Abstract
This paper describes three methods for carrying out non-asymptotic inference on partially identified parameters that are solutions to a class of optimization problems. Applications in which the optimization problems arise include estimation under shape restrictions, estimation of models of discrete games, and estimation based on grouped data. The partially identified parameters are characterized by restrictions that involve the unknown population means of observed random variables in addition to the structural parameters of interest. Inference consists of finding confidence intervals for the structural parameters. Our theory provides finite-sample lower bounds on the coverage probabilities of the confidence intervals under three sets of assumptions of increasing strength. With the moderate sample sizes found in most economics applications, the bounds become tighter as the assumptions strengthen. We discuss estimation of population parameters that the bounds depend on and contrast our methods with alternative methods for obtaining confidence intervals for partially identified parameters. The results of Monte Carlo experiments and empirical examples illustrate the usefulness of our method.
Highlights
Our method applies to models that impose shape restrictions (e.g., Freyberger and Horowitz, 2015; Horowitz and Lee, 2017), a variety of partially identified models (e.g., Manski, 2007a; Tamer, 2010), and models in which a continuous function is inferred from the average values of variables in a finite number of discrete groups (e.g., Blundell, Duncan, and Meghir, 1998; Kline and Tartari, 2016)
Bounds on the partially identified function f (ψ) under the restrictions g1(ψ, μ) ≤ 0 and g2(ψ, μ) = 0, where ψ is a vector of structural parameters; μ is a vector of unknown population means of observable random variables; f is a known, real-valued function; and g1 and g2 are known possibly vector-valued functions
The hypercube makes it possible to replace the multivariate Bentkus (2003) inequality with the one-dimensional Berry-Esseen inequality, which reduces the upper bound on the difference between the true and nominal coverage probabilities of the confidence interval
Summary
We present a method for carrying out inference about a partially identified function of structural parameters of an econometric model. In contrast to existing asymptotic methods, it provides a finite-sample bound on the difference between the true and nominal coverage probabilities of a confidence interval for f (ψ). A second method consists of using a finite-sample concentration inequality to obtain a confidence interval This method is useful for applications only if the inequality provides a bound that does not depend on unknown population parameters. In contrast to conventional asymptotic inference approaches, our theory provides a finite-sample bound on the difference between the true and nominal coverage probabilities of a confidence interval for the partially identified function f (ψ). Tamer, and Ziani (2018) consider finite-sample inference in auction models Their framework and method are very different from those in this paper. Appendix C describes Minsker’s (2015) median of means method
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
