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.
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Abrevaya, Jason, and Wei Jiang. "A nonparametric approach to measuring and testing curvature." Journal of Business and Economic Statistics 23, no. 1 (January 2005): 1-19.
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