A Simple, Short, but Never-Empty Confidence Interval for Partially Identified Parameters

TL;DR

This paper introduces a simple, reliable confidence interval for partially identified parameters that remains valid even when the identified set is empty, requiring minimal computation and tuning.

Contribution

It proposes a novel confidence interval method that is never empty, valid under model misspecification, and easy to compute, improving inference for partially identified parameters.

Findings

Interval is never empty, even when the identified set is empty.

Method achieves valid coverage without tuning parameters.

Simulations show excellent length and size control.

Abstract

This paper revisits the simple, but empirically salient, problem of inference on a real-valued parameter that is partially identified through upper and lower bounds with asymptotically normal estimators. A simple confidence interval is proposed and is shown to have the following properties: - It is never empty or awkwardly short, including when the sample analog of the identified set is empty. - It is valid for a well-defined pseudotrue parameter whether or not the model is well-specified. - It involves no tuning parameters and minimal computation. Computing the interval requires concentrating out one scalar nuisance parameter. In most cases, the practical result will be simple: To achieve 95% coverage, report the union of a simple 90% (!) confidence interval for the identified set and a standard 95% confidence interval for the pseudotrue parameter. For uncorrelated estimators -- notably if bounds are estimated from distinct subsamples -- and conventional coverage levels, validity of this simple procedure can be shown analytically. The case obtains in the motivating empirical application (de Quidt, Haushofer, and Roth, 2018), in which improvement over existing inference methods is demonstrated. More generally, simulations suggest that the novel confidence interval has excellent length and size control. This is partly because, in anticipation of never being empty, the interval can be made shorter than conventional ones in relevant regions of sample space.