Inference on sets in finance

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 the Hansen-Jagannathan sets of admissible stochastic discount factors, Markowitz-Fama mean-variance sets for asset portfolio returns, and the set of structural elasticities in Chetty's (2012) analysis of demand with optimization frictions. The econometric structure of the problem allows us to construct convenient and powerful confidence regions based on the weighted likelihood ratio and weighted Wald statistics. Our statistics differ 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. We show that the resulting inference procedures are more powerful than the structured projection methods. Last, our framework 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 showing how the new econometric methods are able to generate sharp economic conclusions.