[Paper Review] Inference on Functionals of Set-Identified Parameters Defined by Convex Moments

Thomas M. Russell|arXiv (Cornell University)|Oct 7, 2018

Statistical Methods and Inference32 references4 citations

TL;DR

This paper proposes a uniformly valid inference method for continuous convex functionals of set-identified parameters defined by convex moment inequalities. By bootstrapping the Lagrangian of a convex optimization problem, it constructs confidence sets without repeatedly inverting hypothesis tests, offering a less conservative alternative to existing subvector inference methods in high-dimensional settings.

ABSTRACT

Many inference procedures in the literature on partial identification are designed for when the inferential object of interest is the entire (partially identified) vector of parameters. However, when the researcher's inferential object of interest is a subvector or functional of the parameter vector, these inference procedures can be highly conservative, especially when the dimension of the parameter vector is large. This paper considers uniformly valid inference for continuous functionals of partially identified parameters in cases where the identified set is defined by convex (in the parameter) moment inequalities. We use a functional delta method and propose a method for constructing uniformly valid confidence sets for a (possibly stochastic) convex functional of a partially identified parameter. The proposed method amounts to bootstrapping the Lagrangian of a convex optimization problem, and subsumes subvector inference as a special case. Unlike other proposed subvector inference procedures, our procedure does not require the researcher to repeatedly invert a hypothesis test. Finally, we discuss sufficient conditions on the moment functions to ensure uniform validity.

Motivation & Objective

• To address the conservativeness of existing inference procedures when the inferential object is a subvector or functional of a high-dimensional partially identified parameter vector.
• To develop a uniformly valid confidence set construction method for continuous convex functionals of parameters identified via convex moment inequalities.
• To eliminate the need for repeated hypothesis testing inversion, which is computationally burdensome and conservative in subvector inference.
• To establish sufficient conditions on moment functions ensuring uniform validity of the proposed inference procedure.

Proposed method

• The method employs a functional delta method to derive asymptotic distributions of convex functionals of the identified set.
• It constructs confidence sets by bootstrapping the Lagrangian dual of a convex optimization problem that characterizes the identified set.
• The bootstrap procedure is designed to preserve the convex structure of the moment inequalities, ensuring valid inference under weak regularity conditions.
• The approach is applicable to both deterministic and stochastic functionals of the parameter vector.
• The method avoids the need for repeated inversion of hypothesis tests, which is a key computational and inferential limitation in existing subvector inference techniques.
• Sufficient conditions on the moment functions—such as smoothness and local convexity—are derived to ensure uniform validity of the confidence sets.

Experimental results

Research questions

• RQ1 How can uniformly valid inference be constructed for convex functionals of partially identified parameters when the identified set is defined by convex moment inequalities?
• RQ2 What is a computationally efficient alternative to repeated hypothesis testing inversion in subvector inference under partial identification?
• RQ3 Can a bootstrap-based method be uniformly valid for functionals of set-identified parameters without requiring repeated optimization?
• RQ4 What regularity conditions on the moment functions ensure the uniform validity of the proposed inference procedure?

Key findings

• The proposed method constructs uniformly valid confidence sets for convex functionals of partially identified parameters by bootstrapping the Lagrangian of the dual optimization problem.
• The method is less conservative than existing subvector inference procedures because it avoids repeated hypothesis testing inversion.
• The approach is applicable to both deterministic and stochastic functionals, extending its utility beyond point estimation.
• Sufficient conditions on the moment functions—such as continuous differentiability and local convexity—are provided to guarantee uniform validity of the inference procedure.
• The method subsumes subvector inference as a special case, offering a unified framework for functional inference under convex moment inequalities.
• The functional delta method is leveraged to derive the asymptotic distribution of the functional, enabling valid inference under weak regularity conditions.