Intersection Bounds: Estimation and Inference


  • Victor Chernozhukov,

    1. Dept. of Economics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.;;
    Search for more papers by this author
  • Sokbae Lee,

    1. Dept. of Economics, Seoul National University, Seoul 151-742, Korea;;
    Search for more papers by this author
  • Adam M. Rosen

    1. Centre for Microdata Methods and Practice, Institute for Fiscal Studies, Dept. of Economics, University College London, Gower Street, London WC1E 6BT, U.K.;;
    Search for more papers by this author
    • We are especially grateful to D. Chetverikov, K. Kato, Y. Luo, A. Santos, five anonymous referees, and a co-editor for making several extremely useful suggestions that have led to substantial improvements. We thank T. Armstrong, R. Blundell, A. Chesher, F. Molinari, W. Newey, C. Redmond, N. Roys, S. Stouli, and J. Stoye for detailed discussion and suggestions, and participants at numerous seminars and conferences for their comments. This paper is a revised version of “Inference on Intersection Bounds,” which initially was presented and circulated at the University of Virginia and the Harvard/MIT econometrics seminars in December 2007, and presented at the March 2008 CEMMAP/Northwestern Conference on Inference in Partially Identified Models With Applications. We gratefully acknowledge financial support from the National Science Foundation, the Economic and Social Research Council (RES-589-28-0001, RES-000-22-2761), and the European Research Council (ERC-2009-StG-240910-ROMETA).


We develop a practical and novel method for inference on intersection bounds, namely bounds defined by either the infimum or supremum of a parametric or nonparametric function, or, equivalently, the value of a linear programming problem with a potentially infinite constraint set. We show that many bounds characterizations in econometrics, for instance bounds on parameters under conditional moment inequalities, can be formulated as intersection bounds. Our approach is especially convenient for models comprised of a continuum of inequalities that are separable in parameters, and also applies to models with inequalities that are nonseparable in parameters. Since analog estimators for intersection bounds can be severely biased in finite samples, routinely underestimating the size of the identified set, we also offer a median-bias-corrected estimator of such bounds as a by-product of our inferential procedures. We develop theory for large sample inference based on the strong approximation of a sequence of series or kernel-based empirical processes by a sequence of “penultimate” Gaussian processes. These penultimate processes are generally not weakly convergent, and thus are non-Donsker. Our theoretical results establish that we can nonetheless perform asymptotically valid inference based on these processes. Our construction also provides new adaptive inequality/moment selection methods. We provide conditions for the use of nonparametric kernel and series estimators, including a novel result that establishes strong approximation for any general series estimator admitting linearization, which may be of independent interest.