Simple misspecification adaptive inference for interval identified parameters

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. In general, computing the interval requires concentrating out one scalar nuisance parameter. For uncorrelated estimators of bounds --notably if bounds are estimated from distinct subsamples-- and conventional coverage levels, this step can be skipped. The proposed $95\%$ confidence interval then simplifies to the union of a simple $90\%$ (!) confidence interval for the partially identified parameter and an equally simple $95\%$ confidence interval for a point-identified pseudotrue parameter. This case obtains in the motivating empirical application, in which improvement over existing inference methods is demonstrated. More generally, simulations suggest excellent length and size control properties.