Bayesian and frequentist tests of sign equality and other nonlinear inequalities

Testing whether two parameters have the same sign is a nonstandard problem due to the non-convex shape of the parameter subspace satisfying the composite null hypothesis, which is a nonlinear inequality constraint. We describe a simple example where the ordering of likelihood ratio (LR), Wald, and Bayesian sign equality tests reverses the “usual” ordering: the Wald rejection region is a subset of LR’s, as is the Bayesian rejection region (either asymptotically or with an uninformative prior). Under general conditions, we show that non-convexity of the null hypothesis subspace is a necessary but not sufficient condition for this asymptotic frequentist/Bayesian ordering. Since linear inequalities only generate convex regions, a corollary is that frequentist tests are more conservative than Bayesian tests in that setting. We also examine a nearly similar-on-the-boundary, unbiased test of sign equality. Rather than claim moral superiority of one statistical framework or test, we wish to clarify the regrettably ineluctable tradeoffs. JEL classification: C11, C12