We would like to thank Michael Wolf for a careful reading of the paper and useful suggestions. We also thank Nese Yildiz for pointing out the need for the nonzero variance condition in Example 2.1.
Inference for the Identified Set in Partially Identified Econometric Models
Article first published online: 8 FEB 2010
© 2010 The Econometric Society
Volume 78, Issue 1, pages 169–211, January 2010
How to Cite
Romano, J. P. and Shaikh, A. M. (2010), Inference for the Identified Set in Partially Identified Econometric Models. Econometrica, 78: 169–211. doi: 10.3982/ECTA6706
- Issue published online: 8 FEB 2010
- Article first published online: 8 FEB 2010
- Manuscript received September, 2006; final revision received September, 2009.
- Partially identified model;
- incomplete model;
- identified set;
- identifiable parameter;
- uniform coverage;
- confidence region;
- moment inequalities
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∈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 Θ0(P)=argminθ∈ΘQ(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 Θ0(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.