Testing for a mediation effect is important in many disciplines, but is made difficult – even asymptotically – by the inﬂuence of nuisance parameters. Classical tests such as likelihood ratio (LR) and Wald tests have very poor size and power properties in some parts of the parameter space, and many attempts have been made to produce improved tests, with limited success. In this paper we show that augmenting the critical region of the LR test can produce a test with much improved behaviour everywhere. In fact, we ﬁrst show that there exists a test of this type that is (asymptotically) exact for certain test sizes α, including the common choices α = .01, .05, .10. This is evidently an important result, but we also observe that the critical region of this exact test has some undesirable properties. Thus, we then go on to show that there is a very simple class of augmented LR critical regions which provides tests that, while not exact, are very nearly so, and which avoid the issues inherent in the exact test. We suggest an optimal member of this class, and provide the tables needed to implement it. Although motivated by a simple two-equation linear model, the results apply to any model structure that reduces to the same testing problem asymptotically. A short application of the method to an entrepreneurial attitudes study is included for illustration.
10 January 2022
Working Paper (CWP01/22)