Optimal Minimax Rates for Nonparametric Specification Testing in Regression Models

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.

Reference

Pascal Lavergne, and Emmanuel Guerre, “Optimal Minimax Rates for Nonparametric Specification Testing in Regression Models”, Econometric Theory, vol. 18, n. 5, October 2002, pp. 1139–1171.