  In this paper we consider the problem of inference on a class of sets
describing a collection of admissible models as solutions to a single smooth
inequality. Classical and recent examples include, among others, the
Hansen-Jagannathan (HJ) sets of admissible stochastic discount factors,
Markowitz-Fama (MF) sets of mean-variances for asset portfolio returns, and the
set of structural elasticities in Chetty (2012)'s analysis of demand with
optimization frictions. We show that the econometric structure of the problem
allows us to construct convenient and powerful confidence regions based upon
the weighted likelihood ratio and weighted Wald (directed weighted Hausdorff)
statistics. The statistics we formulate differ (in part) from existing
statistics in that they enforce either exact or first order equivariance to
transformations of parameters, making them especially appealing in the target
applications. Moreover, the resulting inference procedures are also more
powerful than the structured projection methods, which rely upon building
confidence sets for the frontier-determining sufficient parameters (e.g.
frontier-spanning portfolios), and then projecting them to obtain confidence
sets for HJ sets or MF sets. Lastly, the framework we put forward is also
useful for analyzing intersection bounds, namely sets defined as solutions to
multiple smooth inequalities, since multiple inequalities can be conservatively
approximated by a single smooth inequality. We present two empirical examples
that show how the new econometric methods are able to generate sharp economic
conclusions.
