Inference for Linear Systems with Unknown Coefficients

TL;DR

This paper develops novel hypothesis testing methods for linear systems with unknown coefficients, applicable in various econometric models, allowing high-dimensional problems and avoiding simulation for critical values.

Contribution

It introduces sample-splitting based tests for unknown coefficient systems, with validity under high-dimensional settings and no need for simulation.

Findings

Tests are valid under weak, interpretable conditions.

Method accommodates rapidly growing dimensions with sample size.

No simulation needed for critical value computation.

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.