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

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 models with one inequality and proposes an inequality-imposed confidence interval (IICI) that has particularly attractive features. The IICI is simple in that it does not require simulations or tuning parameters. Also, the IICI is adaptive to the slackness of the inequality, uniformly valid, and never longer than the usual confidence interval. We demonstrate the IICI in two empirical applications. The first empirical application considers a regression when a coefficient is known to be nonpositive. The second empirical application considers an instrumental variables regression when the endogeneity of a regressor is known to be nonnegative.