Short and Simple Confidence Intervals when the Directions of Some Effects are Known

TL;DR

This paper introduces adaptive confidence intervals for linear regression parameters that leverage known signs of nuisance parameters, resulting in shorter intervals near zero effects while maintaining validity.

Contribution

It proposes a simple, practical method for constructing adaptive confidence intervals that improve length properties when some nuisance parameters have known signs.

Findings

Intervals are shorter near zero effects.

Method is easy to compute and applicable in practice.

Validated through empirical and simulation studies.

Abstract

We provide adaptive confidence intervals on a parameter of interest in the presence of nuisance parameters when some of the nuisance parameters have known signs. The confidence intervals are adaptive in the sense that they tend to be short at and near the points where the nuisance parameters are equal to zero. We focus our results primarily on the practical problem of inference on a coefficient of interest in the linear regression model when it is unclear whether or not it is necessary to include a subset of control variables whose partial effects on the dependent variable have known directions (signs). Our confidence intervals are trivial to compute and can provide significant length reductions relative to standard confidence intervals in cases for which the control variables do not have large effects. At the same time, they entail minimal length increases at any parameter values. We prove that our confidence intervals are asymptotically valid uniformly over the parameter space and illustrate their length properties in an empirical application to a factorial design field experiment and a Monte Carlo study calibrated to the empirical application.