# Inference for the Identified Set in Partially Identified Econometric Models

*Joseph P. Romano* and
*Azeem Shaikh* ()

*Econometrica*, 2010, vol. 78, issue 1, 169-211

**Abstract:**
This paper provides computationally intensive, yet feasible methods for inference in a very general class of partially identified econometric models. Let P denote the distribution of the observed data. The class of models we consider is defined by a population objective function Q(θ, P) for θ is an element of Θ. The point of departure from the classical extremum estimation framework is that it is not assumed that Q(θ, P) has a unique minimizer in the parameter space Θ. The goal may be either to draw inferences about some unknown point in the set of minimizers of the population objective function or to draw inferences about the set of minimizers itself. In this paper, the object of interest is Θ 0 (P)=argmin θ is an element of Θ Q(θ, P), and so we seek random sets that contain this set with at least some prespecified probability asymptotically. We also consider situations where the object of interest is the image of Θ 0 (P) under a known function. Random sets that satisfy the desired coverage property are constructed under weak assumptions. Conditions are provided under which the confidence regions are asymptotically valid not only pointwise in P, but also uniformly in P. We illustrate the use of our methods with an empirical study of the impact of top-coding outcomes on inferences about the parameters of a linear regression. Finally, a modest simulation study sheds some light on the finite-sample behavior of our procedure. Copyright 2010 The Econometric Society.

**Date:** 2010

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