  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 (Sobel) tests have very
poor power properties in 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 behavior everywhere. In fact, we first show that there exists a
test of this type that is (asymptotically) exact for certain test levels
$\alpha $, including the common choices $\alpha =.01,.05,.10.$ The critical
region of this exact test has some undesirable properties. We go on to show
that there is a very simple class of augmented LR critical regions which
provides tests that are nearly exact, and avoid the issues inherent in the
exact test. We suggest an optimal and coherent member of this class, provide
the table needed to implement the test and to report p-values if desired.
Simulation confirms validity with non-Gaussian disturbances, under
heteroskedasticity, and in a nonlinear (logit) model. A short application of
the method to an entrepreneurial attitudes study is included for illustration.
