Estimation and Inference for Set-identi…ed Parameters Using Posterior Lower Probability Toru Kitagawa CeMMAP and Department of Economics, UCL First Draft: September 2010 This Draft: March, 2012 Abstract In inference about set-identi…ed parameters, it is known that the Bayesian probability state- ments about the unknown parameters do not coincide, even asymptotically, with the frequen- tist’scon…dence statements. This paper aims to smooth out this disagreement from a multiple prior robust Bayes perspective. We show that there exist a class of prior distributions (am- biguous belief), with which the posterior probability statements drawn via the lower envelope (lower probability) of the posterior class asymptotically agree with the frequentist con…dence statements for the identi…ed set. With such class of priors, we analyze statistical decision problems including point estimation of the set-identi…ed parameter by applying the gamma- minimax criterion. From a subjective robust Bayes point of view, the class of priors can serve as benchmark ambiguous belief, by introspectively assessing which one can judge whether the frequentist inferential output for the set-identi…ed model is an appealing approximation of his posterior ambiguous belief. Keywords: Partial Identi…cation, Bayesian Robustness, Belief Function, Imprecise Probability, Gamma-minimax, Random Set. JEL Classi…cation: C12, C15, C21. Email: [email protected]. I thank Andrew Chesher, Siddhartha Chib, Jean-Pierre Florens, Charles Manski, Ulrich Müller, Andriy Norets, Adam Rosen, Kevin Song, and Elie Tamer for valuable discussions and comments. I also thank the seminar participants at Academia Sinica Taiwan, Cowles Summer Conference 2011, EC2 Conference 2010, MEG 2010, RES Annual Conference 2011, Simon Fraser University, and the University of British Columbia for helpful comments. All remaining errors are mine. Financial support from the ESRC through the ESRC Centre for Microdata Methods and Practice (CEMMAP) (grant number RES-589-28-0001) is gratefully acknowledged. 1 1 Introduction In the usual situations of inferring identi…ed parameters, the Bayesian probability statements about the unknown parameters are similar, at least asymptotically, to the frequentist con…dence state- ments for the true value of the parameters. In partial identi…cation analysis initiated by Manski (1989, 1990, 2003, 2008), it is known that such asymptotic harmony between the two inference paradigms breaks down (Moon and Schorfheide (2011)). The Bayesian interval estimates for the set-identi…ed parameter are shorter, even asymptotically, than the frequentist ones, and they as- ymptotically lie strictly inside of the frequentist’scon…dence intervals. Frequentists might interpret this phenomenon as that the Bayesian’s over-con…dence in their inferential statement is …ctitious. Bayesians, on the other hand, might reply that the frequentist’scon…dence statements su¤er from an extreme conservatism that lacks any posterior probability justi…cation. The main aim of this paper is to smooth out such disagreement between the two schools of statistical inference from a perspective of robust Bayes inference: a third paradigm of statistical inference, in which we can incorporate researcher’spartial prior knowledge into posterior inference. Among various robust Bayes approaches, this paper in particular focuses on a multiple prior Bayes analysis, where the partial prior knowledge or the robustness concern against prior misspeci…cation is modeled by a class of priors (ambiguous belief). The Bayes rule is applied to each prior to form the class of posteriors. Posterior inference procedures considered in this paper operate on the thus-constructed class of posteriors by focusing on its lower and upper envelopes, the so-called posterior lower and upper probabilities. Along this robust Bayes approach, this paper considers the following questions. First, are there any classes of priors, with which the probabilistic statements made via the posterior lower probability can approximate the frequentist’s con…dence statements for set-identi…ed parameters? The answer turns out to be yes. When the parameters are not identi…ed, a prior distribution of the model parameters is decomposed into two components: one that can be updated by data (revisable prior knowledge) and one that can never be updated by data (unrevisable prior knowledge). We show that, if a prior class is designed in such way that it shares a single prior distribution for revisable prior knowledge, but allows for arbitrary prior distributions for the unrevisable prior knowledge, then, under certain regularity conditions, the posterior interval estimates constructed based upon the posterior lower probability asymptotically attains the correct