This paper argues that the standard test for a U-shaped relationship in econometric models is flawed and proposes a more accurate test. Non-linear relationships are common in economics, and the paper focuses on testing for a U-shaped relationship, which is often used to describe phenomena where a variable first increases and then decreases with another variable. The authors show that the usual approach of including a quadratic term in a regression model and checking the significance of the coefficient is insufficient, as it can falsely indicate a U-shape when the true relationship is convex but monotonic. The paper proposes a more rigorous test that checks whether the relationship is decreasing at the start and increasing at the end of a chosen interval. This test involves checking the sign of the slope at the endpoints of the interval and is based on a composite null hypothesis. The authors use a likelihood ratio test and show that it can be implemented as an intersection-union test. They also demonstrate that constructing a confidence interval for the minimum point of the relationship and checking whether it lies within the data range is an equivalent method. The paper compares this test to the methods used in seven recent studies that have identified U-shaped relationships. It finds that most of these studies rely on the significance of the coefficients of the quadratic term and the estimated extreme point, which is not sufficient for a proper test of a U-shape. The authors provide an example using a Kuznets curve, which is a classic example of a U-shaped relationship between income and inequality. They show that the conventional test may not detect the significance of the U-shape as accurately as the proposed test. The paper concludes that the proposed test provides the exact necessary and sufficient conditions for testing a U-shaped relationship and is more reliable than the commonly used methods. It also highlights the importance of considering the interval of the data and the distribution of the estimated slope when testing for a U-shape.This paper argues that the standard test for a U-shaped relationship in econometric models is flawed and proposes a more accurate test. Non-linear relationships are common in economics, and the paper focuses on testing for a U-shaped relationship, which is often used to describe phenomena where a variable first increases and then decreases with another variable. The authors show that the usual approach of including a quadratic term in a regression model and checking the significance of the coefficient is insufficient, as it can falsely indicate a U-shape when the true relationship is convex but monotonic. The paper proposes a more rigorous test that checks whether the relationship is decreasing at the start and increasing at the end of a chosen interval. This test involves checking the sign of the slope at the endpoints of the interval and is based on a composite null hypothesis. The authors use a likelihood ratio test and show that it can be implemented as an intersection-union test. They also demonstrate that constructing a confidence interval for the minimum point of the relationship and checking whether it lies within the data range is an equivalent method. The paper compares this test to the methods used in seven recent studies that have identified U-shaped relationships. It finds that most of these studies rely on the significance of the coefficients of the quadratic term and the estimated extreme point, which is not sufficient for a proper test of a U-shape. The authors provide an example using a Kuznets curve, which is a classic example of a U-shaped relationship between income and inequality. They show that the conventional test may not detect the significance of the U-shape as accurately as the proposed test. The paper concludes that the proposed test provides the exact necessary and sufficient conditions for testing a U-shaped relationship and is more reliable than the commonly used methods. It also highlights the importance of considering the interval of the data and the distribution of the estimated slope when testing for a U-shape.