The paper "With or Without U? The appropriate test for a U shaped relationship" by Jo Thori Lind and Halvor Mehlum addresses the common issue of testing non-linear relationships, particularly U-shaped curves, in economic theory and empirical research. The authors argue that the standard approach of using a quadratic term in regression models to detect U-shaped relationships is flawed, as it can incorrectly identify monotone relationships as U-shaped. They derive an appropriate test for a U-shaped relationship, which involves testing whether the relationship is decreasing at low values and increasing at high values within a specified interval. The test is based on the likelihood ratio principle and is applicable to various models, including generalized linear models. The paper also provides a detailed example using the Kuznets curve and compares the results with those from other studies, highlighting the limitations of traditional methods. The authors conclude that their test provides exact necessary and sufficient conditions for detecting U-shaped relationships, offering a more robust alternative to existing methods.The paper "With or Without U? The appropriate test for a U shaped relationship" by Jo Thori Lind and Halvor Mehlum addresses the common issue of testing non-linear relationships, particularly U-shaped curves, in economic theory and empirical research. The authors argue that the standard approach of using a quadratic term in regression models to detect U-shaped relationships is flawed, as it can incorrectly identify monotone relationships as U-shaped. They derive an appropriate test for a U-shaped relationship, which involves testing whether the relationship is decreasing at low values and increasing at high values within a specified interval. The test is based on the likelihood ratio principle and is applicable to various models, including generalized linear models. The paper also provides a detailed example using the Kuznets curve and compares the results with those from other studies, highlighting the limitations of traditional methods. The authors conclude that their test provides exact necessary and sufficient conditions for detecting U-shaped relationships, offering a more robust alternative to existing methods.