  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.