frequentist coverage probability for the identi…ed set. 2 The second question we examine is, with such class of priors, is it possible to formulate and solve statistical decision problems including point estimation of the set-identi…ed parameter? We approach this question by adapting the gamma-minimax decision analysis, which can be seen as a minimax analysis in the multiple prior setup. We demonstrate that the proposed prior class leads us to an analytically tractable and numerically solvable formulation of the gamma-minimax decision problem, provided that the identi…ed set for the parameter of interest can be computed for each value of the identi…ed parameters We believe our analyses are insightful not only from the "objective" Bayesian point of view but also from the subjective robust Bayesian point of view, because the classes of priors that can asymptotically match with the frequentist’s inference can serve as benchmark ambiguous belief. That is, by subjectively assessing if the ambiguous belief represented by the benchmark prior class appears reasonable or not, one can judge if the frequentist’s inferential output is too conservative or not in view of his posterior ambiguous belief. On this issue, we argue through an example that some priors in the benchmark prior class appear less appealing than some others, so that subjective robust Bayesians would consider that the frequentist inference statements for identi…ed set are too uninformative to interpret it as an approximation of their posterior ambiguous belief. 1.1 Literature Review Estimation and inference in partial identi…ed models have been a growing area of research in econometrics. From the frequentist perspective, Horowitz and Manski (2000) construct con…dence sets for the interval identi…ed set. Imbens and Manski (2004) propose uniformly asymptotically valid con…dence sets for an intervally-identi…ed parameter. Stoye (2009) extends their analysis to a di¤erent set of conditions. Chernozhukov, Hong, and Tamer (2007) develop a way to construct asymptotically valid con…dence sets for an identi…ed set based on the criterion function approach, which can be applied to the wide range of partially identi…ed models including moment inequality models. Related to the criterion function approach, the literatures on construction of con…dence sets by inverting test statistics include, Andrews and Guggenberger (2009), Andrews and Soares (2010), and Romano and Shaikh (2010), to list a few. From the Bayesian perspective, Neath and Samaniego (1997), Poirier (1998), and Gustafson (2009, 2010) analyze how the Bayesian updating operates when a model lacks identi…ability. Liao and Jiang (2010) conduct Bayesian inference for moment inequality models based on the pseudo- 3 likelihood. Moon and Schorfheide (2011) compares asymptotic properties of frequentist and Bayesian inference for set-identi…ed models, and our robust Bayes analysis is motivated by their important …nding on the asymptotic disagreement between them. The lower and upper probabilities originate from Dempster (1966, 1967a, 1967b, 1968) in his …ducial argument of drawing posterior inferences without specifying a prior distribution. Dem- spter’sanalyses have given a large impact to the …eld of statistics, arti…cial intelligence, and decision theory, such as the belief function analysis of Shafer (1976, 1982), the imprecise probability analysis of Walley (1991). The lower and upper probabilities have been playing the major role also in the robust Bayes analysis for measuring the global sensitivity of the posterior (Berger (1984), Berger and Berliner (1986)) and for characterizing a class of posteriors (DeRobertis and Hartigan (1981), Wasserman (1989, 1990), and Wasserman and Kadane (1990)). In econometrics, the pioneering work using multiple priors was carried out by Chamberlain and Leamer (1976), and Leamer (1982), who obtained the bounds for the posterior mean of the regression coe¢ cients when a prior varies over a certain class. These previous studies do not explicitly considers non-identi…ed models, while this paper focuses on non-identi…ed models, and aims to clarify a link between an early idea of the lower and upper probabilities and a recent issue on inference in set-identi…ed models. The posterior lower probability that we shall obtain with our speci…cation of prior class turns out to be an in…nite-order capacity, or equivalently, a containment functional in the random set theory (see, e.g., Molchanov (2005)). Beresteanu and Molinari (2008) and Bereseanu, Molchanov, and Molinari (2012) show the usefulness and wide applicability of random set theory to partially identi…ed models by viewing an observation as a random set and the estimand as its Aumann expec- tation. Galichon and Henry (2006, 2009) proposed