Approximate Bayesian Computation for Partially Identified Models

Partial identification is a prominent feature of several economic models. Such prevalence has spurred a large literature on valid set estimation under partial identification from a frequentist viewpoint. From the Bayesian perspective, it is well known that, under partial identification, the asymptotic validity of Bayesian credible sets in conducting frequentist inference, which is ensured by several Bernstein von-Mises theorems available in the literature, breaks down. Existing solutions to this problem require either knowledge of the map between the distribution of the data and the identified set -- which is generally unavailable in more complex models --, or modifications to the methodology that difficult the Bayesian interpretability of the proposed solution. In this paper, I show how one can leverage Approximate Bayesian Computation, a Bayesian methodology designed for settings where evaluation of the model likelihood is unfeasible, to reestablish the asymptotic validity of Bayesian credible sets in conducting frequentist inference, whilst preserving the core interpretation of the Bayesian approach and dispensing with knowledge of the map between data and identified set. Specifically, I show in a simple, yet encompassing, setting how, by calibrating the main tuning parameter of the ABC methodology, one could hope to achieve asymptotic frequentist coverage. Based on my findings, I then propose a semiautomatic algorithm for selecting this parameter and constructing valid confidence sets. This is a work in progress. In future versions, I intend to present further theoretical results, Monte Carlo simulations and an empirical application on the Economics of Networks.