With or Without U? The Appropriate Test for a U-Shaped Relationship*


  • *

    While carrying out this research, the authors have been associated with the ESOP centre at the Department of Economics, University of Oslo. ESOP is supported by The Research Council of Norway. The authors are grateful for comments from two anonymous referees, Kalle Moene, and seminar participants at the University of Oslo and PRIO. They also thank Dustin Chambers for making his data available. A companion STATA module utest is available from: http://ideas.repec.org/c/boc/bocode/s456874.html


Nonlinear relationships are common in economic theory, and such relationships are also frequently tested empirically. We argue that the usual test of nonlinear relationships is flawed and derive the appropriate test for a U-shaped relationship. Our test gives the exact necessary and sufficient conditions for the test of a U shape in finite samples in a large class of models.