its critical value. We prove the interesting theoretical result that no similar test exists, even if the asymptotic normal approximation is used. Despite this negative theoretical finding, we are able to construct a critical region that is all but similar and can be used in practice. We extend this result to three dimensions and show how it can be extended to higher dimensions for null hypotheses of the form H₀:θ₁θ₂⋯=0 with singularities in the origin.
