Optimal Inference in a Class of Regression Models

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 substantively 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.

Simple and honest confidence intervals in nonparametric regression

“…Remark Theorem does not address whether further efficiency improvements are possible by using estimators that do not fall into the class , or by using variable length CIs. However, it follows from Donoho () and Armstrong and Kolesár () that, in typical settings where our results hold, little further improvement is possible. In particular, these papers give efficiency bounds that, applied to our setting, yield asymptotic lower bounds for , where is the optimal estimator or CI among all procedures (for CIs, this includes variable length CIs, with performance measured in terms of expected length), and and are the optimal bandwidth and kernel.…”

“…Our asymptotic approach allows for simplifications that deliver our main relative efficiency results. These efficiency results are different from and complementary to the asymptotic form of the efficiency bounds given in Donoho () and Armstrong and Kolesár (): whereas we consider relative efficiency of estimators and fixed‐length CIs based on different kernels and bandwidths, Donoho () and Armstrong and Kolesár () bound the scope for efficiency gains from CIs that do not fall into this class. Donoho () and Armstrong and Kolesár () found that the scope for further improvement is small, which motivates our focus on this class of estimators and CIs.…”

“…When the parameter space places a bound M on a derivative of f , our CIs require this bound to be specified explicitly. While this may appear to be a disadvantage of our particular approach, due to impossibility results of Low (), Cai and Low (), and Armstrong and Kolesár (), this cannot be avoided, regardless of how one forms the CI, without making further restrictions on the function f . In particular, these papers show that, without additional assumptions on the parameter space, one cannot use a data‐driven method to estimate M and maintain coverage over the whole parameter space—any other method that appears to avoid making this choice must do so implicitly .…”

“…Regardless of how one chooses M , the fixed‐length CIs we propose are more efficient than undersmoothed or bias‐corrected CIs that use the same (implicit or explicit) choice of M . In fact, it follows from the calculations in Donoho () and Armstrong and Kolesár () that our CIs, when constructed using a length‐optimal or MSE‐optimal bandwidth, are highly efficient among all honest CIs: no other approach to inference can substantively improve on their length, while still maintaining coverage.…”

“…The CIs we consider in this paper are applications of the fixed‐length CIs proposed in the context of inference on linear functionals in Gaussian nonparametric regression by Donoho (), which have also been studied recently in Armstrong and Kolesár (), and in contemporaneous and subsequent work by Kolesár and Rothe () and Imbens and Wager (). In contrast to the finite‐sample approach taken in these papers, we focus on asymptotic results, and we also allow to be nonlinear.…”