Inference for Large-Scale Linear Systems with Known Coefficients

67 Pages Posted: 21 Sep 2020

See all articles by Zheng Fang

Zheng Fang

Emory University - Department of Economics

Andres Santos

University of California, Los Angeles (UCLA) - Department of Economics

Azeem Shaikh

University of Chicago

Alexander Torgovitsky

University of Chicago

Date Written: September 17, 2020

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.

Keywords: linear programming, linear inequalities, moment inequalities, random coefficients, partial identification, exchangeable bootstrap, uniform inference

Suggested Citation

Fang, Zheng and Santos, Andres and Shaikh, Azeem and Torgovitsky, Alexander, Inference for Large-Scale Linear Systems with Known Coefficients (September 17, 2020). University of Chicago, Becker Friedman Institute for Economics Working Paper No. 2020-134, Available at SSRN: https://ssrn.com/abstract=3695284 or http://dx.doi.org/10.2139/ssrn.3695284

Zheng Fang

Emory University - Department of Economics ( email )

1602 Fishburne Dr
Atlanta, GA Georgia 30322
United States
30005 (Fax)

Andres Santos

University of California, Los Angeles (UCLA) - Department of Economics ( email )

8283 Bunche Hall
Los Angeles, CA 90095-1477
United States

Azeem Shaikh (Contact Author)

University of Chicago ( email )

1101 East 58th Street
Chicago, IL 60637
United States

Alexander Torgovitsky

University of Chicago ( email )

1101 East 58th Street
Chicago, IL 60637
United States

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