A nonparametric approach to measuring and testing curvature.

This article considers the problem of testing curvature (e.g., linearity, concavity, convexity) in a multivariate nonparametric regression model. A measure of curvature, called the simplex statistic, that does not require bandwidth choice and is easy to compute, is introduced. A global test of curvature based on the simplex statistic is also introduced. Localized versions of the test, which require smoothing parameters, are shown to be consistent against more general alternatives than the global test. In the univariate case, the local test of concavity (convexity) is consistent against all nonconcave (nonconvex) alternatives. The simplex statistic can also be used in the context of a partially linear regression model. Applications to examining the curvature of the experience-earnings profile and testing the "style timing" of mutual funds are considered.

KEY WORDS: Concavity; Convexity; Nonparametric test; U-statistics.

1. INTRODUCTION

Concavity and convexity are important concepts in microeconomic theory. The concavity or convexity of a relationship (function) may be predicted by theory; for example, theory tells us that cost functions are concave in input price and that profit functions are convex in output price. The concavity or convexity of a function can have important implications for the way in which economic agents will behave; for example, concavity (convexity) of an individual's utility function corresponds to risk aversion (loving). By definition, concavity or convexity of a function is directly associated with whether there are decreasing returns or increasing returns to a given factor. For instance, to determine whether the returns to education (e.g., on wages) are increasing or decreasing in the level of education, we would want to know if the relationship between wages and education is convex or concave.

Tests of convexity or concavity have appeared in a wide range of empirical studies. For example, there have been studies on testing the curvature between investment and Tobin's q (e.g., Barnett and Sakellaris 1998), between firm value and product price (Borenstein and Farrell 1999), and between mutual fund performance and subsequent money inflow (Chevalier and Ellison 1997). The convex or concave relationship in these examples have important implications for corporate managers' investment strategies and money managers' risk-taking behavior.

Despite the important role of concavity and convexity in economic theory and applications, there has been surprisingly little work in the econometrics literature on statistical testing of concavity and convexity. Although a large literature on nonparametric specification testing has emerged, only a few studies have considered tests for curvature. Moreover, the existing nonparametric methods for curvature are applicable only for the univariate case. There appear to be no nonparametric methods for testing concavity and convexity of a regression function having more than one covariate.

In empirical work, the general approach for testing curvature is to allow nonlinearity in the regression function and perform appropriate hypothesis tests after estimation. Natural candidates for parametric methods that allow nonlinearity are polynomials or linear splines. Although easy to implement, these methods are subject to model misspecification when researchers do not have clear prior information about the shape of the underlying functions. Because a parametric test is essentially a joint test of curvature and parametric functional form, the conclusion of such a test is unclear due to possible misspecification of functional form.