Lind, Jo Thori & Halvor Mehlum (2010) With or Without U? The Appropriate Test for a U-Shaped Relationship, Oxford Bulletin of Economics and Statistics 72(1): 109–118.
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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.