Minimax Rates for Nonparametric Specification Testing in Regression Models

Abstract

We deal with the issue of testing the specification of a regression function. As a leading case, we consider testing for a pure noise model. We study the smallest local alternatives that can be detected asymptotically in a minimax sense. We propose a simple testing procedure that has asymptotic optimal minimax properties for regular alternatives. We then adapt this procedure to testing the specification of a nonlinear parametric regression model. As a by-product, we obtain the rate of the optimal smoothing parameter that ensures optimal minimax properties for the test. We show that, by contrast, non-smoothing tests, such as Bierens' (1982) integrated conditional moment test, have undesirable minimax properties.