Bayesian inference in a class of partially identified models

Abstract

This paper develops a Bayesian approach to inference in a class of partially identified econometric models. Models in this class are characterized by a known mapping between a point identified reduced-form parameter μ and the identified set for a partially identified parameter θ. The approach maps posterior inference about μ to various posterior inference statements concerning the identified set for θ, without the specification of a prior for θ. Many posterior inference statements are considered, including the posterior probability that a particular parameter value (or a set of parameter values) is in the identified set. The approach applies also to functions of θ. The paper develops general results on large sample approximations, which illustrate how the posterior probabilities over the identified set are revised by the data, and establishes conditions under which the Bayesian credible sets also are valid frequentist confidence sets. The approach is computationally attractive even in high-dimensional models, in that the approach avoids an exhaustive search over the parameter space. The performance of the approach is illustrated via Monte Carlo experiments and an empirical application to a binary entry game involving airlines.

Inference in structural vector autoregressions when the identifying assumptions are not fully believed: Re-evaluating the role of monetary policy in economic fluctuations

“…In addition we have direct beliefs about the lagged structural coefficients as captured by the terms r i and V i in equations (34) and (35). This added information is used only for i = 3, the monetary policy rule, where the expectation is that the third element of b 3 should be close to ρ.…”

“…5 Moon et al (forthcoming) and Gafarov et al (2018) developed algorithms to estimate the identified set or its bounds using a frequentist approach, while Kline and Tamer (2016) discussed Bayesian posterior inference about the identified set in a general context. Giacomini and Kitagawa (2015) and Gafarov et al (2016) noted that the identified set could be interpreted and calculated as robust Bayesian posterior inference across the set of all possible Bayesian prior distributions.…”

“…This added information is used only for i = 3, the monetary policy rule, where the expectation is that the third element of b 3 should be close to ρ. This is implemented by taking e i in equation (35) to be column 3 of I 13 and r i in equation (34) equal to ρ. Our confidence in this prior information is captured by the value of V i which we set to 0.1, with a smaller value for V i representing greater confidence in the prior information.…”

MCMC confidence sets for identified sets

“…See the 2009 NBER working paper version of Moon and Schorfheide (2012), Kitagawa (2012), Kline and Tamer (2015), Liao and Simoni (2015) and the references therein. 7,8 Theoretically, all these papers consider separable models and use various renderings of a similar intuition.…”

Confidence intervals for projections of partially identified parameters

“…Other related papers that explicitly consider inference on projections include Andrews, Berry, and Jia (2004), Beresteanu and Molinari (2008), Bontemps, Magnac, and Maurin (2012), Chen, Tamer, andTorgovitsky (2011), Kaido (2012), Kitagawa (2012), Kline and Tamer (2015) and Wan (2013). However, some are Bayesian, as opposed to our frequentist approach, and none of them establish uniform validity of confidence sets.…”