🤖 AI Summary
This study addresses the limitations of conventional econometric inference, which often relies on pre-specified clustering, factor, or sparsity assumptions about dependence structures without data-driven validation. The authors model diverse dependence structures as covariance geometries in a Hilbert space and construct low-dimensional “dependence profiles” via projection-based similarity scores, enabling data-adaptive learning of dependence. They establish identifiability conditions for these profiles and derive finite-sample classification error bounds, revealing that indistinguishability arises from overlapping tangent spaces of the covariance geometries. Building on this insight, they propose an oracle-adaptive inference framework guided by dependence profiles, which yields consistent and asymptotically normal estimators of the underlying dependence structure and achieves inference performance equivalent to that of an oracle with prior knowledge of the optimal structure.
📝 Abstract
We develop a framework for learning dependence structures from empirical dependence operators. Rather than treating cluster, factor, and sparse dependence as maintained assumptions, we represent them as covariance geometries in a common Hilbert space and summarize dependence through a low-dimensional dependence profile based on projection similarity scores. We establish identification under a principal-angle separation condition, prove consistency and asymptotic normality of the estimated profile, and derive finite-sample classification error bounds. We further show that when covariance-geometry tangent spaces overlap, no statistical procedure can distinguish the geometries at first order, providing a formal characterization of ambiguous dependence structures. Projection-residual diagnostics assess absolute goodness-of-fit and detect misspecified covariance dictionaries. Finally, we establish oracle adaptivity of profile-guided inference: dependence profiles can be used to select dependence-robust procedures in a data-driven manner, yielding inference that is asymptotically equivalent to an infeasible oracle that knows the dominant covariance geometry in advance.