A Simple and Adaptive Confidence Interval when Nuisance Parameters Satisfy an Inequality

TL;DR

This paper introduces a simple, adaptive confidence interval method for models with a single inequality constraint, which adjusts based on the slackness of the inequality and is valid across scenarios.

Contribution

It proposes the inequality-imposed confidence interval (IICI), a new method that is simple, tuning-free, adaptive, and shorter than traditional intervals, applicable when a nuisance parameter satisfies an inequality.

Findings

The IICI reduces to the standard confidence interval when the inequality is slack.

The IICI becomes an equality-imposed interval when the inequality is tight.

The IICI is uniformly valid and never longer than the usual confidence interval.

Abstract

Inequalities may appear in many models. They can be as simple as assuming a parameter is nonnegative, possibly a regression coefficient or a treatment effect. This paper focuses on the case that there is only one inequality and proposes a confidence interval that is particularly attractive, called the inequality-imposed confidence interval (IICI). The IICI is simple. It does not require simulations or tuning parameters. The IICI is adaptive. It reduces to the usual confidence interval (calculated by adding and subtracting the standard error times the $1 - \alpha/2$ standard normal quantile) when the inequality is sufficiently slack. When the inequality is sufficiently violated, the IICI reduces to an equality-imposed confidence interval (the usual confidence interval for the submodel where the inequality holds with equality). Also, the IICI is uniformly valid and has (weakly) shorter length than the usual confidence interval; it is never longer. The first empirical application considers a linear regression when a coefficient is known to be nonpositive. A second empirical application considers an instrumental variables regression when the endogeneity of a regressor is known to be nonnegative.