A Simplex Statistic for Testing Joint Curvature

This paper proposes a simplex statistic for testing the joint curvature (e.g., linearity, concavity, or convexity) of a nonparametric regression function. The statistic is based on Jensen’s inequality applied to all (m+ 2)-tuples in the data having one data point in the interior of the (m+1)-dimension simplex spanned by the remaining data points (m is the dimension of the covariate vector). The test does not require choice of smoothing parameters for the global version of the statistic or its asymptotic standard errors, making it easy to implement. The test of linearity is consistent against the alternative of strict convexity or concavity, and the test of convexity (concavity) is consistent against the alternative of strict concavity (convexity). Localized versions of the test are consistent against more general alternatives. We apply the simplex statistic to test the performance of style timing by mutual funds and find that on average the funds’ style timing performance is close to being neutral. ∗Department of Economics, Purdue University, 1310 Krannert Building, West Lafayette, Indiana 47907; e-mail: [email protected]. †Finance and Economics Division, Columbia University Graduate School of Business, 3022 Broadway, New York, NY 10027; e-mail: [email protected].