Non-Parametric Testing of U-Shapes, with an Application to the Midlife Satisfaction Dip

Abstract

Many theories in economics predict U-shaped relationships between variables. However, satisfactory tools to examine U-shapes are lacking. After explaining the limitations of the commonly employed quadratic specification, I propose a non-parametric test of U-shaped regression functions based on critical bandwidth. The test allows one to determine whether an inherent U-shape exists between two variables or the relationship is instead caused by correlation with other variables. I apply the test to the commonly observed U-shape of life satisfaction in age, and find that much of the U-shape can be explained by the increase in financial satisfaction that typically occurs later in life. This novel insight into a long-studied puzzle is not revealed by using a quadratic specification. A user-friendly and efficient R package is provided.