Uncertain identification

Uncertainty about the choice of identifying assumptions is common in causal studies, but is often ignored in empirical practice. This paper considers uncertainty over models that impose different identifying assumptions, which, in general, leads to a mix of point- and set-identified models. We propose performing inference in the presence of such uncertainty by generalizing Bayesian model averaging. The method considers multiple posteriors for the set-identified models and combines them with a single posterior for models that are either point-identified or that impose non-dogmatic assumptions. The output is a set of posteriors (post-averaging ambiguous belief) that are mixtures of the single posterior and any element of the class of multiple posteriors, with weights equal to the posterior model probabilities. We suggest reporting the range of posterior means and the associated credible region in practice, and provide a simple algorithm to compute them. We establish that the prior model probabilities are updated when the models are "distinguishable" and/or they specify different priors for reduced-form parameters, and characterize the asymptotic behavior of the posterior model probabilities. The method provides a formal framework for conducting sensitivity analysis of empirical findings to the choice of identifying assumptions. In a standard monetary model, for example, we show that, in order to support a negative response of output to a contractionary monetary policy shock, one would need to attach a prior probability greater than 0.32 to the validity of the assumption that prices do not react contemporaneously to such a shock. The method is general and allows for dogmatic and non-dogmatic identifying assumptions, multiple point-identified models, multiple set-identified models, and nested or non-nested models.