Optimal inference in a class of regression models

TL;DR

This paper develops finite-sample and asymptotic optimal confidence intervals for linear functionals in regression models with known convex function classes, applicable to various econometric parameters.

Contribution

It derives sharp efficiency bounds and optimal CIs under convex function class assumptions, extending to uniform asymptotic results and regression discontinuity inference.

Findings

Finite-sample optimal confidence intervals derived.

Minimax CIs are nearly efficient for smooth functions.

Data-dependent tuning cannot improve CI tightness while maintaining coverage.

Abstract

We consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression. Our main assumption is that the regression function is known to lie in a convex function class, which covers most smoothness and/or shape assumptions used in econometrics. We derive finite-sample optimal CIs and sharp efficiency bounds under normal errors with known variance. We show that these results translate to uniform (over the function class) asymptotic results when the error distribution is not known. When the function class is centrosymmetric, these efficiency bounds imply that minimax CIs are close to efficient at smooth regression functions. This implies, in particular, that it is impossible to form CIs that are tighter using data-dependent tuning parameters, and maintain coverage over the whole function class. We specialize our results to inference on the regression discontinuity parameter, and illustrate them in simulations and an empirical application.