Confidence intervals for linear unbiased estimators under constrained dependence

We propose an approach for conducting inference for linear unbiased estimators applied to dependent outcomes given constraints on their independence relations, in the form of a dependency graph. We establish the consistency of an oracle variance estimator when a dependency graph is known, along with an associated central limit theorem. We derive an integer linear program for finding an upper bound for the estimated variance when a dependency graph is unknown, but topological or degree-based constraints are available on one such graph. We develop alternative bounds, including a closed-form bound, under an additional homoskedasticity assumption. We establish a basis for Wald-type confidence intervals that are guaranteed to have asymptotically conservative coverage.