Pairwise - slopes statistics for testing curvature

This paper introduces nonparametric U-statistics for testing the curvature (e.g., linearity, concavity, or convexity) of an unknown regression function. The U-statistics are based on pairwise slopes in the data. No choice of smoothing parameters is required for the global version of the statistics or their (asymptotic) standard errors, making them simple to compute. The proposed tests are analogous to the test of independence based on the rank correlation statistic (“tau”) of Kendall (1938). Consistent versions of the tests are constructed using a discretized process of localized versions of the (global) U-statistics. Though they require bandwidth choice, these tests are consistent against general alternatives. Monte Carlo simulations investigate the small-sample properties of the statistics along with the size and power of the associated hypothesis tests. An empirical application to earnings equations is considered. Modified versions of the test statistics are developed for other models, including a semilinear regression model and a nonparametric fixed-effects model.