Can we have it all? Non-asymptotically valid and asymptotically exact confidence intervals for expectations and linear regressions

TL;DR

This paper develops confidence intervals that are both valid in finite samples and exact asymptotically for expectations and linear regressions, bridging the gap between large-sample and small-sample inference.

Contribution

It introduces a generic condition linking uniform asymptotic exactness to non-asymptotic validity, and constructs closed-form NAVAE confidence intervals under minimal moment conditions.

Findings

Constructed NAVAE confidence intervals for expectations and OLS coefficients.

Enlarged CLT-based CIs ensure non-asymptotic guarantees under heteroskedasticity.

Simulation study demonstrates the approach's potential and limitations.

Abstract

We contribute to bridging the gap between large- and finite-sample inference by studying confidence sets (CSs) that are both non-asymptotically valid and asymptotically exact uniformly (NAVAE) over semi-parametric statistical models. NAVAE CSs are not easily obtained; for instance, we show they do not exist over the set of Bernoulli distributions. We first derive a generic sufficient condition: NAVAE CSs are available as soon as uniform asymptotically exact CSs are. Second, building on that connection, we construct closed-form NAVAE confidence intervals (CIs) in two standard settings -- scalar expectations and linear combinations of OLS coefficients -- under moment conditions only. For expectations, our sole requirement is a bounded kurtosis. In the OLS case, our moment constraints accommodate heteroskedasticity and weak exogeneity of the regressors. Under those conditions, we enlarge the Central Limit Theorem-based CIs, which are asymptotically exact, to ensure non-asymptotic guarantees. Those modifications vanish asymptotically so that our CIs coincide with the classical ones in the limit. We illustrate the potential and limitations of our approach through a simulation study.