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 in a linear regression, and inference on the regression discontinuity parameter, and illustrate them in simulations and an empirical application.
Armstrong, Timothy B. and Kolesár, Michal, "Optimal Inference in a Class of Regression Models" (2016). Cowles Foundation Discussion Papers. 2494.