Minimax rates for nonparametric speci cation testing in regression models

In the context of testing the speci cation of a nonlinear parametric regression function, we study the power of speci cation tests using the minimax approach. We determine the maximum rate at which a set of smooth local alternatives can approach the parametric model while ensuring consistency of a test uniformly against any alternative in this set. We show that a smooth nonparametric testing procedure has optimal minimax asymptotic properties for regular alternatives. As a by-product, we obtain the rate of the smoothing parameter that ensures optimality of the test. By contrast, many non-smooth tests, such as Bierens' (1982) integrated conditional moment test, have suboptimal minimax properties.