Testing the Solvability of Systems of Linear Inequalities

TL;DR

This paper introduces a bootstrap-based testing method for determining the solvability of systems of linear inequalities, applicable to partially identified models, with demonstrated finite-sample effectiveness.

Contribution

It provides a novel characterization of hypothesis testing for linear systems using linear programming and develops valid bootstrap procedures for practical inference.

Findings

Bootstrap tests are valid across large classes of data-generating processes.

Simulation shows good finite-sample performance of the proposed tests.

Applications demonstrate the method's usefulness in empirical contexts.

Abstract

This paper studies the problem of testing whether a system of linear equality and inequality constraints admits a solution when the coefficients of that system may have to be estimated. We show that a wide range of inferential questions in partially identified models can be formulated as hypotheses of this form. Our approach exploits an alternative characterization of the hypothesis based on whether the value of a certain linear program is equal to zero. Building on this characterization, we develop bootstrap-based testing procedures and establish their uniform validity over large classes of data-generating processes. Simulation results demonstrate good finite-sample performance, even for moderate sample sizes. We illustrate the usefulness of the approach in two empirical applications.