Testing Sign Congruence Between Two Parameters

We test the null hypothesis that two parameters (μ1,μ2)$$ \left({\mu}_1,{\mu}_2\right) $$ have the same sign, assuming that (asymptotically) normal estimators (μ^1,μ^2)$$ \left({\hat{\mu}}_1,{\hat{\mu}}_2\right) $$ are available. Examples of this problem include the analysis of heterogeneous treatment effects, causal interpretation of reduced‐form estimands, meta‐studies, and mediation analysis. A number of tests were recently proposed. We recommend a test that is simple and rejects more often than many of these recent proposals. Like all other tests in the literature, it is conservative if the truth is near (0,0)$$ \left(0,0\right) $$ and therefore also biased. To clarify whether these features are avoidable, we also provide a test that is unbiased and has exact size control on the boundary of the null hypothesis, but which has counterintuitive properties and hence we do not recommend. We show how to improve p$$ p $$‐values in an existing paper from information contained in that paper's main text, and we revisit an empirical analysis of the effect of trade on voter behavior.