Sharp Identification Regions in Models With Convex Moment Predictions


  • Arie Beresteanu,

    1. Dept. of Economics, University of Pittsburgh, 4510 Posvar Hall, Pittsburgh, PA 15260, U.S.A.;
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  • Ilya Molchanov,

    1. Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland;
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  • Francesca Molinari

    1. Dept. of Economics, Cornell University, 492 Uris Hall, Ithaca, NY 14853-7601, U.S.A.;
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    • This paper supersedes two earlier working papers (Beresteanu, Molchanov, and Molinari (2008, 2009)). We thank a co-editor and three anonymous reviewers for very useful suggestions that substantially improved the paper. We are grateful to H. Bar, T. Bar, L. Barseghyan, I. Canay, G. Chamberlain, P. Ellickson, Y. Feinberg, K. Hirano, T. Magnac, T. O'Donoghue, M. Peski, A. Rosen, X. Shi, J. Stoye, A. Sweeting, and, especially, L. Blume, D. Easley, and C. Manski for comments. We also thank seminar participants at many institutions and conferences where this paper was presented. We are grateful to M. Grant for advice on the use of the CVX software, which we used to run simulations in MatLab. Financial support from the U.S. NSF Grants SES-0617559 and SES-0922373 (Beresteanu) and SES-0617482 and SES-0922330 (Molinari); from the Swiss National Science Foundation Grants 200021-117606 and 200021-126503 (Molchanov); and from the Center for Analytic Economics at Cornell University is gratefully acknowledged. Ilya Molchanov gratefully acknowledges the hospitality of the Department of Economics at Cornell University in October 2006.


We provide a tractable characterization of the sharp identification region of the parameter vector θ in a broad class of incomplete econometric models. Models in this class have set-valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous-move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted ΘI, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in ΘI. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method.