Essays on Testing Non-Monotone Relationships

Abstract

These three chapters discuss problems and propose solutions to exploring potentially non-monotone regression functions.

Chapter 1 focuses on a specific class of non-monotone regression functions, U-shapes. I propose a non-parametric test of U-shaped regression functions based on critical bandwidth (CB), first introduced by Silverman (1981). The tests in this chapter are extended to work in generalized additive models, which allows investigation of whether there is an inherent U-shape between two variables or if the relationship is actually caused by correlation with other variables. I give sufficient conditions for consistency of the test statistic and show that the rate of convergence under the null is at least as fast as any bandwidth sequence leading to pointwise-consistent estimates of the regression function.

Chapter 2 combines three new elements to take a closer look at whether there is an inherent U-shape between subjective well-being and age. First, the latest waves of data from the surveys are used; second, semi-parametric methods are used in addition to the commonly used OLS and ordered logit quadratic specifications; and third, financial satisfaction is added as a control variable which changes the relationship between life satisfaction and age from a U-shape to monotonically decreasing. The U-shape of financial satisfaction explains the U-shape of life satisfaction, which suggests evidence against the theory that the U-shape of life satisfaction is due to a midlife crisis that is hard-coded in DNA.

Chapter 3 provides an intuitive way to explore non-monotone relationships in a multivariate context. It is well-known that OLS is sensitive to a minority's strong positive effect drowning out a majority's weak negative effect. This chapter proposes estimation of a new parameter, the Partial Monotonicity Parameter (PMP), which improves on the common practice of testing predictions of monotonicity with OLS, and provides a single number that summarizes the degree to which two random variables have a non-monotone relationship. By combining OLS which is sensitive to magnitude, with PMP which is not sensitive to magnitude, a rich summary of the regression function can be succinctly reported. A plug-in estimator based on local polynomial regression is proposed.