With or without u? the appropriate test for a U-shaped relationship

Abstract

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. © 2009 Blackwell Publishing Ltd and the Department of Economics, University of Oxford.