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Unformatted text preview: Intersection bounds: estimation and inference Victor Chernozhukov Sokbae Lee Adam M. Rosen The Institute for Fiscal Studies Department of Economics, UCL cemmap working paper CWP19/09 INTERSECTION BOUNDS: ESTIMATION AND INFERENCE VICTOR CHERNOZHUKOV, SOKBAE LEE, ADAM M. ROSEN Abstract. 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. Our approach is especially convenient in mod- els comprised of a continuum of inequalities that are separable in parameters, and also applies to models with inequalities that are non-separable in parameters. Since analog estimators for intersection bounds can be severely biased in finite samples, routinely underestimating the length of the identified set, we also offer a (downward/upward) me- dian unbiased estimator of these (upper/lower) bounds as a natural by-product of our inferential procedure. Furthermore, our method appears to be the first and currently only method for inference in nonparametric models with a continuum of inequalities. We develop asymptotic theory for our method based on the strong approximation of a sequence of studentized empirical processes by a sequence of Gaussian or other piv- otal processes. We provide conditions for the use of nonparametric kernel and series estimators, including a novel result that establishes strong approximation for general series estimators, which may be of independent interest. We illustrate the usefulness of our method with Monte Carlo experiments and an empirical example. Key words. Bound analysis, conditional moments, partial identification, strong ap- proximation, infinite dimensional constraints, linear programming, concentration in- equalities, anti-concentration inequalities. JEL Subject Classification. C12, C13, C14. AMS Subject Classification. 62G05, 62G15, 62G32. Date : 20 July 2009. We thank R. Blundell, A. Chesher, F. Molinari, W. Newey, and J. Stoye for detailed discussion and suggestions, and participants at numerous seminars and conferences for their comments. We thank Nicolas Roys for providing excellent research assistance. This paper is a revised version of Inference on Intersection Bounds initially presented at the University of Virginia and the Harvard/MIT econo- metrics seminars in December 2007, as well as the March 2008 CEMMAP/Northwestern conference on Inference in Partially Identified Models with Applications. Financial support from the Economic and Social Research Council for the ESRC Centre for Microdata Methods and Practice (RES-589-28-0001) and the small research grant (RES-000-22-2761) is gratefully acknowledged....
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