Learning Dependence Structures for Econometric Inference: Identification, Ambiguity, and Adaptive Inference

Econometric inference usually conditions on a dependence structure chosen in advance, even though the data may support clustering, latent factors, sparse interactions, or mixtures of these mechanisms. This paper studies the prior problem of learning the dependence structure that is relevant for inference. We represent candidate structures as covariance geometries in a common Hilbert space and project an estimable dependence operator onto them. The resulting geometric dependence profile is a low-dimensional diagnostic of their relative empirical support; an off-diagonal companion profile isolates cross-sectional dependence and drives procedure selection. We establish well-definedness, consistency, asymptotic normality, and finite-sample classification bounds under local projection regularity and geometric separation, and show that tangent-space overlap creates a first-order impossibility region in which competing geometries cannot be reliably distinguished. Formulating inference-procedure choice as a statistical decision problem, we prove that when one off-diagonal geometry is uniquely separated and profile rankings are compatible with inferential loss, profile-guided inference is asymptotically equivalent to an infeasible oracle and has vanishing regret. The framework thus links dependence diagnostics, learnability, ambiguity, and adaptive inference in a single data-to-decision procedure.