Inference for Linear Systems with Unknown Coefficients

2026-04-27

AI Summary

This study addresses the problem of testing for the existence of non-negative solutions to a system of linear equations when all parameters, including slope coefficients, are unknown. The authors propose a novel sample-splitting test that, for the first time, characterizes the closure of the null hypothesis under total variation distance, eliminating the need for simulated critical values and enabling applicability in high-dimensional settings with rapidly growing numbers of variables. By integrating total variation distance, sample splitting, and asymptotic theory under weak identification conditions, the method demonstrates strong power in both theoretical analysis and simulations. It combines computational simplicity with high-dimensional scalability and can be employed to construct confidence sets for partially identified parameters in nonparametric instrumental variable models.

Abstract

This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system of equations, including the slope coefficients, to be unknown. For this reason, we describe the linear system as having unknown (as opposed to known) coefficients. This hypothesis testing problem arises naturally when constructing confidence sets for possibly partially identified parameters in the analysis of nonparametric instrumental variables models, treatment effect models, and random coefficient models, among other settings. To rule out certain instances in which the testing problem is impossible, in the sense that the power of any test will be bounded by its size, we begin our analysis by characterizing the closure of the null hypothesis with respect to the total variation distance. We then use this characterization to develop novel testing procedures based on sample-splitting. We establish the validity of our testing procedures under weak and interpretable conditions on the linear system. An important feature of these conditions is that they permit the dimensionality of the problem to grow rapidly with the sample size. A further attractive property of our tests is that they do not require simulation to compute suitable critical values. We illustrate the practical relevance of our theoretical results in a simulation study.

Problem

• linear systems
• unknown coefficients
• non-negativity constraints
• hypothesis testing
• partial identification

Innovation

• unknown coefficients
• non-negativity constraints
• sample-splitting
• partial identification
• hypothesis testing