Inference for Large-Scale Linear Systems with Known Coefficients

TL;DR

This paper develops a computationally feasible test for the existence of non-negative solutions in large-scale linear systems with known coefficients, applicable in various economic and statistical models.

Contribution

It introduces a geometric characterization of the null hypothesis and a linear programming-based test suitable for high-dimensional settings.

Findings

Test maintains correct size asymptotically.

Method is computationally feasible for large systems.

Applicable to diverse models like treatment effects and linear programming.

Abstract

This paper considers the problem of testing whether there exists a non-negative solution to a possibly under-determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of settings, including random coefficient, treatment effect, and discrete choice models, as well as a class of linear programming problems. As a first contribution, we obtain a novel geometric characterization of the null hypothesis in terms of identified parameters satisfying an infinite set of inequality restrictions. Using this characterization, we devise a test that requires solving only linear programs for its implementation, and thus remains computationally feasible in the high-dimensional applications that motivate our analysis. The asymptotic size of the proposed test is shown to equal at most the nominal level uniformly over a large class of distributions that permits the number of linear equations to grow with the sample size.