Robustness? Range Tests for Equality and Equivalence Across Specifications

Applied economists routinely compare estimates across specifications, observe that they are "similar," and conclude that their results are "robust.'" This common procedure makes an implicit inferential claim about the range of estimates, but usually does not account for their joint sampling distribution. I formalize informal practice with two bootstrap statistics. The minimum equivalence bound, $R^*_{1-\alpha}$, is the smallest tolerance within which the estimates can be judged equivalent. The range-based equality $p$-value, $p_R$, tests whether the estimates are statistically distinguishable. Together they distinguish failure to detect differences from affirmative evidence of agreement. Simulations show approximately correct size and coverage. Applications to five prominent papers validate some robustness claims while revealing cases in which apparent agreement reflects imprecision rather than stability. A survey of CEPR and NBER affiliates shows that expert judgments aligns with the framework in obvious cases but diverges in intermediate cases. I suggest that $R^*_{.95}$ and $p_R$ be reported whenever multiple specifications are presented as evidence of robustness.