A model is said to be partially identified when the sampling process and the maintained assumptions restrict the value of the parameter of interest to a set, called the identified set, which is smaller than the logical range of the parameter but potentially larger than a single point. Partially identified models arise naturally in economic models when strong and usually unrealistic assumptions are traded by weaker and more credible ones. Since their relatively recent introduction, partially identified models have become increasingly popular in many areas of economics and other social sciences.
The objective of this research proposal is to extend the scope of inference in partially identified models and it is divided into three related research projects. The first project studies the behavior of several confidence sets commonly used in the literature on inference in moment inequality models when we allow for the possibility of making small mistakes in the specification of the model (i.e. local misspecification). The motivation for this project stems from the fact that econometric models are only approximations to the underlying phenomenon of interest and are therefore intrinsically misspecified. There are different inference procedures available in the literature that have been compared in terms of asymptotic size and power properties under the assumption of correct model specification. This project proposes the amount of distortion to asymptotic confidence size as a criterion to choose among competing inference methods, and applies this criterion to compare across critical values and test statistics employed in the construction of confidence sets in partially identified models. As a result, the applied researcher will be aware of the problems caused by these mistakes and will be able to choose a methodology that minimizes these problems.
The second project addresses inference in models where there is a multi-dimensional confidence set for a parameter vector of interest. At the time of the presentation of the results, researchers typically resort to the use of a projection of the identified set in each of its individual coordinates, generating a multi-dimensional hyper-rectangle. While researchers typically know the properties of the original confidence sets very well, little is known about the properties of these hyper-rectangles. This project has the objective of filling in this void.
The third project proposes an inference approach for partially identified models based on moment equalities with unknown/unidentified functions as opposed to the now standard approach of deriving moment inequality restrictions. The approach can handle models that are not easily framed into moment inequality models (e.g. missing covariates in a regression model).
The three projects in this proposal are motivated by problems that applied researchers face when working with partially identified models. The ultimate objective of this proposal is to provide applied researchers and policy makers with a better understanding of the statistical properties of these models. For example, when analyzing data for policy recommendations, researchers typically make assumptions to simplify the analysis and the exposition of the results. This proposal aims to help the researcher choose tools that are less sensitive to those assumptions. Furthermore, the implementation of these projects requires developing computational tools that will be of interest to computational economists and computer scientists. Finally, the developments of this research agenda are incorporated into the curriculum of courses targeted to graduate students interested in econometrics and even outstanding undergraduate students.