Dual Validity of Robust Standard Errors under Complex Dependence

Regression is a workhorse for analyzing data from randomized experiments, adaptive trials, and related designs, where valid inference typically relies on {\it robust standard errors}. These adjustments are usually motivated as correcting dependence in unobserved errors, dependence driven by shocks, interference, or hidden common causes that is rarely observable or reliably modeled in practice. By contrast, the dependence structure of the regressor of interest, most notably treatment assignment, is often known, testable, and engineered by design. Motivated by this asymmetry, we develop a general asymptotic framework for regression inference under unknown and potentially unstructured dependence and prove a {\it dual validity principle}: various robust $t$-statistics are all asymptotically valid if the correlation structure encoded by the chosen standard-error adjustment is correct for either the regressor of interest or the unobserved errors, even when the other component is misspecified. Establishing this predictor-error symmetry is mathematically nontrivial and, to our knowledge, has not been formalized in the classical robust-inference literature despite its centrality to empirical practice. For example, cluster-robust inference remains valid when either errors are clustered or assignment is clustered.

The result reframes robust standard errors as design-aligned devices: when error dependence is uncertain, robust adjustments should be chosen to match the known assignment correlation, rather than as catch-all corrections for unspecified error structure.