Lavergne, Pascal and Guerre, Emmanuel (2002) Optimal Minimax Rates for Nonparametric Specification Testing in Regression Models. Econometric Theory, vol.18 (n°5). pp. 1139-1171.

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Abstract

In the context of testing the specification of a nonlinear parametric regression function, we adopt a nonparametric minimax approach to determine the maximum rate at which a set of smooth alternatives can approach the null hypothesis while ensuring that a test can uniformly detect any alternative in this set with some predetermined power. We show that a smooth nonparametric test has optimal asymptotic minimax properties for regular alternatives. As a by-product, we obtain the rate of the smoothing parameter that ensures rate-optimality of the test. We show that, in contrast, a class of nonsmooth tests, which includes the integrated conditional moment test of Bierens (1982, Journal of Econometrics 20, 105–134), has suboptimal asymptotic minimax properties.