Sup-tests for linearity in a general nonlinear AR(1) model when the supremum is taken over the full parameter space

We consider linearity testing in a general class of nonlinear time series model of order 1, involving a nonnegative nuisance parameter which (i) is not identified under the null hypothesis and (ii) gives the linear model when equal to zero. This paper studies the asymptotic distribution of the Likelihood Ratio test and asymptotically equivalent supremum tests. The asymptotic distribution is described as a functional of chi-square processes and is obtained without imposing a positive lower bound for the nuisance parameter. The finite sample properties of the sup-tests are studied by simulations.